Embed Size (px)
Citation preview
1
Thermopower measurements and
Electron-Phonon coupling in
molecular devices
Thesis submitted for the degree of "Doctor of Philosophy"
By
Ilan Yutsis
Submitted to the Senate of Tel-Aviv University
July 2010
2
This work was carried out under the supervision of
Doctor Yoram Selzer
School of Chemistry, Tel Aviv University
3
Abstract
Measurements of thermoelectric transport yield important information
regarding fundamental properties of a system in addition to the information supplied
from electronic transport measurements. In this thesis we report our measurements of
Thermoelectricpower (TEP) in graphene flakes. Graphene is single atom layer of
carbon atoms arranged in honeycomb lattice, recently isolated and attracted enormous
attention in the scientific community due to its exceptionally high crystal and
electronic quality1. We modulated both the conductance and TEP in graphene by
electric field in wide temperature range, and obtain a relation between the two. We
experimentally show that around the Dirac point (interface point of valence and
conduction bands) the well known Mott formula does not describe accurately the
Seebeck coefficient of graphene. Disorder affects the thermopower properties of
graphene around room temperatures. Using a phenomenological treatment based on
the presence of disorder-induced electron and hole puddles we provide a description
of the behavior of the Seebeck coefficient around zero gate voltage.
To further investigate the relation between electrons and phonon we conducted
Raman spectroscopy measurements on graphene flakes under potential bias. Increase
in the population of excited optical phonons is observed and based on their Raman
Anti-Stokes/Stokes signals ratio we infer their effective mode temperature. We prove
that the conductivity of single-layer graphene under high bias (>0.2V) depends on
direct coupling between charge carriers and optical phonons of the flake and not due
to coupling with modes of the underlying SiO2 layer, as suggested by others. These
results therefore suggest that there should be an intrinsic limitation to the performance
of graphene-based electronic devices.
In addition, we present our measurements in a new model for highly stable
metallic quantum point contact (MQPC) invented by us for investigation of current
induced local heating effects in atomic junctions. Our devices show controllable and
stable conductance plateaus typical for few atoms junctions and posses healing
capability once broken. We estimate the temperature increase just prior to junction
formation by Electrormigration to be in the order of 150K, and once the metallic
atom bridges the gap the temperature induced by the applied voltage to be in the order
of 60K. This is surprising because the size of the contacts is much smaller than the
electron mean free path. It is argued that the high electric current density in our highly
stable junctions increase dramatically the probability for electronic induced heating in
the junction even if the scale is shorter than the mean free path.
4
Contents
Contents ......................................................................................................................... 4
List of figures ................................................................................................................. 5
Acknowledgments.......................................................................................................... 8
1 Thermoelectric transport ............................................................................................. 9
1 Thermoelectric transport ............................................................................................. 9
1.1 Electronic properties of Graphene ....................................................................... 9
1.1.1 Graphene Band structure............................................................................... 9
1.1.2 Graphene Electric Field Effect dependence ................................................ 10
1.2 Thermoelectricity ............................................................................................... 11
1.2.1 The Seebeck effect ...................................................................................... 13
1.2.2 The Peltier effect ......................................................................................... 13
1.3 Motivation .......................................................................................................... 16
1.4 Methods and Materials ....................................................................................... 17
1.4.1 Identification and characterization of prospect flakes for measurement .... 17
1.4.2 The experimental structure and it‟s calibration .......................................... 18
1.5 Results and Discussion ...................................................................................... 21
1.6 Conclusions ........................................................................................................ 24
2 Electron-Phonon coupling ........................................................................................ 26
2.1 Raman Spectroscopy .......................................................................................... 26
2.2 Motivation .......................................................................................................... 28
2.3 Methods and Materials ....................................................................................... 30
2.3.1 Device Fabrication ...................................................................................... 30
2.3.2 Measurement Technique ............................................................................. 31
2.3.3 Evaluation of heat loss in joule heated graphene due to convection and
radiation ............................................................................................................... 32
2.4 Results and Discussion ...................................................................................... 33
2.5 Conclusions ........................................................................................................ 43
3 Energy dissipation in Quantum Point Contact .......................................................... 44
3.1 Electrical Transport theory in Quantum point contact ....................................... 44
3.2 Motivation .......................................................................................................... 46
3.3 Methods and Materials ....................................................................................... 49
3.3.1 Device Fabrication ...................................................................................... 49
3.3.2 Measurements Technique ........................................................................... 50
3.4 Results and Discussion ...................................................................................... 50
5
3.5 Conclusions ........................................................................................................ 58
Appendix A- Preparation of Graphene flakes .............................................................. 59
List of figures
Figure 1: (a) Real space and (b) reciprocal space structure of graphene. ................... 10
Figure 2: Electronic Energy bands of graphene. ........................................................ 10
Figure 3: Schematic representation of a 4 probe conductance measurement. Measured
and calculated behavior of vs gate voltage (b). ....................................... 11
Figure 4: The Seebeck effect ...................................................................................... 13
Figure 5: The Peltier effect ......................................................................................... 13
Figure 6: Scheme of charge puddles in a small segment of graphene……………....17
Figure 7: Raman spectrum and AFM measurement of graphene flake……. ............. 18
Figure 8: Optical picture of the experimental setup. .................................................. 19
Figure 9: Comparison between simulated and measured temperature gradients
between two electrodes. .............................................................................. 20
Figure 10: Representative report of Numerical simulations results of the temperature
gradient on graphene flake .......................................................................... 20
Figure 11: Outside and inside view of our probe station. ........................................... 21
Figure 12: Seebeck versus gate voltage curves measured at three bath temperatures22
Figure 13: Fitting of the Mott formula to measured Seebeck coefficients as a function
of gate voltages. .......................................................................................... 23
Figure 14: Fitting of equations 1.5.1 and 1.5.2 to the results at T=290K. .................. 24
Figure 15: Illustration of Rayleigh and Raman scattering processes. ........................ 27
Figure 16: Scheme of a graphene device, I-V curve of a representative device,
conductivity of gate voltage VG and Raman spectrum of a device. ............ 31
Figure 17: Temperature distribution in a heated graphene flake in Vacuum (a) and
under Argon gas flow (b). ........................................................................... 33
Figure 18: Scheme diagram of the atomic displacements in the graphene plane for the
E2g mode at the point .............................................................................. 34
Figure 19: dI/dV vs. V4p of the I-V curve in Figure 16. Schematic presentation of the
carrier concentration at three different points a long the I-V curve ............ 34
6
Figure 20: Anti-Stoks and Stoks signals of the Gmode phonon and 55mev SiO2
phonon at V4p voltages, effective temperature of the G mode phonon and
the conductance as a function of applied source-drain bias. ....................... 37
Figure 21: Effective temperature of the G mode and of the 55meV mode of SiO2 as a
function of source-drain bias. ..................................................................... 38
Figure 22: Effective G mode temperature as function of electric power,. ................. 39
Figure 23: Full width at half maximum of the G band as a function of bias.. ............ 40
Figure 24: Scheme for a ballistic two-terminal conductance problem ....................... 45
Figure 25: Conductance (in units of G0) as a function of time for a typical MQPC. . 49
Figure 26: An optical image, schematic presentation of the structure, and SEM
picture showing a typical gap ..................................................................... 49
Figure 27: Behavior of conductance as a result of applied voltage and as a function of
time. ............................................................................................................ 52
Figure 28: Field, and drift velocity distributions in close proximity to a protrusion
located 50Å from the other lead.................................................................. 53
Figure 29: Switching between quantized conductance values in two MQPCs. ......... 56
Figure 30: Conductance histograms of two MQPCs, based on break-and-make
cycles. ......................................................................................................... 58
7
This work is dedicated to the people who waited
"so long" for this final result,
To my mother, Viki and my father, Michael, who taught
me the value of hard work, and provided me the
opportunity to acquire higher education.
To my loving, supporting family, Regina, Yechiel, Revital,
Moran, Eli, Yaffa, Ahuva, Sigal, Yariv, Oren, Tal, Guy, and
Nir.
To my true partner Dalit, and to my joy Yonatan.
8
Acknowledgments
I would like to express my deepest gratitude and appreciation to my advisor Dr.
Yoram Selzer for providing me the opportunity to do my Ph.D. dissertation under his
supervision, for his constant help, support, encouragement and endless optimism.
Thank you for teaching me how to do science and for the patience until I understood.
I would like to thank my friends and colleagues at the laboratory: Eyal, Tamar, Dana,
Gilad, Dvora, Naomi, Matan and Ayelet for their pleasant company, helpful advice
and fruitful discussions.
I would like to thank Denis Glozmann for his help with the E-beam work.
9
1 Thermoelectric transport
In this chapter we report Thermopower (TP) measurements of single layer
graphene flakes. Graphene is single atom layer of carbon atoms arranged in
honeycomb lattice, recently isolated, attracting enormous attention in the scientific
community due to its exceptionally high crystal and electronic quality2. We
modulated both the conductance and TP in graphene by electric field, (applied by a
gate electrode) over a wide temperature range, and obtain a relation between the two.
We experimentally show that disorder due to charged impurities and the co-
existence of electron-hole puddles in graphene lead, at high temperatures (where
charged impurities are unscreened), to failing of the well known Mott formula to
describe accurately the Seebeck coefficient of graphene as a function of Fermi level
position close to the Dirac point. Instead, using a very simple phenomenological
treatment to modify the Mott formula we prove the existence of electron-hole puddles
with characteristic length similar to the length of measured samples, suggesting mean
free path contained within puddles.
In this chapter we begin with introduction to electronic properties of graphene
(1.1), and to Thermoelectricity (1.2), followed by motivation to this research (1.3),
methods and materials (1.4), results and discussion (1.5), and conclusions (1.6).
1.1 Electronic properties of Graphene
1.1.1 Graphene Band structure
Graphene is a two-dimensional array of carbon atoms arranged in a hexagonal
structure. Figure 1 shows a diagram of graphene structure in real space (a) and
reciprocal space (b). The four valance electrons of the carbon atom hybridize in a sp2
configuration: three in-plane s bonds form the strong hexagonal lattice and one
covalent, out-of-plane bond ultimately determines the electric transport properties
of the system. The band structure is calculated within a tight binding model. Two
carbon atoms per unit cell allow us to compose Bloch functions out of the atomic
orbitals of the two carbon atoms. By only considering nearest-neighbor interactions it
is possible to derive the -* band structure.
Within the tight-binding scheme, solving the secular equation leads to the
following equation for the band dispersion relation, obtained as a function of the 2D
wave vectors, (kx, ky):
11
Figure 1: (a) Real space and (b) reciprocal space structure of graphene (reproduced from3).
2cos4
2cos
2
3cos41),( 2
)1.1.1(akakak
tkkEyyx
yx
Here the nearest neighbor hopping integral is eVt 5.2 and the length of the lattice
vector is o
a 46.2 .
The electronic band structure of a single layer graphene is shown in Figure 2
depicting energy dispersion along the high symmetry points4. Because there are two
electrons per unit cell, these electrons exactly fill up the E<0 band. One can also
calculate the band structure for the d band of graphene, but these bands are located
further away from EF=0. Thus, the low energy transport properties are completely
determined by the bands.
The low energy band structure of graphene is in fact quite unique. The energy bands
are linear around the charge neutrality Dirac point. The bands cross at two equivalent
points K and K' as a result of the two atoms per unit cell. Thus, graphene can behave
as a metal or as a semiconductor.
Figure 2: Electronic Energy bands of graphene.
1.1.2 Graphene Electric Field Effect dependence
In single layer graphene flake, the typical dependence of its sheet resistivity
on gate voltage VG (Figure 3b) exhibits a sharp peak with a value of several kilohms
which decays to ~100 ohms at high VG (2D resistivity is given in units of ohms). This
11
behavior can be explained quantitatively by a model of a 2D metal with a small
overlap between conductance and valence bands5. The gate voltage induces a
surface charge density teVng/
0 and, accordingly, shifts the position of the Fermi
energy F. (0 and are the permittivities of free space and SiO2, respectively; e is the
electron charge; and t is the thickness of the SiO2 layer (300 nm)). For a typical VG
=100V, the surface charge density is n ~7.2x1012
cm-2
.
If the Fermi level F lies between 0 and , there are both electrons and holes
present (a mixed state). In this case, the standard two-band model for a metal
containing both electrons and holes in concentrations ne and nh and with nobilities e
and h describes the conductance, by:
)(/1)2.1.1(hhee
nne
If F is shifted by electric-field doping below (above) the Dirac point, only holes
(electrons) are present and, instead of a semimetal, we get a completely hole
(electron) conductor. Then, the metal‟s conductance is described simply by:
heheen
,,/1)3.1.1(
To calculate the dependence of on VG for the whole range of gate voltages
(such that F varies from well below zero to well above ), we combine equations
(1.1.2-3) with the equation for induced (uncompensated) charge nh – ne = n =
0VG/te. The result of this calculation is shown in Figure 3.
Figure 3: Schematic representation of a 4 probe conductance measurement (a). Measured
(squares) and calculated (blue line) behavior of vs gate voltage in graphene using the setup in
(a). We assume e = h=7,000cm2V
-1s
-1
1 . The asymmetry is due to different electron and hole
masses (measured in ShdH experiments6) (b).
1.2 Thermoelectricity
The broad topic of thermoelectricity describes the direct relationship between
heat and electrical energy. From a fundamental point of view, the study of
thermoelectric effects can help elucidate the electronic structure and the basic
(a) (b)
12
scattering processes in a system. Aside from its fundamental importance, the
phenomenon of thermoelectricity is also relevant to applications such as refrigeration.
This field is currently based on semiconducting materials which have sufficiently
large thermoelectric coefficients to be of practical interest. Measurement of
thermoelectric effects has significantly advanced our understanding of metals, due to
the unique dependence of these effects on the energy derivative of the mean free path,
as well as on the electrical and thermal conductivities. Scientists continuously search
for and work to develop efficient thermoelectric materials. Nanoscale materials, with
their reduced geometries, offer an enormous variety of new and interesting materials
to study.
In attempting to understand transport properties of a system, one normally
applies external force to the material and measures the systems response. In the linear
response regime by applying a potential gradient E
, one can measure the electrical
current, J
, that flows and calculate the conductance, , governing electron transport
in the system. By applying a temperature gradient, T, one can measure the heat
which flows and calculate the thermal conductance, , using the relation TQ
,
governing thermal transport in the system. We can combine these equations of
transport into a matrix equation relating generalized forces and their corresponding
currents.
T
V
LL
LL
Q
J
2221
1211)1.2.1(
here V is the electric potential and Lij are constants independent of V and T. We
immediately recognize L11 as the electrical conductance ,and L22 as the thermal
conductance . The non-zero off-diagonal components, on the other hand, serve to
mix heat and charge, and give rise to two interrelated thermoelectric phenomena: the
seebeck and Peltier effects.
In the following section we discuss these thermoelectric effects, Onsager's
relations and relationship between the off-diagonal components and the Mott formula.
The derivations presented here follow Barnard7.
13
1.2.1 The Seebeck effect
Figure 4: The Seebeck effect
If we take a conductor and heat one end, electrons at the hot end will acquire
increased energy relative to the cold end and will diffuse to this end where their
energy may be lowered. This results in accumulation of negative charge at the cold
end, thus setting up an electrical field or a potential difference between the ends of the
material. This electric field will build up until a state of dynamic equilibrium is
established between electrons urged down the temperature gradient and electrostatic
repulsion due to the excess charge at the cold end. The ratio of the potential difference
V to the applied temperature difference T is called the thermoelectric power (TEP),
or equivalently, the Seebeck coefficient, S:
T
VS
)1.1.2.1(
The number of electrons passing through a cross section normal to the flow per
second in both directions will be equal, but the velocities of electrons proceeding from
the hot end will be higher than those velocities of electrons passing through the
section from the cold end. This difference ensures that there is a continuous transfer
of heat down the temperature gradient without actual charge transfer, i.e 0I
.
1.2.2 The Peltier effect
Figure 5: The Peltier effect
When an electric current flows across a junction of two dissimilar conductors,
heat is liberated or absorbed. Figure 5 shows the immediate environment of one such
junction at constant temperature. The electric current through the junction is
comprised of N conduction electrons per unit volume possessing a mean energy E per
particle and moving with a mean drift velocity v. The current in conductor A brings
up energy at the rate NAEAvA per unit area to the junction and the current takes away
14
energy from the junction in B at the rate NBEBvB per unit area. There is thus a net rate
of release or absorption of heat energy at the junction given by,
BBBAAAUvENvENJ )1.2.2.1(
where JU is the total heat flux density. The currents densities in A and B at the
junction are identical and are given by
BBAAveNveNI )2.2.2.1(
where e is the electronic charge and I is the common current density. Thus,
)()3.2.2.1(BAU
EEe
IJ
and, if EA≠EB, an absorption or release of heat will take place at the junction via the
interaction of the electrons with the lattice and will be proportional to the current
flowing. This reversible heat absorption process is known as the Peltier effect and
arises from the fact that the two metals have different electronic properties and, in
particular, their conduction electrons have different heat capacities. The Peltier
coefficient is defined for two metals A and B as the rate of heat absorption or
evolution per unit current.
)(1
)4.2.2.1(BA
U
ABEE
eI
J
We again refer to equation 1.2.1 and relate the Pletier coefficient to the
quantity L21/L11.
1.2.3 Onsager relation and Mott formula
In a thermoelectric circuit the Seebeck and Thomson effects take place
simultaneously with Joule heating and thermal conduction processes. In deriving
relationships between S, and (conductivity) Onsager (1931) established a method
that relates the flow of heat and charge under the action of electric potential and
temperature gradients.
T
V
LL
LL
Q
J
2221
1211)1.3.2.1(
when implying thermodynamics principles the obtained onsager relations are:
15
TeS
eSTL
eSTL
TTe
SL
L
22
21
12
11)2.3.2.1(
where is the chemical potential.
The Mott and Jones treatment of thermoelectricity is a quantum mechanics approach
to this phenomenon. The electric and heat currents are given respectively by:
vdNeJ
)3.3.2.1(
where N is the number of conduction electrons per unit volume. Calculation of the
density of states and application of the Fermi-Dirac distribution function yields the
following expression for the electric current:
dEfEv
meJ
2/3
22
2
2)4.3.2.1(
where v is the velocity of an electron and E is the energy of an electron. If we assume
that a relaxation time, , exists that:
)()5.3.2.1( 0
ff
t
f
then the influence of the electric field and the temperature gradient upon the Fermi-
Dirac function could be described by the equation:
x
fv
v
f
m
eff 00
0)6.3.2.1(
where, is the electric field and 0
f is the Fermi-Dirac distribution in equilibrium.
The expression for the electric current becomes:
x
TH
TH
T
E
T
EeHeJ FF
211
2 1)7.3.2.1(
where
dE
E
fE
h
mH n
n
02/1
3
2/1
3
216)8.3.2.1(
Equation (1.2.3.7) gives the current density j which results from an electric field and
a temperature gradient xT / applied to a free-electron metal obeying Fermi-Dirac
statistics. It can be compared with the Onsager relations to give the conductivity
16
1
2)9.3.2.1( He
and the thermopower S is obtained from:
T
HH
T
E
T
EHe
Te
ES FFF 2
11)10.3.2.1(
to give
FnE
H
H
TeS
1
21
)11.3.2.1(
Numerical evaluation of the Hn integrals to the first order gives the following
expression for the Seebeck coefficient:
2/3
2
2
2/5
2
2
2/3
2/122 1
6)12.3.2.1(
FE
F
FE
FFFE
EdE
dE
dE
d
EEe
TmkS
FF
F
The above Mott expression (1.2.3.12) for the seebeck coefficient S depends on the
functional form of )(E , the relaxation time. If we assume that )(E =constant
(independent of E) then the expression for the Seebeck coefficient becomes:
FEe
TkS
2)13.3.2.1(
22
1.3 Motivation
The problem of understanding the electrical conductivity of graphene, has
generated a large experimental and theoretical effort, given its fundamental
importance and its technological relevance2. The transport properties of graphene
derive mainly from its linear zero-gap dispersion relation of fermion carriers
(electrons and holes) with a charge neutral singular point, named the Dirac point,
where the electron and hole bands touch with essentially zero density of states2. In the
limit of vanishing disorder the Fermi energy of graphene lies exactly at the Dirac
point, and conductivity should have a universal value of /4 2eD 8,9,10,11,12,13,14
.
Experimentally however, conductivity at the Dirac point is found to be bigger (by a
varying factor of 2-20) than σD15,16
. Disorder in the form of either ripples17,18,19
or
random charged impurities in the substrate20
is invoked to explain this discrepancy.
Disorder locally shifts the Dirac point moving the Fermi energy from the charge
neutrality point21,22,23
. As a result an inhomogeneous 2D charge density landscape is
developed with electron-hole puddles, which have recently been observed
experimentally24,25
. Figure 6 exemplifies the main qualitative properties of the carrier
distribution close to the Dirac point: (i) The distribution is characterized by electron-
17
hole puddles, (ii) the typical size of the majority of puddles, defined as regions with
same-sign charges, is of the order of the sample size, as expected for a semimetal
close to the neutrality point.
This inhomogeneity is considered to dominate graphene physics at low (n 1012
cm-2
) carrier density with density fluctuations becoming larger than average density15
.
It is therefore clear that in order to be able to design future graphene-based
thermoelectric devices, it is essential to know the properties, origin, and effects of
extrinsic disorder in graphene.
The effect of puddles on electronic transport properties is a matter of active
debate24
. Specifically, one of the important questions is whether the p-n junctions
formed between puddles dominate the transport properties of samples or alternatively
is the mean free path of charge carriers smaller than the average size of puddles.
Here we wish to show that semi-quantitative insight regarding this issue can be
achieved by thermopower measurements of graphene devices in a field-effect
configuration. Though these measurements were published in the course of our work
by two other groups26,27
. They didn‟t address the influence of intrinsic disorder and
charge puddles distribution on the thermoelectric behavior at room temperature as the
source for deviation from Mott formula, but rather focused on measurements in the
presence of magnetic fields and at cryogenic temperatures, what is known as the
Nernest effect.
Figure 6: Schematic of charge puddles in a small segment of graphene: the charge carriers in the
grey and white puddles are electrons and holes, respectively. Two representative dispersion
curves are shown above the two types of puddles. The Fermi level in electron and hole puddles
reside in the dispersion curves above and below the Dirac point, respectively.
1.4 Methods and Materials
1.4.1 Identification and characterization of prospect flakes for
measurement
In order to obtain graphene flakes on Si/SiO2 we tried different methods and
directions and came to the conclusion that the method that gives the best quality
18
pristine graphene flakes in the simplest way is by rubbing with tweezers a scotch tape
with graphite flakes on it onto Si/SiO2 (300nm) substrates and searching for flakes
under an optical microscope. The various methods explored and the one used
eventually are described in detail in appendix A.
The position of chosen flakes was assigned relative to prefabricated alignment
marks for E-beam writing of electrodes. The validation that these flakes are truly one
atomic layer thick was done by Raman spectroscopy28
and Atomic Force Microscope
measurements, Figure 7.
Figure 7: (a) Raman fingerprint of single atomic layer of a graphene flake28
. (b) Atomic force
microscope image of a graphene flake attached to multi layer graphene and it’s obtained height
of about 0.8nm in accordance with previous works5.
1.4.2 The experimental structure and it’s calibration
The measurement configuration, presented in Figure 8, is fabricated by electron
beam lithography, followed by evaporation of Cr(3nm)/Au(100nm) features and a lift-
off procedure in acetone. The Cr/Au leads on the flake are used for the measurement
of both the conductance σ, and the thermovoltage, VTH. The latter is induced by a
temperature gradient caused by a micro-heater adjacent to one end of the flake, but
not in contact with.
Joule heat generated at the heater propagates through the 300nm thick silicon
oxide layer, which has a relatively low thermal conductivity (~ 0.5 W/m K @ 300k),
creating a temperature gradient, ΔT, across the Graphene flake and the surface of the
substrate to which it is thermally anchored. Heat dissipates quickly once it reaches the
underlying silicon substrate, which has higher thermal conductivity (150W/mK @
300k). The configuration presented in Figure 8 enables to determine the temperature
gradient between the two thermovoltage probes. This is done by measuring the
7006005004003002001000
2.5
2
1.5
1
0.5
0
X[nm]
Z[n
m]
(a) (b)
19
resistance change in the electrode adjacent to the heater line and of the one distant
from the heater as a function of heater power at different bath temperatures. The
electrode's temperature rise is obtained from the linear relation TRR )1(0
,
which holds for metals. Because the resistance change is very small, it was
determined by a Lock in amplifier (LIA) technique (Stanford SR830, V=50mV,
f=18Hz) in a 4 probe configuration. The LIA operates in constant current mode of
<1A by connecting a large resistor in series (1M). In this current range the
resistors don‟t self heat and the signal is prominent enough to be detected. Electrodes'
temperature coefficient was obtained by calibration of resistance versus bath
temperature in the range 4-300k which was done in advance and found to
be 13104.1 C .
Figure 8: Optical picture of the experimental setup.
The determined temperature gradients at various bath temperatures was
compared to simulated results based on FlexPDE (a commercial partial differential
equations solver), Figure 929
. Figure 10 shows that each resistor feels a uniform
temperature along its 4m length, proving that the measured temperature reflects the
temperature at the graphene flake ends.
21
Figure 9: Comparison between simulated (red) and measured (black) temperature gradients
between two probes (5m apart) as function of bath temperature for constant heater power of
300W and SiO2 thickness of 1m. the heater is located 0.5m from one electrode.
Y(
m)
Z(
m)
X(m) X(m)
Y(
m)
Z(
m)
X(m) X(m)
Figure 10: Representative report of Numerical simulations results of the temperature gradient on
graphene flake on 300nm SiO2 on 450m Si substrate. The temperature gradient is established
by microfabricated metal line adjacent to but not touching the graphene flake. The Heater power
was set to 80W. In order to include graphene contribution into the simulation we upscale its
height from one atom thick to the height of the metallic electrodes and changed the thermal
conductivity constant in proportion. (Scaling tests validates this correction)
1.4.3 Measurement Technique
At each gate voltage, VG applied to the underlying Si substrate versus one of the
voltage-probing leads, conductance is determined by a standard lock-in procedure.
Subsequently, the measurement setup is switched (using a Keithley 707A switching
matrix) to measure ΔVTH (using Keithley 2002 multimeter). Care was taken to use low
heating powers in order to maintain a temperature gradient across the flake that is
much less than bath temperature, as only then the Seebeck coefficient described by S=
X(m)
21
VTH / ΔT is thermodynamically well defined. All experiments were computer
controlled with labView 7 via GPIB communication protocol. In general, all programs
needed, setups and instruments used in the experiments, and fabrication procedures
presented in this thesis were prepared and assimilated from scratch by us.
Measurements were conducted over a temperature range of 77–300K using a
continuous flow cryogenic probe station with a base pressure of 10-6
torr (Desert
Cryogenics), Figure 11. Overall 6 different samples were measured with characteristic
mobility of ~3000 cm2/Vs.
Figure 11: Outside view of the probe station (Left), and inside view of the probe station (Right).
1.5 Results and Discussion
Figure 12 shows a set of Seebeck versus gate voltage curves (S-VG), determined
as described above at different bath temperatures. The maximum Seebeck at room
temperature is ~80μV/K. This value is typical to all measured samples. The sign of S,
which indicates the sign of the majority charge carrier, changes from positive to
negative as VG crosses the charge neutrality point (in this specific sample at VD =
16V). Some electron-hole asymmetry is observed probably due to charged impurities
in the substrate or formation of a p-n junction at the Au contacts30,31,32
. The inset of
Figure 12 shows a linear increase of S with the bath temperature T, extracted from the
peak values of the S-VG curves. The linearity suggests that the developed
Thermovoltage originates from a diffusive process. It also suggests that contribution
of phonon-drag is negligible33
, which is expected considering the weak electron-
phonon coupling in graphene34,35
. The measured thermopower contains contributions
from the seebeck coefficient of the gold electrodes (~0.7V/K). These effects are
negligible relative to the thermopower of the graphene flake.
Chamber
Probes
Microscope
Heated Platform
22
Figure 12: Seebeck versus gate voltage curves measured at three bath temperatures. Inset: The
absolute value of S (measures at two different gate voltages) increases with T.
Considering the apparent diffusing behavior and within the Boltzmann
formalism, the Seebeck coefficient can be described by the semiclassical Mott
formula36
, which for graphene has the form37
:
F
B
Ee
TkS
3
222
)1.5.1(
where kB is the Boltzmann constant, e is the electron charge, and EF is the Fermi
energy. EF depends on the carrier density n according to nE FF where is
Planck‟s constant, and the Fermi velocity is smF /106 . The carrier density is
defined by e
VVCn DGG )( where the gate capacitance in our device geometry is
CG~100 aF/cm2. Thus S is expected to be proportional to DG VV /1 . Indeed, Figure
13 shows that far away from the Dirac point (VD), this proportionality prevails.
It also describes accurately the behavior around the Dirac point at 100K.
However it fails to do so at 290K. In the course of our work, similar observation was
recently published by two other groups26,27
. This is not necessarily surprising if one
considers that the Mott relation, which is based on the Sommerfeld expansion, is valid
only when the temperature is much lower than the Fermi temperature, where the
scattering time is essentially energy independent.
23
Figure 13: Fitting of the Mott formula to measured Seebeck coefficients as a function of gate
voltages at two different bath temperatures. Far from the Dirac point the Mott relation is
accurate suggesting S n-1/2
. At room temperature the Mott relation diverges and does not
describe accurately the Seebeck coefficient around the Dirac point.
Around room temperature, in the absence of screening of charged impurities,
inhomogeneity near the Dirac point which affects transport properties22
need to be
considered as well in order to describe accurately the transport properties38
also the
effects thermopower.
A simple semi-quantitative treatment can be developed to modify the Mott
formula is the presence of electron-hole puddles. When electron-hole puddles exist
close to the neutrality point, the density of impurities, ni, is equal to the density of
excess electrons (ne) and holes (nh) according to: ni=ne+nh. We can assume that
initially ne=nh=ni/2. Upon application of gate voltage, n carriers are introduced into
the flake, or in other words, n/2 charges are added to one type of carrier, say the
electrons and n/2 are reduced from the other type, i.e., holes. As a result the following
balance is achieved: ne=ni/2+n/2 and nh=ni/2-n/2. These relations hold as long as
n<ni. Beyond this point the Mott relation is applicable.
The Fermi level of the puddles can be described by “electrons-Fermi
level”, eFeF nE , and the “holes-Fermi level”, hF
hF nE . If we assume
that the charge puddles extend across the devices, then all electronic transport
24
properties should be a weighted average of electrons and holes contributions. Around
the Dirac point, the Seebeck coefficient, under mixed contribution of electrons and
holes is therefore defined as7:
(1.5.2)
eh
hheeSS
S
where σe,h the conductivity (with the subscripts e and h referring to electrons and
holes, respectively) is defined as23
:
(1.5.3) i
he
hen
n
h
e ,2
,
20
Combining equations 1.5.1-3, it can easily be shown that under the above conditions
the Seebeck coefficient is defined as:
(1.5.4)he
he
F
B
nn
nn
e
TkS
3
222
Figure 14 displays the behavior of equations 1.5.1 and 1.5.4 as a function of VG.
Indeed, equation 1.5.4 appears to describe well the results at 290K close to the Dirac
point, where the Mott formula (equation 1.5.1) fails. Fitting is based on one free
parameter ni=41010
cm-2
.
Figure 14: Fitting of equations 1.5.1 (line+open circles) and 1.5.4 (red continuous line) to the
results (black squares) at T=290K.
1.6 Conclusions
In the first part of this thesis we have shown that the Mott formula diverges and
does not describe well the Thermopower properties of graphene at low carrier density.
Under these conditions, the role of inhomogeneity becomes dominant and the
presence of electron-hole puddles needs to be considered in order to describe
accurately the thermopower properties. We used a very simple model to account for
this effect, which appears to describe well the behavior of the Seebeck coefficient as a
25
function of gate voltage near the Dirac (electroneutrality) point. This comes as
another proof for the unique ground state properties of graphene patterned on
dielectric materials.
26
2 Electron-Phonon coupling
In this chapter we present our experiments in Raman spectroscopy in order to
prove that the conductivity of single-layer graphene under high bias (>0.2V) depends
on direct coupling between charge carriers and optical phonons of the flake and not
due to coupling with modes of the underlying SiO2 layer. These results therefore
suggest that there should be an intrinsic limitation to the performance of graphene-
based electronic devices.
In this chapter we begin with short introduction to Raman spectroscopy (2.1),
followed by the motivation to this research (2.2), methods and materials (2.3), results
and discussion (2.4), and conclusions (2.5).
2.1 Raman Spectroscopy
When a monochromatic light with wavelength i~ is scattered from an atom or a
molecule, most photons are elastically scattered (Rayleigh scattering). The scattered
photons have the same energy and therefore the same wavelength as the incident
photons. That is why Rayleigh scattering is also called elastic scattering. However, a
small fraction of scattered light (approximately 1 out of 1000 photons) is scattered
from exitations with wavelengths slightly different from that of the incident photons,
in the form of mi ~~'~ , when m~ corresponds to transitions between vibrational
energy levels of the molecules in the medium (Raman scattering). In the Raman
scattering process, energy passes between the photon and the molecule, making it an
inelastic scattering process.
The photon can give energy to the molecule, and then the scattered light would
be of lower energy mi ~~'~ , or receive it from the molecule, and the scattered light
would be of higher energy mi ~~'~ . The lines in the spectrum, corresponding to
the situation when the photon loses energy are called Stokes lines, and the lines
corresponding to the situation when the photon receives energy are called anti-Stokes
lines. The Rayleigh scattering can always be observed when the Raman scattering is,
making it easy to measure m~ instantly and of course the Raman spectrum is
symmetric relative to the Rayleigh band, but the downside is that it disturbs the ability
to observe Raman peaks of low frequencies.
27
A Raman transition from one state to another, and therefore a Raman shift, can
occur only when the polarizability changes during the process under consideration, i.
e. during the vibration of the molecule (Raman selection rule). But the main problem
of Raman spectroscopy is the very weak signal, as a result of the small operation cross
section for Raman. Raman scattering is only 0.1% of all the scattered light. Thus, for
an efficient collection of a Raman spectrum, powerful light sources are needed,
sensitive detectors and a large number of scattering molecules.
Figure 15: Illustration of Rayleigh and Raman scattering processes. The incident light excites the
molecule to a virtual excited state, where the molecule spends very little time in, and goes back to
the electronic ground state. In most cases, the molecule will go back to it's previous state,
resulting in scattered photons with the same wavelength as the incident photons (Rayleigh
scattering). Another option is that the molecule will go to an excited vibrational state, and the
photon will scatter with less energy (Stokes lines). If the molecule was originally in an excited
vibrational state, it can go to an energetic state lower that the one it was in before, resulting in a
scattered photon with more energy than the incident photon (anti-Stokes).
Since a photon can receive energy from a molecule only if the molecule is in
an excited state (returning to ground state following the interaction), and since there is
only a small population of molecules in excited states at room temperature, the Stokes
spectrum is more intense than the anti-Stokes spectrum. In addition, since the
intensities of the Raman lines are dependent on the number of molecules occupying
the different vibration states, Bose-Einstein distribution teaches us that more
molecules occupy the lower energy levels at room temperature, and can assist us in
determining the effective temperature of the molecules in a junction, by the ratio of
the intensity of the Stokes and anti-Stokes lines:
))(/exp()(
)()1.1.2(
4
4
effBv
vL
vL
S
AS TkhI
I
where IAS(S) is the intensity of the anti-Stokes (Stokes) Raman mode, νL(ν) is the
frequency of the laser (Raman mode), σAS(S) is the anti-Stokes (Stokes) scattering
28
cross section of the adsorbed molecules, and AAS(S) is the average local field
enhancement at the molecules at the anti-Stokes (Stokes) frequency.
2.2 Motivation
Graphene is a promising material for high-speed nanoscale electronic devices39
.
The realization of graphene-based devices depends on deep understanding of how
their conductivity, σ, is affected by the contribution of various scattering
mechanisms40,41,42,43
:
(2.2.1)σ-1
= σ-1
ci + σ-1
sr + σ-1
LA + σ-1
OP+ σ-1
corr
where the subscripts indicate contributions of charged impurities (ci), short range
scatterers (sr), longitudinal phonons (LA), optical phonons (OP), and surface
corrugations (corr).
Elucidating the role of underlying dielectrics, most commonly SiO2, on the
various scattering mechanisms is a subject of active research44,45,46,47
. Such an
understanding is important since SiO2 affects the performance of graphene devices
quite substantially, decreasing room temperature mobility from a typical value for
graphite of ~100m2V
-1s
-1 to a mobility in the range of 0.1-2 m
2V
-1s
-1 39
.
The effect of trapped charges in the oxide and of surface corrugations on the
values of σci, σsr, and σcorr is quite understood48
. The main current controversy is as to
the exact nature of scattering events that govern the magnitude of σOP. High energy
carriers could either couple to in-plane optical phonons of graphene49,50,51,52,53,54,55,56
or remotely to the field induced by surface polar modes of the underlying
SiO244,45,46,47
. In either case, the resulting inelastic scattering events are expected to
dominate device operations and limit their operational time scales. With characteristic
inelastic mean free path of several hundreds of nanometers, this limit is in the range of
~0.1psec based on a Fermi velocity of ~106 m/sec.
If scattering of high energy carriers is dominated by SiO2, then better
performance of devices could be achieved by using underlying materials with higher
dielectrics. However, if carriers are coupled more efficiently to the vibrational modes
of graphene itself, then the limit set by this interaction is intrinsic and cannot be
improved. Hence, elucidating the exact mechanism that limits σOP is critical for the
prospective future of graphene-based electronic technology.
Currently it is suggested that coupling of carriers to SiO2 modes is dominant as
determined by analyzing high bias saturation currents in graphene devices using
29
simulations46,47
and analytical approximations in which the energy of coupled
phonons is a free parameter45
. Better fit is achieved using the 55meV mode of SiO2
instead of the 200meV mode, which is associated with the longitudinal optical phonon
at the Brillouin zone center of graphene, commonly referred to as the G mode28
. In
addition, temperature dependent resistivity measurements suggest that coupling of
charge carriers takes place with two optical modes of SiO2 59meV and 155meV,
with a ratio of coupling of 1:6.5 40
.
Here, because of the critical importance of the issue, we wish to reexamine the
question of dominant carrier-phonon coupling by combining I-V measurements with
Raman spectroscopy. We show that changes in the effective temperature of the optical
G mode of graphene correlate well with changes in the conductance of devices, while
at the same time the effective temperature of the 55meV SiO2 mode appears just to
follow temperature changes of the G mode. From these results we conclude that SiO2
modes do not interact directly with charge carriers in graphene and that its
conductivity is inherently limited.
Raman has proven to be a valuable tool to study graphene and carbon
nanotubes. It can be used to determine the number of layers in thin flakes28
, degree of
doping57
, and phonon dispersion58,59,60
. In relation to charge-carrier optical phonon
coupling, Raman measurements have shown that in graphene the adiabatic Born-
Oppenheimer approximation fails50
and that under high bias generation of hot
phonons, i.e., phonons that are not at equilibrium with their surroundings, can be
detected 45,54,61
.
Raman can be used to monitor mode-specific effective temperature, Teff(), of
phonons by two methods: by probing the softening of modes as a function of
temperature45,62
, or by measuring spectra both in the Stokes (S) and anti-Stokes (AS)
regimes, facilitating direct probing of changes in phonons population as a function of
temperature or source-drain bias54,63,64
.
The first approach was used to measure the thermal conductivity of suspended
graphene flakes65
. Its applicability to determine Teff(), under voltage bias, i.e., far
from equilibrium is currently questionable. The second approach was recently used to
probe the effective temperature of molecular junctions under bias64
and also of carbon
nanotubes and graphene61,63
. We therefore use in this study the AS/S ratios of both the
G mode of graphene and the 55meV mode of SiO2 to determine their Teff() as a
function of voltage bias. Though AS/S signals of the Gmode were reported in the
course of this work by Cahe et al54
, they didn‟t address the delicate role of the 55mev
31
mode of the underlying SiO2, and didn‟t present a comprehensive discussion in
relation with the conductivity behavior.
2.3 Methods and Materials
2.3.1 Device Fabrication
Fabrication of typical devices began by placing graphene flakes on a p-type
degenerate Si wafer covered with 300nm of SiO2 by standard micromechanical
cleavage of graphite, as described before in section 1.4.1 and in detail in appendix A.
Single layer graphene flakes were identified by their characteristic color under an
optical microscope, and by their Raman spectroscopy fingerprint. Graphene flakes
with typical dimensions of ~5m1m were chosen to enable patterning of electrodes
for four-probe measurements by e-beam lithography. The potential difference
between the two inner electrodes, V4p, will be referred below, for simplicity, as either
the four-probe potential or as the source-drain bias. Following metal evaporation of Ti
(3nm)/Au (80nm) and liftoff (see Figure 16), the resulting devices were annealed at
400K under vacuum (10-5
torr) for few hours prior to their further characterization by
plotting their conductivity, σ, versus back gate voltage, VG,. Altogether, three devices
were measured, all showing comparable results, with detectable anti-stokes signal at
zero bias, with drift-less, non-fluorescent Raman spectrum and with no indication of
graphene being graphitized. We focus below on one of these devices (Figure 16) to
present the results and facilitate discussion. The conductivity curve (Figure 16c) is
largely symmetric around a particular gate voltage VG = VDirac and shows a minimum
at this value. The nonzero value of VDirac (+7V) indicates that there is unintentional
doping of the graphene sample whose origin may be caused by random charged
impurities, located in the graphene environment, that could also induce disorder10
.
This disorder is characterized by electron-hole puddles in quantitative agreement with
surface probe experiments, as well as recent thermopower measurements. The typical
size of an electron-hole puddle, defined as a region with same-sign charges, is of the
order of the sample size, L, as expected for a semimetal close to the neutrality point.
The open rectangular points in Figure 16c present a fit to the experimental
results based on a previously presented self-consistent Boltzmann transport theory
with charged impurity scattering66
. In this specific sample, the density of charged
impurities is ni ~ 7.51011
cm-2
at VG = 0, the hole carriers density is nh ~ 51011
cm-2
31
and the Fermi energy is 83meV calculated by hFF nE , where the Fermi velocity
νF ~ 106 m/sec.
2.3.2 Measurement Technique
Raman measurements of devices were taken by a home-built scanning Raman
confocal microscope operating in a reflection mode, with through the objective
illumination from a fiber-coupled 532 nm solid-state laser with an effective power on
the samples of ~1mW and beam diameter of <0.5m. The reflected Raman scattered
light was collected with a 0.7 NA 100 objective and analyzed by a spectrometer
equipped with an electron multiplying CCD camera (Andor Newton EM, DU971N-
BV). In order to capture the Raman Stokes and Anti Stokes signals simultaneously,
the spectrometer resolution was set to 6±1cm-1
. All measurements were conducted at
room temperature under a blanket of Ar flow in order to maintain the stability of
devices under the laser beam over the period of time necessary for a full set of bias-
dependent Raman spectra measurements, using an integration time of 60-120s for
each bias value. Obtained spectra were baseline corrected by polynomial fit in the
regions of interests. The Raman signals were obtained from integration over 6 CCD
camera channels, the fluctuations are in the order of ~5% should result in apparent
Teff
(ν) variations of ~3K, which are presented in the figures as the error bars.
Figure 16: (A) Scheme of a graphene device with four probes Au contacts. (B) I-V curve of a
representative device. (C) Conductivity as a function of gate voltage VG. (D) Raman spectrum of
a device the 2D/G intensity ratio is typical for a single layer flake.
32
2.3.3 Evaluation of heat loss in joule heated graphene due to
convection and radiation
In this section, we would like to address the issue of heat loss in the joule heated
graphene flake due to convection in the presence of Argon gas flow over the sample
and radiation heat losses.
When the temperature of a plate Ts and the temperature of the running gas flow TF are
different, the heat transfer process occurs between the plate and the gas. According to
Newton law:
)()1.3.3.2(FS
TThAq
where q is the quantity of heat transferred, h is the heat transfer coefficient and A is
the surface area.
The heat flux is proportional to the temperature difference Ts – TF. The
coefficient of heat transfer h depends on the hydraulic picture and regime of the
medium flow (laminar or turbulent), distance x from the front edge of the plate and
thermo-physical properties of the running gas.
Convective heat transfer coefficients at forced flow are commonly represented
by the following parameters (The values obtained below are for Argon and in MKS
units67
):
A non dimensional parameter that unifies the effects of flow, distance, and fluid
properties called Reynolds number Re:
6281007.2
1011013.1)2.3.3.2(
5
53
vLR
e
where is the gas density, v is gas velocity, L is sample length, and is viscosity.
Prandtl number Pr that is the ratio of kinematic viscosity to thermal diffusivity:
67.0101.3
1007.2)3.3.3.2(
5
5
r
P
and the non dimensional Nusselt number Nu:
Lk
hN
U)4.3.3.2(
where h is the heat transfer coefficient, k is the thermal conductivity, and L is sample
length. The average of the nusselt number Nu over the length of the plate for laminar
flow is given by68
:
14646.0)5.3.3.2(2131
erU
RPN
And the heat transfer coefficient is then obtained:
33
KmWkL
Nh U 2
5/2800002.0
101
14)6.3.3.2(
Once we obtained the heat transfer coefficient of the running gas- h. we simulated the
experimental setup in FlexPDE numerical solver, in order to estimate the heat loss in
the sample due to convection. The simulation results are presented in Figure 17.
Figure 17: Temperature distribution in a heated graphene flake (L=10m) in Vacuum (a) and
under Argon gas flow (b). The Argon gas flow was introduced to the simulation through the
boundary condition of convection heat loss (2.3.3.1) with the proportionality factor h=28000, that
was calculated in equation (2.3.3.6), and the thermal conductivity coefficient k=0.02W/mK. We
upscale the graphene flake height from one atom thick to the height of 300nm and changed the
thermal conductivity constant in proportion. (Scaling tests validates this correction).
From the results presented in Figure 17 it is seen that the Argon flow cools the
graphene by roughly 50K in the experiment temperature range (300-600K).
We estimate the radiation heat loss of the graphene flake at a uniform temperature of
600K (the highest temperature reached in the experiment) using the Stefan-Bolzmann
law:
4)37.3.2( ATI
where =2.3% is the emissivity of graphene69
, is the Boltzmann constant and A
is the area of the graphene flake. The calculated radiation heat loss is only ~6nW,
which is minor relative to the dissipated power of ~1mW.
2.4 Results and Discussion
The two most intensive features in the Stokes regime (figure 16D) are the G
peak at 1580 cm−1
and the 2D band at ~2700 cm−1
, where the latter is a second order
of the D peak (at ~1350 cm-1
)28
, see Figure 16D. The G mode in graphene is due to
the -E2g phonon (1580cm-1
/0.196eV). The E2g is a mode consisting in an anti-phase
34
movement of the two carbon atoms in the unit cell, Figure 18. The D peak is never
observed directly in all our samples providing excellent proof for their high structural
sp2 uniformity with no nanocrystalline structures or presence of amorphous sp
3
carbon. The G/2D peak ratio is indicative of a single layer flake.
Figure 18: Schematic diagram of the atomic displacements in the graphene plane for the E2g
mode at the point
Figure 20 plots the conductance (dI/dV) of the same device shown in Figure 16
as a function of V4p, measured at VG = 0. The conductance is stable until a bias of
~0.25V. It then gradually decreases until V4p ~ 1.1V. At this point, it starts to increase
reaching approximately its initial value at a bias of ~2.5V. Junctions were not
measured at bias voltages exceeding the latter value, since then based on their Raman
spectra, their structure starts to deteriorate.
Figure 19: dI/dV vs. V4p of the I-V curve in figure 16. Schematic presentation of the carrier
concentration at three different points a long the curve is shown. (a) channel charge is uniform,
(b) channel charge at the drain end begins to decrease as the minimal density point enters the
channel, (c) An electron channel forms at the drain.
The above behavior can be semi-quantitatively explained in the following way:
With VDirac = 7V (see Figure 16), when VG=0, the device is a hole conductor. At low
35
bias, these holes can interact only with charged impurities and acoustic phonons55
.
The elastic scattering time, τel, can be estimated, to a first approximation, by
employing the Einstein relation: )/()/( 22/1Fel en , and the corresponding mean
free path can be obtained from Fl . For σ = 0.8mS, with τel = 8 10-14
sec, the
corresponding mean free path is l ~ 80nm.
Charged impurity scattering rate is described by55
:
E
nimpF
imp20
1 2
)1.4.2(
Calculating for E = EF ~90mV, τimp is 8.7 10-14
sec.
Acoustic phonon scattering rate can be calculated according to70
:
ETkD
EphmF
Bac
ac
22
2
3 4
1
)(
1)2.4.2(
where ρm= 7.610-7
Kg/m2 is the mass density, ph = 2 10
4 m/sec is the acoustic
phonon velocity in graphene, kB is the Boltzmann constant, T=300K, and Dac ~ 60 eV
is the acoustic phonon deformation potential. For E = EF, τac is 8.7 10-14
sec.
Thus with increasing bias, since τimp E and τac 1/E, and since the scattering
rate in both mechanisms is similar, the overall scattering rate is constant leading to
constant conductance as a function of bias, within agreement with the experimental
results.
Once a bias threshold of ~0.25V is crossed, the conductance appears to
decrease. This behavior is a direct result of the onset of inelastic scattering of charge
carriers with optical phonons55
. Following references71
and72
, in order for a charge
carrier to emit an optical phonon, , it must first accelerate over a length defined by
LeV )/( , where L is the sample‟s length, to attain sufficient energy for emission. If
once the carriers attaining the needed energy emit phonons instantaneously, any
further increase in their momentum as a function of increasing bias is prevented, and
the velocity of charges should saturate. The observed progressive decrease of
conductance supports a different scenario in which instead of immediate emission of
phonons there is an energy dependent finite scattering rate that is defined by55
:
)()(2)(
1)3.4.2(
2
2
ED
E Fm
OP
op
where DOP is the deformation potential in the order of 25eV/Å.
Based on the above equation, for the G mode phonon with energy = 0.2eV
at the observed threshold of ~0.25V, τop is smaller than τac ~8 10-14
sec, and
36
emission of optical phonons becomes the dominant scattering mechanism of hot
carriers.
Full saturation of current (dI/dV=0) is not reached. Instead an increase of
conductance above a bias voltage of V4p ~1.1V is revealed. This can be explained by
considering an ambipolar behavior of the device43
. Under a gradual channel
approximation the potential along a device can be written as:
C
xVVxV DiracG
)()()4.4.2(
where ρ(x) = nh(x)-ne(x) is the charge density.
When the local potential V(x) satisfies V(x) = VG -VDirac, the Fermi level
coincides with the Dirac point and the local charge density is nullified. Assuming the
drain electrode is grounded, close to the source electrode (x = 0) the Fermi level
should align with the Dirac point for a given gate voltage when Vs satisfies: VS=VG -
VDirac. In addition, when VS >VG -VDirac then ρ(0)>0 and when VS <VG -VDirac then
ρ(0)<0. This sign change of ρ(x) implies that as long as V4p<1.1V, current is carried
by holes throughout the length of the device. At V4p=1.1V, a „punch-through‟ region
is formed at the drain where the carrier density is nullified (see Figure 19). Above this
bias, the „punch-through‟ point starts to move into the device. While the charge
carriers on one side of the „punch-through‟ point remain holes, those on the other side
are now electrons. Both types of charge carriers contribute to the measured current.
Having semi-quantitatively understanding the behavior of conductance as a
function of bias, it is imperative to compare between the conductance pattern
appearing in Figure 19 and the corresponding effective temperature of the G and SiO2
modes as a function of voltage bias. The effective temperature of each mode is
calculated according to54
:
1
4
4
)(
)(ln)()5.4.2(
L
L
S
AS
B
effI
I
k
hT
where IAS (IS) is the intensity of the anti-Stokes (Stokes) Raman mode, L() is the
frequency of the laser (Raman mode), Figure 20.
37
0.0 0.5 1.0 1.5 2.0 2.50.5
0.6
0.7
0.8
dI/
dV
(m
S)
Bias(V)
0.0 0.5 1.0 1.5 2.0 2.5
400
500
600
TG
(K)
Figure 20: Anti-Stokes (left) and Stokes (right) signals of the G mode phonon (a) and 55mev SiO2
phonon (b) at four V4p voltages. (c) Effective temperature of the G mode phonon and the
conductance (d) as a function of applied source-drain bias. Dashed vertical lines indicate points
where changes in heating rate are accompanied by changes in conductance.
Figure 20c,d plot both the conductance and the effective temperature of the G
mode as a function of V4p. At low bias, the effective temperature of the mode is
-450 -4000
50
100
150
200
Inte
ns
ity
(a
.u)
Raman shift (cm-1)
(a)
(b)
1500 1600 1700
0
500
1000
1500
2000
2500
Inte
ns
ity
(a
.u)
Raman shift(cm-1)
0.15v
0.3v
1v
1.5v
400 450
0
100
200
300
400
500
Inte
ns
ity
(a
.u)
Raman shift (cm-1)
-1800 -1700 -1600 -1500 -14000
20
40
60
80
100
Inte
ns
ity
(a
.u)
Raman shift (cm-1)
(c)
(d)
38
~400K as a result of laser heating above room temperature. Effectively no heating of
the mode is taking place until a bias of ~0.3V is reached. At this point, the charge
carriers have enough energy to excite G phonons, and heating becomes more
pronounced with the commencing of inelastic scattering events as marked by the
decrease of conductance at V4p ~ 0.25V. Changes in conductivity are more sensitive to
commencing of inelastic processes than changes in the Raman signal.
Similar behavior is observed just after the „punch-through‟ point at V4p~1.1V,
where a change in the heating rate of the G mode is initiated at ~1.25V. If before this
point all applied potential is dropped entirely across the hole conducting region, then
after this point part of the potential is dropped also across the electron conducting
region (see Figure 19) giving the additional carriers sufficient energy to excite the G
mode as well. The direct result of increased conductance is additional heating leading
to enhanced rate of effective temperature increase (starting at ~1.25V).
Above V4p~2.0V, there is no further heating of the G mode. Such a behavior has
never been reported before, and further work is required to elucidate its possible
origin.
0.0 0.5 1.0 1.5 2.0 2.5300
320
340
360
T55m
eV(K
)
Bias (V)
0.0 0.5 1.0 1.5 2.0 2.5
400
500
600
TG
(K)
Figure 21: Effective temperature of the G mode (top) and of the 55meV mode of SiO2 (bottom) as
a function of source-drain bias.
The double graph in Figure 21 shows in addition to the G mode, also the
effective temperature of the 55meV SiO2 mode as a function of bias. It is clear that
Teff(55meV)<Teff(G). Most importantly and unlike in the case of the G mode, no
39
heating of the 55meV mode is observed once the applied bias crosses this value.
Heating appears to commence only when the G mode is heated as well. The same
behavior is also observed above the „punch-through‟ point at V4p~1.1V. The
temperature of the SiO2 mode does not change independently as a function of bias,
but instead follows temperature changes in the G mode.
All this is clear indication that the carriers in graphene are coupled directly to
the G mode and not to underlying SiO2 modes making inelastic scattering with the
former mode the dominant mechanism to affect σop.
The G mode is weakly coupled to substrate phonons73
. This is rationalized by
the fact that G mode frequency is much higher than the phonon frequencies of the
substrate. According to first principle calculations, out-of-plane vibrations in
graphene are not coupled to the in-plane motion, which define the G band. The out of
plane vibrations are expected to be more susceptible to the substrate influence. This is
verified by the fact that the Raman spectra of suspended and on-substrate graphene
are similar74
. Figure 22 shows that the G mode temperature is proportional to the
normalized electrical power P=IV/Area for energies above the mode energy
threshold.
PrTTGO)6.4.2(
Where To is G mode temperature prior to onset of heating. Linear fit to the data
gives thermal interface resistance of Gr =6±1·10-8
Km2W
-1, which is in good
agreement to the value obtained from simulation done by Freitag et al75
.
0 1 2 3 4 5
400
450
500
550
600
650
TG
(K)
Normalized Power(109Wm-2
)
Figure 22: Effective G mode temperature as function of electric power, the red line is linear fit to
the data.
To understand why based only on I-V measurements it has been suggested that
σop is governed by coupling to SiO2 modes, it is important to examine previous studies
carefully. In some of these studies, the determination of the most effective coupled
41
mode relies on fitting of the saturated current of devices under high bias to the
following analytical argument for the saturated velocity of carriers45,47
:
F
F
sat
E
2)7.4.2(
As has already been noted by others55
, this analytical approximation is accurate only
if 2/FE . Thus for example in our case, since EF ~89mV, Equation 2.4.7 is by
definition biased towards the 55meV mode than for the G mode (~200meV).
To critically discuss other previous studies that are not based on fitting of
saturation currents40
, it is imperative to examine the coupling of G mode to substrate
phonons via low energy modes of graphene. For this purpose we monitor the full
width at half maximum of the G line, γG, as a function of V4p, as depicted in Figure
23. The width has a constant value up to ~0.7V, and then it decreases by ~1.5cm-1
until a potential bias of ~1.5V. From this point the width increases. This non-
monotonic behavior, also observed by others5554
, can be explained quantitatively.
The width of the G line describes the decay rate of the G phonons, which has
two scattering contributions, electron-phonon (γe-ph) and phonon-phonon (γph-ph): γG =
γe-ph + γph-ph .
Figure 23: Full width at half maximum of the G band as a function of bias. The continuous (red)
curve is a fit to equation 9 (see text for details).
Large G line width values, similar to ours, have previously been reported from
experiments done under Ar flow (after prolonged annealing) similarly to our
experimental procedure. It has been suggested that Ar improves annealing by
desorption of contaminants76
. Heat transport simulations presented before in section
2.3.3 suggest that gas flow above a sample decreases its temperature, for a heating
41
power within the range of our experiments, by 50K. This suggests that scattering of
Ar atoms could open additional route of phonon decay, by interacting with vibrational
modes perpendicular to the surface, which by anharmonic interactions should
effectively increase γph-ph.
Quantitative interpretation of Figure 23 is based on the dependence of γph-ph and
γe-ph on the temperature, assuming for simplicity that the lattice and electrons are at
thermal equilibrium.
The behavior of the anharmonic contribution, γph-ph, as a function of temperature
can be treated by a very simple model describing G mode decaying into two
phonons77
:
)1
1
1
11()8.4.2(
//
0
21
TkTkphphphphBB ee
where G=1+2 with 1=0.14eV and 2 = 0.06eV, based on DFT calculations of
this process60
. We used γph-ph0 = 9 cm
-1 implicitly taking Ar scattering into account.
More elaborate treatment of anharmonic processes, considering 3 and 4 phonon
interactions exist60
, however they are found to be not essential to explain the results.
The value of γe-ph results from G phonons annihilation by creation of electron-
hole pairs. Considering the Pauli exclusion principle and the requirement of energy
and momentum conservation, this process is possible only if the Fermi level is
residing within an energy window [2
,2
GG ] around the Dirac point. The decay
rate is given analytically from Fermi‟s golden rule in the vicinity of the Γ point of the
Brillouin zone under the assumption of linear energy-momentum dispersion of the
charge carriers78
:
)
2()
2(),( 0
)9.4.2(GG
pheeFpheffTE
where f(E) = )1/(1/)(
eBf TkEE
e is the Fermi-Dirac distribution and γe-ph0, Ef, and
Te are the zero temperature width, Fermi energy, and electronic temperature,
respectively. γe-ph0, the fwhm at zero temperature due to electron-phonon coupling is
given by the following expression79
:
0
0)10.4.2( phe
where 0
=20cm-1
is a constant term and
is the intrinsic fwhm of G mode and is
related to electron phonon coupling parameter2
D 79
:
42
2
22
4
3)11.4.2(
D
M
ao
where o
a = 2.46 Å is the graphite lattice spacing, = 5.52 Å eV is the slope of the
electron band, and M is the carbon atomic mass. The electron-phonon coupling
parameter is calculated from the Fermi golden rule80
to be 2
D = 47 (eV/A)
2. This
results in intrinsic fwhm cm-1
.
When Ef is located close to the Dirac point, thermal broadening reduces the
number of occupied electronic states near 2
G and the number of empty states near
2
G . With increasing temperature, the probability for phonon annihilation by
electron-hole pair generation becomes smaller, and γe-ph decreases. With temperature
increase proportional to the applied source-drain bias VSD, γe-ph 1/VSD.
Application of source-drain bias affects equation 2.4.9 not just via temperature
increase but also by the effect on Ef. The applicability of equation 2.4.9 has been
verified by measuring changes in γG upon tuning of the Fermi energy by changing the
gate voltage in a field effect device81
. In these measurements, since no source-drain
bias was applied, the Fermi energy was uniform across each device. In contrast, in our
experiments, as discussed above, with increasing VSD, the Fermi energy distribution
becomes steeper, and G phonon annihilation can take place within a progressively
smaller area of the device with characteristic length, parallel to the applied electric
field, that is proportional to 1/VSD.
At high VSD values, on the other hand, electron-phonon coupling becomes more
efficient according to recent calculations52
. Thus, in contrast to the above two effects,
it is also expected that γe-ph VSD. Thus, in order to account for the possible effect of
VSD on γG via its effect on γe-ph we use the following relation:
SD
eFphephphG
V
CTE
),()12.4.2(
with C a (proportion) free fitting factor.
The continuous red line in Figure 23 is a fit of equation 2.4.12 to the experimental
data using C=0.8.
From the above discussion it is apparent that the life time of an excited G mode
depends on the thermal activation of out of plane modes (taken here to be 60meV, and
140meV) that are expected to be effectively coupled to SiO2 modes because of
symmetry and energy matching78
. The effect of SiO2 modes on current is according to
43
the following route: Using former analytical expressions for inelastic scattering time,
τin, it can be shown that τin (1+Nop)-1
, where Nop is the Bose-Einstein occupation
number of an optical phonon47
. With increasing bias, and consequently increasing
temperature, the life time of the G modes decreases. The resulting decrease in Nop
leads to longer τin which increases the current density according to j E τin, where E is
the applied field across the device.
2.5 Conclusions
From the Raman Spectroscopy measurements which were presented in this
chapter, we deduce that the effective temperature of optical G and SiO2 modes could
be determined as a function of voltage bias in current carrying graphene devices. We
suggest that high energy carriers in these devices are coupled more effectively to the
G mode than to the SiO2 beneath, in contrast to what was suspected before. This
implies that the mobility of graphene transistors is inherently limited.
44
3 Energy dissipation in Quantum Point Contact In this chapter we present our measurements in a new model for highly stable
metallic quantum point contact (MQPC) invented by us for investigation of current
induced local heating effect in atomic junctions. Our devices show controllable and
stable conductance plateaus typical for few atoms junctions and posses healing
capability once broken. We estimate the temperature increase just prior to junction
formation by Electrormigration to be of the order of 150K, and once the metallic atom
bridges the gap the temperature induced voltage to be of the order of 60K. This is
surprising because the size of the contacts is much smaller than the electron mean free
path. It has been proposed that the oscillating atoms heat up in the high local current
because the excited state of the current-carrying electrons is seen by the atoms as
having an elevated effective temperature82
.
We begin with introduction to Quantum point contact electrical transport theory
(3.1), followed by motivation to this research (3.2), methods and materials (3.3),
results and discussion (3.4), and conclusions (3.5).
3.1 Electrical Transport theory in Quantum point contact
Macroscopic conductors are characterized by Ohm‟s law, which establishes that
the conductance G of a given sample is directly proportional to its transverse area S
and inversely proportional to its length L, i.e.
2/)1.2.3( LSG
where is the conductivity of the sample.
Although the conductance is also a key quantity for analyzing atomic-sized
conductors, simple concepts like Ohm‟s law are no longer applicable at the atomic
scale. Atomic-sized conductors are a limiting case of mesoscopic systems in which
quantum coherence plays a central role in the transport properties.
When the dimensions of a contact are much smaller than their mean free path,
the electrons will pass through ballistically. In such contacts there will be a large
potential gradient near the contact, causing the electrons to accelerate within a short
distance. The conduction through this type of contacts was first considered by
Sharvin83
, who pointed out the resemblance to the problem of the flow of a dilute gas
through a small hole. The current density is written as
45
k
kk fvL
ej
2)2.2.3(
where k
f is the distribution function and gives the occupation of state k at position r
and k
v is the group velocity of the electrons.
In a typical transport experiment on a mesoscopic device, the sample (which in
our case is an atomic-sized constriction) is connected to macroscopic electrodes by a
set of leads which allow us to inject currents at fix voltages. The electrodes act as
ideal electron reservoirs in thermal equilibrium with a well-defined temperature and
chemical potential. The basic idea of the scattering approach is to relate the transport
properties (conductance) with the transmission and reflection probabilities for carriers
incident on the sample. In this one-electron approach phase-coherence is assumed to
be preserved on the entire sample and inelastic scattering is restricted to the electron
reservoirs only.
Figure 24 depicts a two terminal configuration where the sample is just a perfect
one-dimensional conductor, having a single mode, occupied between two metallic
electrodes and a voltage difference V applied between the electrodes.
Figure 24: Schematic representation for a ballistic two-terminal conductance problem. The circle
represents the one level sample. The reservoirs (electrodes) are the left and right gray triangular.
Electrons are omitted onto the sample from the left lead with an energy distribution
corresponding to the electrochemical potentials L and collected at the right lead with
electrochemical potentials R, due to potential difference imposed by a battery.
A net current will arise from the imbalance between the population of the mode
moving from left to right (fixed by the Fermi distribution on the left electrode, L
f )
and the population of the mode moving in the opposite sense (fixed byR
f ). The
current is then simply given by:
46
)()(())()(()3.2.3( kRkL
k
kkRkLk ffvdke
ffvL
eI
where L is the length of the conductor. For a long conductor one can replace the sum
over allowed k values by an integral over k. In a one-dimensional system the density
of states is k
v/1)( and the current can be written as:
)()((2
)4.2.3( kRkL ffdh
eI
The factor 2 in this expression is due to spin-degeneracy. At zero temperature )(L
f
and )(R
f are step functions, equal to 1 below 2/eVF and 2/eV
F ,
respectively, and 0 above this energy. Thus the expression leads to I = GV, where the
conductance is ./2 2 heG
This simple calculation demonstrates that a perfect single mode conductor
between two electrodes has a finite resistance, given by the universal
quantity keh 9.122/ 2. This is an important difference with respect to
macroscopic leads, where one expects to have zero resistance for the perfectly
conducting case.
3.2 Motivation
Electromigration and local heating are known to be important considerations in
the design of conventional electronics. It is natural to ask how important this effect is
in nanoelectronics. Current-induced instability and local heating arise from energy
exchanged between electrons and phonons84
. In an atomic scale junction the inelastic
electron mean free path is large compared with the junction size, so that each electron
should release only a small fraction of its energy during transport in the junction.
However, substantial effects still arise due to the large current density in the
nanojunction. Realization of such atomic junctions has been mainly achieved by three
experimental techniques, the scanning tunneling microscope (STM)85
, mechanically
controlled break junction (MCBJ)85
, and electrochemical deposition/dissolution into
lithography defined gaps86
. All these techniques, however, are of no significant
importance in terms of technological applications such as atomic switches87
or
coherent controlled devices88,89
as they do not enable to integrate MQPCs into
electronic circuits. It is therefore essential to develop a different method for their
realization which would be compatible with current large-scale fabrication
technologies.
47
The resulting MQPCs need to be highly stable at room temperature. Contacts
based on STM or MCBJs are reported to have only limited mechanical stability at
room temperature, degrading in best cases, within tens of seconds89,90,91,92,93
.
Complimentary to this high-stability demand, the new fabrication route of MQPCs
should also posses fault-correcting capabilities, i.e., to be able to “heal” damaged
MQPCs when needed.
Recent studies suggest that Electromigration (EM) could be an alternative
fabrication method of MQPCs. EM is the directed migration of atoms caused by a
large electric current density. It proceeds by momentum transfer from electrons to
atoms and requires sufficient atom mobility to occur,94
. Recently, a surge of activity
has been focusing on ways to precisely control EM in order to fabricate very small
gaps between metal leads which can then accommodate single molecules in order to
explore their conductance properties95,96,97,98,99,100,101,102,103,104
. The mechanical
stability of these molecular junctions has been shown to be superior to that of MCBJs
104.
A typical process to form an electromigrated gap is based on passing high
current density (typically >108 A/cm
2) through a lithography defined metal (typically
Au) wire with a constriction (junction) which roughly determines the position of the
EM-induced failure. The resistance of the leads to the junction is RL, and the
resistance of the junction itself is RJ. Since RL>>RJ, the junction is effectively current
biased through RL. Consequently, as EM starts shrinking the junction and RJ
increases, the power dissipated on the junction grows proportionally to RJ, causing
thermal runaway and formation of gaps that are too often larger than a typical length
of a molecule.
Several feedback-controlled algorithms have been developed to prevent this
from happening95,97,98,99
. All of them try to dissipate a constant critical power at the
junctions while breaking in order to minimize thermal runaway. With these methods,
nanoconstrictions may be narrowed to a target conductance with high reproducibility,
often within 10%, provided the target conductance is greater than 2-3G0. Control of
conductance below this level is apparently not trivial. Transmission electron
microscopy imaging of gap formation reveal sudden (within less than 50msec) large
deformation of junctions once they become transgranular, suggesting that EM no
longer dominates the very last moment of breaking105
. The rate of such a process can
be estimated in the following way: the frequency of diffusion steps of metal atoms at
temperature T can be written as νexp(-Ea/kBT), where Ea is an activation energy
barrier for a diffusing step, and ν is an effective vibrational frequency. For self
48
diffusion on metal surfaces, the prefactor ν is in the order of 1012
s-1
, which implies
that a process with an energy barrier of ~0.5eV106
occurs within ~l ms at room
temperature. It is clear then that devising a feedback circuit to halt such a (thermally
activated) processes is very challenging. Thus currently, reproducible and robust
fabrication of MQPCs by EM with conductance in the range of 1-4G0 remains a
formidable task.
We developed a new method to fabricate MQPCs. The advantages of our
approach are depicted in Figure 25, which shows the conductance behavior of a
typical MQPC as a function of time. Each of the four zones in this Figure reveals a
different advantage of the fabrication approach and of the resulting contacts. Zone I
shows the formation of a gap in a junction using EM. This step is performed without
any feedback control to avoid thermal runaway. Zone II shows tunneling current
across the gap until a „jump to contact‟ process takes place and a MQPC with
conductance of G0 is established. The „jump to contact‟ process is induced by an
appropriate bias voltage (to be discussed). The fluctuations observed below G0 are
discussed later in the text. Zones I and II show that our approach has the capability to
break and make MQPCs. Such a capability is important since conductance
quantization is revealed by statistical averaging, i.e., by repeated cycles of breaking
and making of MQPCs. With STM and MCBJ, these cycles are induced by
mechanical movement of the metal leads. Here it is induced by voltage biases (electric
force), allowing gathering of the same statistics from fully anchored junctions. Zone
III shows that the formed MQPC is very stable maintaining its conductance over tens
of minutes at room temperature. The transition to a higher conductance, in this case to
2G0, which marks the beginning of zone IV, is not arbitrary and the necessary
conditions to deterministically jump between discrete conductance values can be
quantitatively rationalized. The stability of the formed contacts in this zone is also
very high. The necessary experimental conditions to achieve the above capabilities
and advantages and their rationalization are detailed below.
49
Figure 25: Conductance (in units of G0) as a function of time for a typical MQPC. Zone I: shows
an abrupt decrease in conductance as a gap is formed in a junction (initially at 200G0, not
shown). Zone II: Tunneling current (~10-5
G0) until a ‘jump-to-contact’ to 1G0 process occurs.
Zone III: the MQPC is shown to be highly stable at 1G0 for minutes. Subsequently, a transition to
a higher conductance value is induced by a proper bias. Zone IV: switching between quantize
conductance values is demonstrated. Each transition between conductance values occurs by a
‘jump-to-contact’ process, within a time scale of less than ~100msec.
3.3 Methods and Materials
3.3.1 Device Fabrication
We have devised the structure that is schematically presented in Figure 26.
Junctions are fabricated on oxide covered (100nm) Si substrates by two
lithography/evaporation/lift-off procedures. In the first process a thick Cr/Au
(3/200nm) lead is patterned. In the second step a thin Cr/Au lead (3/15nm) is
patterned, partially overlapping the previous lead. The fabrication ends with a final
step to form contact pads to the thin leads. All steps are followed by thorough
cleaning by solvent rinsing and plasma etching to avoid any contamination of the
junctions. Both “wide” and “thin” junctions with typical widths of ~30m and 0.5m
respectively, have been fabricated and tested. The experimental routine described
below has been successfully applied to 80% of the “wide” junctions and to 40% of the
“thin” junctions. All together more than 150 junctions have been tested in this study.
Figure 26: An optical image of a junction (A), along with a schematic presentation of its structure
(B), emphasizing the different thicknesses of the two metal leads. EM results in a gap at the
interface between the two leads. (C) An SEM picture showing a typical gap. EM induces changes
51
of structure on the thin side while the thick electrode appears to be untouched. The signs in this
picture depict the bias polarity applied to induce EM. Formation of a MQPC takes place by
closing the gap between the thick lead and one of the protrusions of the thin lead: under high
enough bias, adatoms on the former electrode are induced to diffuse towards the high field zone
adjacent to the protrusion in order to nucleate and close the gap.
3.3.2 Measurements Technique
All experiments were performed at room temperature under vacuum with base
pressure of <10-6
Torr (Desert cryogenics probe station). The various transitions
shown in Figure 25 are induced by applying relatively high voltages on the junctions,
where Joule heating is expected to affect the measured conductance. Thus all
experiments are performed by applying voltages in a square wave pattern comprises
of 200msec of a certain bias, V, followed by 200msec of 30mV in order to determine
the resulting conductivity. The latter values are the reported and plotted conductance
values in all graphs. The duration of voltage pulses is determined by the time response
of the source measuring units that were used (Kithley 2430, 236).
3.4 Results and Discussion
Previous images of gaps formed by EM reveal in many cases rather chaotic
structures, with large changes in the structure of the leads and variations in gaps
position103,105
. The structure of the junctions makes the interface between the two Au
leads a flux divergence plane of diffusion, forcing the EM gap to be formed at the
interface according to a reasoning that can be described by the following equation of
continuity (in one dimension)93
:
x
T
RT
jeCDZ
TRT
jeCDZ
xt
C
tconsxtconsT
tan
*
tan
*
)1.4.3(
where C is the concentration of metal atoms, D is their diffusion coefficient, Z* is the
effective valance of the metal ions, e is the charge of an electron, is resistivity, j is
current density, R is the gas constant and T is the temperature.
According to equation 3.4.1, EM is expected to be more effective at the thinner
lead due to the higher current density that it carries and its higher temperature under
bias due to its larger resistance. Thus, when current is forced through the junction
51
making the thicker lead a cathode, i.e., forcing electrons to move towards the thinner
lead, voids (flowing in opposite direction to the atoms and electrons) start to
accumulate at the interface since they approach from the thinner lead much faster than
they are dissipated through the thicker lead. Eventually, a large void is formed
resulting in a gap between the leads. Figure 27 shows a sudden drop of conductance
as a gap is formed in a “wide” junction. The potential bias that was used to drive the
EM process is also shown. No feedback protocol is applied to minimize the voltage
drop on the gap upon its formation. Inspection of the junctions after this step by
scanning electron microscope (see Figure 26) reveal two important facts: (i) As
expected EM affects mainly the thinner side of the junction, while the thicker lead
seems to be smooth and „untouched‟. (ii) With no attempt to avoid thermal runaway,
the formed gaps appear to be large with few protrusions of the thinner lead which
form gaps of less than 10nm with the thicker lead. These protrusions are argued to be
essential for the ability to form MQPCs. The lack of feedback is argued to be the
origin of lower yield in “thin” versus “thick” junctions. Thermal runaway in the
former junctions causes more events of catastrophic formation of gaps that cannot be
closed by the processes outlined below.
Formation of a MQPC with conductance of 1G0 is shown in Figure 27. The
polarity of the bias potential necessary to induce this process is opposite to the EM
bias. This observation is general to all formed MQPCs, and is believed to be a direct
outcome of the non-symmetrical structure of the gaps (see Figure 26) and of the
process by which MQPCs are formed.
Considering the value of the applied bias and assuming an initial average gap of
~50Å, we rule out the possibility that formation of MQPCs is a result of field induced
emission or metal deformation, since for Au a threshold field of >1.0V/Å is needed
for both mechanisms107,108,109,110
. While these processes could be effective, perhaps
even dominate when the gap is small, it is conceivable that initiation of contact
formation is by directed diffusion of Au adatoms towards a zone of high field between
a protrusion and the thick lead (see). There, as the density of migrating atoms
increases, the probability of nucleation and cluster growth also increases, until a
growing metal mound is closing the gap111,112,113,114
. Such a mechanism has been
suggested in the past to explain formation of metal structures induced by STM
tips107,115
.
52
Figure 27: Behavior of conductance (black rectangulars) as a result of applied voltage (red line)
as a function of time. After applying increasing voltages on the junction up to approximately -4V,
the junction is broken by EM to form a gap. The formation is manifested as a sudden drop of
conductance from ~200G0 (not shown) to ~10-4
G0. The polarity of the bias is then reversed to
induce directed diffusion of Au adatoms on the thick lead to close the gap (see text for details).
Once some fluctuations in conductance are observed the applied bias is lowered below ~3V to
avoid excessive heating in the MQPC once it is formed (see text for details). Once the MQPC is
formed the bias is maintained at 30mV and the contact appears to be stable with conductance of
1G0 at room temperature.
It is possible to examine main attributes of the process by a simple model
describing the initiation of gap closing, i.e., when the deformation of the gap is
negligible. To a good approximation the field of a protrusion varies as a Lorentzian
function of the form107
:
12
1)()2.4.3( 2
w
rd
VrE
where V is the applied voltage, d is the size of the gap, r is the radius from the center
of symmetry, and w is the Lorentzian width parameter (taken here as 100nm).
Without any bias, the diffusion of adatoms on the metals‟ surface is a random walk
process through small diffusion potential barriers Ud115
. However, since these atoms
have non-negligible atom polarizability, α, in the presence of an inhomogeneous
electric field, E, the diffusion barriers are changed according to:
2
2
1)3.4.3( EUU deff
The resulting net drift velocity under these conditions is described by116
:
Tk
LE
L
D
B
E
2sinh
2)4.4.3( 0
53
where DE=0 is the surface diffusion constant in the absence of a field, L is the jump
length, is the lateral field gradient, kB is the Boltzmann constant, and T is the
temperature.
Figure 28 depicts the distribution of field and drift velocity as a function of
lateral distance for d = 50Å, and V=3V. On the surface (of the thick lead) just
opposite the protrusion center, the drift velocity decreases sharply to zero, marking
the surface area where a metal mound would eventually grow.
Importantly, according to the model, the electric field drops to zero within
~100nm from the protrusion center. Beyond this radius, the drift velocity nullifies.
Assuming adatom surface density of 1012
cm-2
it is evident that within the above
radius the number of available adatoms is not sufficient to form a large enough cluster
to bridge, in this case, a 50Å gap. It is therefore argued that the rate of gap closing is
determined not by the drift velocity but by the rate of diffusing adatoms on the surface
towards the depleted zone adjacent to the protrusion.
According to the model, higher fields should enhance the rate of MQPC
formation. However, the applied voltage to drive this process cannot be arbitrarily
high. To estimate the limit we recall that under high tunneling flux, e.g., short gaps
just prior to contact formation, substantial heating of the leads can take place by
coupling of the tunneling electrons to phonons or by excitation of electron-hole
pairs117
.
Figure 28: Field (black line), and drift velocity (red rectangular) distributions in close proximity
to a protrusion located 50Å from the other lead, under a bias voltage of 3V. A ‘well’ of drift
velocity is located at the center. Here drifting adatoms accumulate to form a mound that bridges
54
the gap. The parameters taken for the calculations are: α = 2.510-24
cm3, D = 3.610
-5 cm
2/sec,
and L = 3Å.
The latter mechanism can be shown to be more effective, and can be analyzed
by assuming that a steady state is reached at the interface, by balancing the energy
deposited into the leads (in regions of radius r~5Å), with the flux of electron-hole
pairs that dissipates heat away from the gap. The increase in temperature can then be
estimated to be117
:
rK
fPT
2~)4.4.3(
where f is the fraction of dissipated power P (=IV), and K is the thermal conductivity
of the leads (KAu=320 W m-1
K-1
). It is important to note that the small contact region
reduces the bulk thermal conductivity K by a factor of r/λ, λ being the bulk mean free
path for electrons (λ~500Å).
The fraction of energy f can be calculated according to117
:
FP
F
k
dkVf
4
)2ln()5.4.3(
2
where kF is the wave vector at the Fermi level (~0.35Å-1
) d is the distance between the
leads (~5Å), ωp is the plasma frequency (~12eV), and is in the order of the metal‟s
work function (~5eV).
Former studies suggest that the temperature of junctions under EM can be in the order
of 450K99,102
. Assuming that thermal runaway takes place at this temperature, we set
ΔT in equation 3.4.4 to be 150K. Assuming G = 0.1G0 (R~105) results in a bias limit
of ~3V, which is set as the limit to the applied bias at this step of the process (see
Figure 27).
Once the gap is closed and an MQPC is formed, the nature of transport is
ballistic, where the length of the constriction is much smaller than the mean free path
of the electrons, so that electrons „fly‟ through the contact almost without scattering.
It may be expected that under these conditions most of the energy of the electrons will
be dissipated not in the contact itself but in the adjoining electrodes where most of the
electron scattering takes place. Since the electrodes are macroscopic and act as heat
reservoirs, it is not at all obvious why there should be any appreciable rise in
temperature anywhere in the system in the presence of the electrical current.
However, substantial local heating can still arise because of the large current density,
and thus power per atom, in the nanojunction compared to the bulk. High bias values
results in a small probability for inelastic scattering of the electrons to phonons in the
55
point contact itself. Taking into account the heat conduction from a nanocontact to
bulk electrodes, Todorov derived the following formula for the effective contact
temperature, Teff:
444)6.4.3( Voeff TTT
where T0 is the ambient temperature and TV is the bias dependent temperature. For
vanishing bias voltages the effective temperature is considered to be the bath
temperature T0. In high bias regime, Teff should rise due to a voltage-driven term, TV.
Thermal conductance to the leads limits the heating considerably and TV is obtained
by the expression with L and V being the contact length and bias
respectively82
.
For our MQPC junctions 103
, and ,
which results in temperature rise in MQPC itself in the order of 60K due to cold
ballistic electrons, which is substantial only in liquid helium temperatures.
Control over the resulting conductance value after a „jump to contact‟ process
appears to be difficult. However, the formed MQPCs appear to be highly stable.
MQPCs with conductance, for example, of 1G0 were found to be stable for many
hours. This stability opens the possibility to deterministically control the conductance
of MQPCs by gently manipulating their configurations using controlled EM. The
issue of EM in ballistic contacts has previously been discussed118,119,120
.
The driving force for EM has two components: the direct field force and the
electron-wind force118,119,120
that is usually dominating. The wind force acting on the
contact can be written as:
FW meJaF )/(2)7.4.3( 2
Where J is the current density, a is the atomic distance, m is the electron mass,
and νF is the Fermi velocity. A constant η depends on the scattering geometry, and
takes values between 0 (forward scattering) and 1 (back scattering). The needed
breaking force of a single-atom contact of gold is 1.5nN121
. Taking νF = 1.410-6
m/s118
gives Ja2 = 9.410
-5A. Since when one gold atom bridges the gap between the
leads, the resistance of the junction is 12.9KΩ, the needed potential bias to drive
electromigration is ~1.2V.
The importance of this limiting bias value is demonstrated in Figure 29 which
depicts switching between quantized conductance values in two different MQPCs. In
both cases a sequence of 1,2,3,4,3,2,1G0 is established. However, Figure 29A shows
56
that when the voltage pulses are not limited in their absolute values some switching
can take place to high conductance values leading to chaotic behavior in the needed
applied biases in order to maintain the sequence. In Figure 29B, the applied voltages
are ~1.2V, leading to switching without abrupt jumps to large (high conductance)
contacts.
Apparently the polarity of the pulses is important as well. In general, if for a
certain MQPC one polarity increases its conductance, the reverse polarity will
decrease it. This is clearly revealed in the two plots of this Figure. The reasons for this
are currently unclear, and will be resolved in future studies. The high stability of the
contacts is once again evident, demonstrated in this case for conductance values larger
than 1G0.
Figure 29: Switching between quantized conductance values in two MQPCs (black line), forming
in both cases a sequence of 1,2,3,4,3,2,1 in units of G0. The voltage bias applied on the junctions to
induce switching is plotted in red. Comparing between the two junctions, it can be seen that in A
when the absolute value of the applied voltage is higher than the voltage limit calculated by
equation 6, more chaotic behavior is observed, with occasion instantaneous jumps of conductance
to values of more than 10G0. In B the applied bias is not more than ~1.2V, maintaining ordered
switching between quantized conductance values. The polarity of the applied voltage determines
if the conductance increases or decreases. Positive bias refers to the thin metal lead as the
cathode.
We now reexamine the behavior in zone II of Figure 25, where fluctuations in
the conductance are observed until a contact of 1G0 is formed. These fluctuations
occur since the applied potential needed to close the gap is much higher than the
approximate voltage limit calculated above to preserve formed contacts. As a result,
rapid (with respect to the sampling rate) processes of MQPCs formation and breakage
take place until at some point a stable enough contact is formed which gives enough
time for the feedback loop to respond and ramp the bias down to 30mV.
57
MQPC formation by the above mechanism is accompanied by negligible EM.
This is mainly because of bias polarity. Contact is established while electrons are
flowing from the thin to the thick lead. This current is accompanied by only negligible
accumulation of voids at the contact since the voids are coming from the thick (small
flux) lead and dissipate away from it through the thin (high flux) lead. The unique
capability of our method to break and make MQPCs enables to explore conductance
quantization, which manifests itself only after statistical averaging122,123,124,125,126,127
.
Averaging is needed mainly because the minimum cross section of the contact is not
the only ingredient that controls the conductance but also the geometry of the
narrowest part and disorder. In practice, contacts with different radii can have similar
values of conductance.
Importantly, the statistical averaging presented here is different than reported
statistics in literature with MCBJs. In these devices, quantization is revealed as
transient (few tens of data points) conductance plateaus while (mechanically)
breaking the contacts. These plateaus result in a higher weight of integer
multiplications of G0 in the final histograms. Here, since mechanical manipulation of
the contacts is not possible, an open-close sequence is driven by appropriate voltages;
few μsec (typically 10) of a breaking voltage (~2V) followed by a closing-gap-voltage
(~-2V with a same duration). Each of these voltages is followed by a measurement at
30mV in order to establish the conductance. This sequence is repeated many times
and the conductance values of the formed contacts are reported in the histogram.
The results of this procedure are demonstrated in Figure 30 which plots
conductance histograms (each based on several thousands of break-and-make cycles)
from two typical junctions. The first junction has a stable 8G0 contact configuration in
addition to peaks at G0, 2G0 and (although less pronounced) also at 3G0. The second
junction appears to adopt only 1-3G0 contact configurations.
58
Figure 30: Conductance histograms of two MQPCs, based on break-and-make cycles. The top
junction reveals in addition to pronounce conductance peaks at 1G0 and 2G0 also a stable
structure with characteristic conductance of 8G0. This demonstrates that deterministic control
over the configuration at the end of a jump-to-contact process is difficult. The lower panel
reveals a different MQPC with conductance quantization at 1G0, 2G0 and 3G0.
3.5 Conclusions
A novel route to fabricate MQPCs has been described in the third section. The
proposed method enables to deterministically form MQPCs with integer conductance
values between G0 to 4G0. It allows reconfiguration of formed MQPCs to switch
between conductance values within this range. The resulting MQPCs appear to be
highly stable at room temperature. Since the formed MQPCs are patterned on Si
substrates, the fabrication method essentially forms a route to embed MQPCs with
deterministic position and conductance within nano-scale circuits and architectures
using current mass-production technologies. Additionally, since degraded MQPCs can
be reformed, one can envision devices with a fault-correcting algorithm that “heals”
damaged MQPCs using the above capability128
. We estimated the electronic
temperature rise prior to contact formation to be ~150K and after, when the bridging
atom is stressed under current flow, to be ~60K.
The End
59
Appendix A- Preparation of Graphene flakes
In order to obtain graphene flakes on Si/SiO2 substrates we tried different
methods and directions: we made mesas in oxygen plasma of various areas between
2-20m and about 5m in height in HOPG, cleaved them off with microscope glass
covered with photoresist, then repeatedly peeled flakes with scotch tape until thin
flakes left attached to the photoresist, afterwards, we immersed this in pettry dish
filled with acetone and tried to fish flakes with Si pieces; we ultrasonicated graphite in
different solvents (dichlorobenzene, benzene, n-hexane, NMP) and dispersed drops
from the solutions on Si substrate; we built an instrument for arc discharge graphene
flakes from graphite on Si substrate with voltages pulses of few kilovolts; we tried to
capture them from a solution drop using dielectrophoresis with electrodes array
patterned on Silicon substrate, similar to work done by Vijayaraghavan et al129
with
dispersion of carbon nanotubes; we tried stamping graphene flakes from HOPG with
home made pressing machine; we tried graphite cleavage in liquid nitrogen. None of
the experiments above showed exceptionally high yield of production or any
advantage.
At the end we used a scotch tape approach directly on the graphite material to
extract single layer graphene flakes in very crude way130
. A small piece of a graphite
(HOPG from SPI131
, natural grahite flakes from Graphitede, or Kish graphite from
Graphitede) is placed on a scotch tape (preferable is the blue sticky tape used in the
clean room because it doesn‟t leave a lot of stains on the Si substrate when the
graphite is transferred) and is peeled off with fresh pieces of tape typically ten times
until thin light gray colored graphite pieces are left on the original piece of tap (A
short clip of principally the same procedure is on Youtube132
) . Then, the piece of
tape with the thin flakes on it is attached to Si/SiO2 piece (10X10mm2 with predefined
Alignment marks separated 500m)133
, and rubbed with plastic tweezers for few
minutes. After careful peeling of the tape a variety of flakes are left attached to the
surface. Among the pieces of graphite, debris and thick flakes left on the surface we
always found single layer flakes. Their preliminary identification amid thicker flakes
and other residue was done in an optical microscope (Olympus M52).
We used Optical Microscope which has an optical camera connected to it. The
video feed from the camera can be a viewed on a computer screen. The sample is
scanned slowly to see whether thinner layers are present. The snapshots of the spots
are taken. Snapshots of surrounding area and features are also taken with all
61
objectives (X10, X20, X50, and X100) to identify these spots later, Figure A1. The
Alignment mark adjacent to the flake is counted relative to a mark labeled in one of
the Si piece corners.
The flakes become visible on top of an oxidized Si wafer, because even a
monolayer adds up sufficiently to the optical path of reflected light so that the
interference color changes with respect to the one of an empty substrate (phase
contrast)134
. Their further analysis was done by atomic force microscopy (AFM).
We used a Veeco Dimension 3100 Atomic Force Microscope. The image of the
particular spot is taken using the AFM. The image is analyzed using the Nanoscope
software. The differential height of the layers is measured by section analysis. The
differential height between two successive graphite layers should be 0.35nm and the
differential height between the dioxide substrate and graphene layer can vary up to
0.8nm due to water layer on the chip and van der Waals interaction between silicon
dioxide and graphene layer. another verification that a flake is truly single atomic
layer of carbon is done by Raman Spectroscopy fingerprint24
.
61
Figure A1: optical pictures of graphene relative to predefined alignment marks sparated 500m
with objective X5 (A), X10 (B), X20 (C), X50 (D), X100 (E). The different colour of substrate
between the magnifications is due to the CCD camera connected to the microscope.
1 A. K. Geim and K. S. Novoselov, Nat. Mat. 6, 183 (2005).
2 A. K. Geim and K. S. Novoselov, Nat. Mat. 6, 183 (2005).
62
3 Datta, S., "Quantum transport : atom to transistor", Cambridge, UK ; New York : Cambridge
University Press, 2005. 4 Dresselhaus M. S.; Eklund P. C.; Adv. Phys. 49, 705, (2000).
5 Novoselov K. S., Geim A. K., Morozov S. V., Jiang D., Zhang Y., Dubonos S. V., Grigorieva I. V.,
and Firsov A., Science 306, 666 (2004). 6 Novoselov K. S., Geim A. K., Morozov S. V., Jiang D., Katsnelson M. I., Grigorieva I. V., Dubonos
S. V., and Firsov A. A., Nature 438, 197, (2005). 7 Barnard, R. D., "Thermoelectricity in Metals and Alloys", Taylor and Francis LTD, London, 1972.
8 Aleiner I. L.; and Efetov K. B.; Phys. Rev. Lett. 97, 236801 (2006).
9A. Atland, Phys. Rev. Lett. 97, 236802 (2006).
10 N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B. 73, 125411 (2006).
11 L. Fritz, J. Schmalian, M. Müller, and S. Sachdev Phys. Rev. B. 78, 085416 (2008).
12 A. Kashuba, Phys. Rev. B. 78, 085415 (2008).
13 J. H. Bardarson, J. Tworzydo, P. W. Brouwer, and C. W. J. Beenakker, Phys. Rev. Lett. 99, 106801
(2007). 14
K. Nomura, M. Koshino, and S. Ryu, Phys. Rev. Lett. 99, 146806 (2007). 15
Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer,
and P. Kim, Phys. Rev. Lett. 99, 246803 (2007). 16
J.-H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D. Williams & M. Ishigami, Nat. Phys. 4, 377
(2008). 17
Juan F. de; Cortijo A.; and Vozmediano M. A. H.; Phys. Rev. B. 76, 165409 (2007). 18
Brey L.; Palacios J. J.; Phys. Rev. B. 77, 041403(R) (2008). 19
F. Guinea, M. I. Katsnelson, M. A. H. Vozmediano, Phys. Rev. B. 77, 075422 (2008). 20
E. H. Hwang, S. Adam, and S. Das Sarma, Phys. Rev. Lett. 98, 186806 (2007). 21
M. I. Katsnelson, K. S. Novoselov and A. K. Geim, Nature Phys. 2, 620 (2006). 22
Rossi E.; and Das Sarma S.; Phys. Rev. Lett. 101, 166803 (2008). 23
S. Adam, E. H. Hwang, V. M. Galitski, and S. Das Sarma, PNAS 104, 18392 (2007). 24
J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing and A. Yacoby, Nat.
Phys. 4, 144 (2008). 25
V. W. Brar, Y. Zhang, and Y. Yayon, T. Ohta, Appl. Phys. Lett. 91, 122102 (2007). 26
P. Wei, W. Bao,Y. Pu, C. N. Lau, J. Shi, Phys. Rev. Lett. 102, 166808 (2009). 27
Y. M. Zuev, W. Chang, P. Kim, Phys. Rev. Lett. 102, 096807 (2008). 28
A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K.
S. Novoselov, S. Roth, and A. K. Geim, Phys. Rev. Lett. 97, 187401 (2006). 29
E. shapira, "Thermoelectric properties of Nickel nanowires", Thesis, (2007). 30
S. Adam, E. H. Hwang, V. M. Galitski, and S. Das Sarma, PNAS 104, 18392 (2007). 31
V. M. Pereira, J. Nilsson, and A. H. Castro Neto, Phys. Rev. Lett. 99, 166802 (2007). 32
J. Cayssol, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. B. 79, 075428 (2009). 33
V. W. Scarola, and G. D. Mahan, Phys. Rev. B. 66, 205405 (2002). 34
K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stromer, and P. Kim, Phys. Rev. Lett. 101, 096802 (2008). 35
E. H. Hwang, and S. Das Sarma, Phys. Rev. B. 77, 115449 (2008). 36
M. Cutler, and N. F. Mott, Phys. Rev. 181, 1336 (1969). 37
N. M. R. Peres, J. M. B. Lopes dos Santos, and T. Stauber, Phys. Rev. B. 76, 073412 (2007). 38
E. Rossi, and S. Das Sarma, Phys. Rev. Lett. 101, 166803 (2008). 39
Avouris , P.; Chen, Z.; Perebeinos, V. Nature Nanotech. 2, 605, (2007). 40
Chen, J. H.; Jang, C.; Xiao, S.; Ishigami, M.; Fuhrer, M. S. Nature Nanotech. 3, 206, (2008). 41
Du, X.; Skachko, I.; Barker, A.; Andrei, E. Y. Nature Nanotech. 3, 491, (2008). 42
Tan, Y. W.; Zhang, Y.; Bolotin, K.; Zhao, Y.; Adam, S.; Hwang, E. H.; Das Sarma, S.; Stormer, H.
L.; Kim, P. Phys. Rev. Lett. 99, 246803, (2007). 43
Meric, I.; Han, M. Y.; Young, A. F.; Ozyilmaz, B.; Kim, P.; Shepard, K. L. Nature Nanotech. 3, 654,
(2008). 44
Fratini, S.; Guinea, F. Phys. Rev. B. 77, 195415, (2008). 45
Freitag, M.; Steiner, M.; Martin, Y.; Perebeinos, V.; Chen, Z.; Tsang, J. C.; Avouris, P. Nano Lett.
5, 1883, (2009). 46
Dasilva, A. M.; Zou, K.; Jain, J. K.; Zhu, J. Phys. Rev. Lett. 104, 236601, (2010). 47
Perebeinos, V.; Avouris, Phys. Rev. B. 81, 195442, (2010). 48
Rossi, E.; Adam, S.; Das Sarma, S. Phys. Rev. B. 79, 245423, (2009). 49
Kampfrath, T.; Perfetti, L.; Schapper, F.; Frischkorn, C.; Wolf, M. Phys. Rev. Lett. 95, 187403,
(2005). 50
Pisana, S.; Lazzeri, M.; Casirahgi, C.; Novoselov, K. S.; Geim, A. K.; Ferrari, A. C.; Mauri, F.
Nature Mater. 6, 198, (2007). 51
Ouyang, Y.; Wang, X.; Dai, H.; Guo, J. Appl. Phys. Lett. 92, 243124, (2008). 52
Tse, W. K.; Hwang, E. H.; Das Sarma, S. Appl. Phys. Lett. 93, 23128, (2008).
63
53
Barreiro, A.; Lazzeri, M.; Moser, J.; Mauri, F.; Bachtold, A. Phys. Rev. Lett. 103, 76601, (2009). 54
Chae, D. H.; Krauss, B.; von Klitzing, K.; Smet, J. H. Nano Lett. 10, 466, (2010). 55
Chauhan, J.; Guo, J. Appl. Phys. Lett. 95, 23120, (2009) 56
Butscher, S.; Milde, F.; Hirtschultz, M.; Malic, E.; Knorr, A. Appl. Phys. Lett. 91, 203103 (2007). 57
Das, A.; Pisana, S.; Chakraborty, B.; Piscanec, S.; Saha, S. K., Waghmare, U. V.; Novoselov, K. S.;
Krishnamurthy, H. R.; Geim, A. K.; Ferrari, A. C.; Sood, A. K. Nature Nanotech. 3, 210, (2008). 58
Saito, R.; Jorio, A.; Souza Filho, A. G.; Dresselhaus, G.; Dresselhaus, M. S.; Pimenta, M. A. Phys.
Rev. Lett. 88, 27401, (2002). 59
Piscanec, S.; Lazzeri, M.; Mauri, F.; Ferrari, A. C.; Robertson, J. Phys. Rev. Lett. 93, 185503, (2004). 60
Bonini, N.; Lazzeri, M.; Marzari, N.; Mauri, F. Phys. Rev. Lett., 99, 176802, (2007). 61
Berciaud, S.; Han, M. Y.; Brus, L. E.; Kim, P.; Heinz T. F. Phys. Rev. Lett. 104, 227401, (2010). 62
Bushmaker, A. W.; Deshpande, V. V.; Bocrath, M. W.; Cronin, S. B. Nano Lett. 7, 3618, (2007). 63
Oron Carl, M.; Krupke, R. Phys. Rev. Lett. 100, 127401, (2008). 64
Ioffe, Z.; Shamai, T.; Ophir, A.; Noy, G.; Yutsis, I.; Kfir, K.; Cheshnovsky, O.; Selzer, Y. Nature
Nanotech. 3, 727 (2008). 65
Calizo, I.; Balandin, A. A.; Miao, F.; Bao, W.; Lau, C. N. Appl. Phys. Lett. 91, 71913, (2007). 66
Adam, S.; Hwang, E. H.; Galitski, V. M.; Das Sarma, A. Proc. Nat. Ac. Sci. 104, 18392, (2007). 67
Варгафтика Н. Б., "Теплофизические свойства веществ", ГЭИ (Государственное
Энергетическое Издательство) Москваб 1956. Thermo-physical properties of substances,
Vargaftik N.B., State Energetic Publishing, Moscow, 1956. 68
Luikov A. V., "Heat and Mass transfer", Mir publishers, Moscow 1980; Chapman A. J., "Heat
transfer", Macmillan publishing, Rice university, 1974. 69
Nair R. R., Blake P., Grigorenko A. N., Novoselov K. S., Booth T. J., Stauber T., Peres N. M., Geim
A. K., Science 320, 1308 (2008) 70
Hwang, E.; Das Sarma, S. Phys. Rev. B. 75, 205418 (2007). 71
Yao, Z.; Kane, C. L.; Dekker, C. Phys. Rev. Lett. 84, 2941 (2000). 72
Park, J. Y.; Rosenblatt, S.; Yaish, Y.; Sazonova, V.; Ustunel, H.; Braig, S.; Arias, T. A.; Brouwer, P.
W.; Mceuen, P. L. Nano Lett. 4, 517, (2004). 73
Mathias Steiner, Marcus Freitag, Vasili Perebeinos, James C. Tsang, Joshua P. Small1,
Megumi Kinoshita, Dongning Yuan, Jie Liu and Phaedon Avouris, Nat. Nano. 4, 320 (2009). 74
A. C. Ferrari, J. C. Meyer, V. Scardaci1, C. Casiraghi1, M. Lazzeri, F. Mauri, S. Piscanec1, D.
Jiang, K. S. Novoselov, S. Roth, and A. K. Geim, Phys. Rev. Lett, 97 187401, (2005). 75
M. Freitag, M. Steiner, Y. Martin, V. Perebeinos, Z. Chen, J. C. Tsang, and P. Avouris Nano Lett. 9
, 1883 (2009). 76
Abdula, D.; Ozel, T.; Kang, K.; Cahill D. G.; Shim, M. J. Phys. Chem. C. 112, 20131, (2008). 77
Kuzmany, H. Solid State Spectroscopy; Springer-Verlag: Berlin, (1998). 78
Lazzeri, M.; Mauri, F. Phys. Rev. Lett. 97, 266407, (2006). 79
M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. B 73, 155426 (2006). 80
M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. Lett. 95, 236802
(2005). 81
Jun, Y.; Zhang, Y.; Goler, S.; Kim, P.; Pinczuk, A. Solid Stat. Comm. 143, 39, (2007). 82
Todorov T N, Phil Mag B. 77 965 (1998). 83
Sharvin Yu. V., Sov. Phys.- JETP, 21, 655 (1965). 84
Chen Y. C., Zwolak M., and Di ventra M., Nano lett 3, 1691 (2003) 85
Agraït , N.; Levi Yeyati, A.; van Ruitenbeek J., Phys. Rep. 377, 81 (2003). 86
Meszaros, G.; Kronholz, S.; Karthäuser, S.; Mayer, D.; Wandlowski T. Appl. Phys. A. 87, 569
(2007). 87
Terabe, K.; Hasegawa, T.; Nakayama, T.; Aono M. Nature 433, 47 (2005). 88
Sokolov, A.; Zhang, C.; Tysmbal, E. Y.; Redepenning, J.; Doudin, B. Nature Nanotechnology 2, 171
(2007). 89
Costa-Krämer, J. L.; Garcia, N.; Garcia-Mochales, P.; Serena P. A.; Marques, M. I.; Correia A. Phys.
Rev. B. 55, 5416 (1997). 90
Abellán J., Chicon. R.; Arenas, A. Surf. Sci., 418, 493 (1998). 91
Abellán, J.; Arenas, A.; Chicon, R.; Reyes, F. Surf. Sci. Lett. 372, L315 (1997). 92
Smit, R. H. M.; Untiedt, C.; van Ruitenbeek, J. M.; Nanotechnology, 15, S472, (2004). 93
Tsutsui, M.; Shoji, K.; Taniguchi, M.; Kawai, T. Nano. Lett., 8, 345 (2008). 94
Ho, P.S.; Kwok, T. Rep. Prog. Phys. 52, 301 (1989). 95
Park, H.; Andrew, K.; Lim, L.; Alivisatos, P.; Park, J.; McEuen P. L. Appl. Phys. Lett. 75, 301
(1999). 96
Wu, Z. M.; Steinacher, M.; Huber, R.; Calame, M.; van der Molen, S. J.; Schönenberger C. Appl.
Phys. Lett. 91, 53118 (2007). 97
Johnston, D. E.; Strachan, D. R.; Charlie Johnson, A. T. Nano Lett. 7, 2774 (2007).
64
98
Strachan, D. R.; Smith, D. E., Johnson, D. E.; Park, T. H., Therien, M. J.; Bonnell, D. A.; Johnston,
A. T. Appl. Phys. Lett. 86, 43109 (2005). 99
Trouwborst, M. L.; van der Molen, S. J.; van Wees, B. J. J. Appl. Phys. 99, 114316 (2006). 100
Ramachandran, G.; Edelstein M. D.; Blackburn, D. L.; Suehle, J. S.; Vogel, E. M.; Richter C. A.
Nanotechnology 16, 1294 (2005). 101
Sordan, R.; Balasubramanian, K.; Burghard, M.; Kern, K. Appl. Phys. Lett. 2005, 87, 13106. 102
Esen, G.; Fuhrer, M. S.; Appl. Phys. Lett. 87, 263101 (2005). 103
Taychatanapat, T.; Bolotin K. L.; Kuemmeth, F.; Ralph, D. C. Nano Lett. 7, 652 (2007). 104
Yang, C. S.; Zhang, C.; Redepenning, J., Doudin, B. J. Mag. Mag. Mat. 286, 186, (2005). 105
Heersche, H. B.; Lientschnig, G.; O‟Neill, K.; van der Zant, H. S. J.; Zandbergen, H. W. Appl. Phys.
Lett. 91, 72107, (2007). 106
Sørensen, M. R.; Jacobsen, K. W.; Jơnsson, H. Phys. Rev. Lett. 77, 5067, (1996). 107
Mayer, T. M.; Houston, J. E.; Franklin, G. E.; Erchak, A. A.; Michalske T. A. J. Appl. Phys. 85,
8170, (1999). 108
Mamin H. J.; Guethner P. H.; Rugar D. Phys. Rev. Lett. 65, 2418, (1990). 109
Csonka, Sz.; Halbritter, A.; Mihaly, G.; jurdik, E.; Shklyarevskii, O. J.; Speller, S.; van Kempen, H.
J. Appl. Phys. 96, 6169, (2004). 110
Hansen, O.; Ravnkilde, J. T.; Quaade U.; Stokbro, K.; Grey F. Phys. Rev. Lett. 81, 5572, (1998). 111
Kellogg, G. L. Phys. Rev. Lett. 70, 1631, (1993). 112
Tsong, T. T.; Kellogg, G. Phys. Rev. B. 12, 1343, (1975). 113
Anaya, A.; Korotkov, A. L.; Bowman, M.; Waddell, J.; Davidovic D. J. Appl. Phys. 93, 3501,
(2003). 114
Bowman, M.; Anaya, A.; Korotkov, A. L.; Davidovic D. Phys. Rev. B. 69, 205405, (2004). 115
Mendez, J.; Gomez-Herrero, J.; Pascual, J. I.; Saenz, J. J.; Soler, J. M.; Baro, A. M. J. Vac. Sci.
Tech. B. 14, 1145, (1996). 116
Tsong, T. T.; Walko, R. J. Phys. Stat. Sol. A. 12, 111, (1972). 117
Flores, F.; Echenique P. M.; Ritchie, R. H. Phys. Rev. B. 34, 2899, (1986). 118
Yasuda, H; Sakai, A. Phys. Rev. B. 56, 1069, (1997). 119
Ralls K. S.; Ralph D. C.; Buhrman, R. A. Phys. Rev. B. 40, 11561, (1989). 120
Holweg, P. A. M.; Caro, J.; Verbruggen, A. H.; Radelaar, S. Phys. Rev. B. 45, 9311 (1992). 121
Torres, J. A.; Pascual, J. I., Saenz, J. J. Phys. Rev. B. 49, 16581, (1994). 122
Garcia-Martin, A.; Torres, J. A.; Saenz, J. J. Phys. Rev. B. 54, 13448, (1996). 123
Lopez-ciudad, T.; Garcia-Martin, A.; Caamano, A. J.; Saenz, J. J. Surf. Sci. 440, L887, (1999). 124
Bascones, E.; Gomez-Santos, G.; Saenz, J. J. Phys. Rev. B. 57, 2541, (1998). 125
Bratkovsky, A. M.; Rashkeev, S. N. Phys. Rev. B. 53, 13074, (1996). 126
Bratkovsky, A. M.; Sutton, A. P., Todorov, T. N. Phys. Rev. B. 52, 5036, (1995). 127
Dreher, M.; Pauly, F.; Heurich, J.; Cuevas, J. C.; Scheer, E.; Nielaba, P. Phys. Rev. B. 72, 75435,
(2005). 128
Heat, J. R.; Kuekes, P. J., Snider, G. S.; Williams, R. S. Science 280, 1716, (1998). 129
A. Vijayaraghavan, S. Blatt, D. Weissenberger, M. Oron-Carl,F. Hennrich, D. Gerthsen, H. Hahn,
and R. Krupke, Nano Lett. 7, 1556 (2007). 130
Novoselov K. S., Jiang D., Schedin F., Booth T. J., Khotkevich V. V., Morozov S. V., Geim A. K.,
PNAS 102, 10451 (2005). 131
http://www.2spi.com/catalog/new/hopgsub.php, grade is not important. 132
http://www.youtube.com/watch?v=rphiCdR68TE. 133
The Si/SiO2 chips are cleaned in acetone and IPA for about five minutes in ultrasonic bath each
before use. 134
Blake P., Hill E. W., Castro Neto A. H., Novoselov K. S., Jiang D., Yang R., Booth T. J., Geim A.
K., Appl. Phys. Lett 91, 063124 (2007).
אלקטריות-תרמומדידות
פונון -וצימוד אלקטרון
מולקולריים צמתיםב
"דוקטור לפילוסופיה"חיבור לשם קבלת התואר
מאת
אילן יוציס
אביב-הוגש לסנאט של אוניברסיטת תל
ע"אב תש 0202יולי
עבודה זו נעשתה בהנחייתו של
דוקטור יורם זלצר
אוניברסיטת תל אביב, בבית הספר לכימיה
תקציר
בנוסף החומרשל פיסיקליותמידע חשוב לגבי תכונות אלקטרית מספקות -הולכה תרמומדידות
אלקטריות של -תרמוידות דעל מ בעבודה זו אנו מדווחים. יתאלקטרונ ולכהלמידע המסופק ממדידות ה
חתיכות . כוורתריג ששל אטומי פחמן המסודרים במית אטו -ה חדשכב, מימדי-חומר דוגרפן הוא .גרפן
איכות תשומת לב עצומה בקהילה המדעית בשל ומשכגרפן בודדות התגלו לראשונה לפני זמן קצר ו
דדנו את המוליכות החשמלית בד בבד עם מ, בעבודתנו .גבוהה של הגביש ותכונות אלקטרוניות ייחודיות
ועמדנו על הקשר בין , אלקטרי תחת השפעת מתחי שער וטמפרטורות שונים-מדידת המקדם התרמו
התנהגות את נכונה תמתאר האינ MOTTשל ידועה הכי הנוסחה מתוך המדידות שבוצעו נובע . השניים
דירק מציינת את המפגש בין פס ההולכה נקודת ) סביב נקודת דיראק של גרפן אלקטרי-התרמו מקדםה
של אלקטריות-התרמותכונות העל גם יםמשפיע רציפויות בשריג-זיהומים ואי. (לפס הערכיות של גרפן
אלקטרונים ושלוליות לוליותשמבוסס על נוכחות של פיתוח שלנו הבעזרת . פרטורת החדרבטמגרפן
.נקודת הדירקסביב אלקטרי-המקדם התרמו נכון יותר להתנהגותאנו מספקים תיאור בגרפן יםחור
עלראמאן באמצעות ספקטרוסקופיית ניםפונוו נשאי מטען חשמליאת הקשר בין נוחקרהמשכנו ו
ומיחס העוצמות של אותות יםונונים אופטיפ תאוכלוסיבגידול עוקבים אחר האנו . חשמלי מתח תחתגרפן
התוצאות שהתקבלו מוכיחות כי. הטמפרטורה האפקטיבית שלהםסטוקס אנו מחשבים את -הסטוקס ואנטי
של יםמטען ופונונים אופטיהבין נשאי שצימוד ב התלויוולט 2.0במתחים שמעל המוליכות של גראפן
. שמתחת SiO2-השל שכבת פונונים אופטייםבשל צימוד עם , כפי ששוער בעבר ,ולא גרפן עצמוה
.גראפן ם של התקני מוליכים מבוססיביצועילית גבלה מהותמצביעות על מתוצאות אלו
ודל ניסיוני חדשני לצומת המבוסס על אטום זהב בודדמבאנו מציגים את המדידות שלנו , בנוסף
המרא מודל שלנוה .אלו כתוצאה ממעבר זרם חשמלי דרכם בכדי לחקור את אופי התחממות צמתים
מוליכות הצמתים . תים המכילים אטומים בודדיםניסויית ערכי מוליכות חשמלית בדידים המתאימים לצמ
אנו מעריכים את העלייה . שלנו נשמרת לאורך זמן וניתן להחיותם על ידי פולס חשמלי במידה וניתקו
לאחר שנוצר צומת של ו, K052 -בכ מיגרציה-אלקטרו על ידיצומת הלפני היווצרות געבטמפרטורה ר
.כתוצאה מחימום חשמלי K02 -אטום בודד העלייה בטמפרטורה מוערכת בכ
הסבר . תוצאה זו מפתיעה משום שגודל הצומת קטן בהשוואה למהלך החופשי של האלקטרון נושא הזרם
צומתבהסיכוי לחימום המושרה אלקטרונית , בצמתים שלנו גבוהה שמליהצפיפות זרם בשל כי לכך הוא
.של נשאי הזרם חופשי הממוצעה המהלךאפילו אם קנה המידה הוא קצר יותר מאשר גדל משמעותית
![Electron–Phonon Coupling in Two-Dimensional ...tion driven superconductivity in doped graphene [49] due to the very low electron– phonon coupling in graphene. Indeed, conventional](https://img.pdfslide.net/doc/110x75/613331aadfd10f4dd73aee43/electronaphonon-coupling-in-two-dimensional-tion-driven-superconductivity.jpg)
