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Metals, Semiconductors and Insulators.
2nd half of the course by D.Krizhanovskii ( E16, [email protected])
Topics to cover1. Optical transitions2. Doping3. Electron statistics4. Hall effect5. Cyclotron resonance6. Scattering and mobility7. Plasma oscillations8. Amorphous materials9. Quantum wells
20nm GaAs
InGaAs
Nanostructures.Single quasiparticleQuantum bit=> novel data storage/memory elements, quantum computing, quantum cryptography…..
Graphene- one monolayer of Graphite. Much faster
Carbon-based nanoelectronics.
Nanophotonics, plasmonics.
Integrated optical nanocircuits
etc, etc and etc……
Topic 10: Optical absorption
1. Optical Absorption-onset at band gap
2. Absorption process for direct gap semiconductors.Vertical process.
3. Absorption coefficient proportional to density of states.For a bulk material ~ E1/2.
4. Excitons. Bound electron-hole state.
5. Role of phonons for indirect gap semiconductors. Temperature dependence of absorption spectra.
6. Optical absorption in doped semiconductors (the shift of the absorption edge- the Moss-Burstein effect)
Onset of optical absorption is at ; is the photon energy
Energy band gaps in various materials
GaN > 3 eV
GaInP 2-2.3 eV
GaAs 1.4 eV
Si 1 eV
Ge 0.7 eV
InSb 0.2 eV
gE>ω ω
Optical absorption in semiconductors
Conduction band(C-band)
Valence band(V-band)
Schematic diagramm of electron energy spectrum
gE
Optical absorption in direct gap semiconductors
One photon with energy creates electron-hole pairConservation of energy and momentum:
Electronic transition due to photon absorptionis vertical on energetic schematic diagramm
*22*22 2/2/ hheeg mkmkE ++=ω
heph kkk += m*e,h-effective electron (e) and hole (h) masses;
ke,h –electron and hole wavevectorskph-photon wavevector
ωh
ke at the edge of BZ ~ 1010~a
πm-1 (a is lattice spacing)
610~2
λπ=phk m-1
|||| he kk ≈
Direct transition
Schematic of vertical electronic transition
Example of energy spectrum ofGaAs crystal, which is direct gap material (minima of C and V bands coincide)
Energy dependence of absorption coefficient in direct band gap material
Absorption coefficient is defined as the relative rateof decrease of light intensity along its propagation path
)( ωα h)(ωI
α is proportional to the density of states associated with optical transition at given frequency
)()(2
2
)2(
8)()( 2/1
32
2/3
**
**
3
2
ωωππ
πωω
dEmm
mm
dkkdN g
he
he
−
+
==
)( ωhN
2/1)(~)( gE−ωωα
)(
)()(
ωωωα
dxI
dIh =
The effect of free carriers on optical absorption spectra
The Moss-Burstein effectConsider semiconductor material, where C band is partially filledwith electrons (n-doped material)
Only optical transitions from V-band to empty states in C-band allowed
*22*22 2/2/ hheFgonset mkmkE ++=ωOnset of optical absortption at
Fk - Fermi wavevector
)/1(2/ ***22heeFgonset mmmkE ++= ω
Indirect optical transitions
Schematic of indirect optical transitions
Indirect gap semiconductors
Conduction band minimum not at k=0
Phonon with momentum kphonon andenergy is involved
Momentum and energy conservationkel+kh=kphonon ;
1. Second order process andthus very weak ~104 times weaker
1. Phonon energy smallcompared to Eg
Ω
Ω±= GEω
Indirect gap semiconductors
Ω+= hEh Gω
Low temperature.Phonons are not present and thus absorptionoccurs with emission of phonon
The absorption threshold is at
High temperaturePhonons are present and a phonon can be absorbed alongwith a photon. Threshold energy Ω−= hEh Gω
Excitons
Formation of stable bound e-h bound state by analogy with Hydrogenatom. This is exciton in semiconductors.
The energy levels of bound states (Hydrogenic model) lie below Eg
So far we assumed thatelectrons interact only with crystal lattice.
Consider attractive interaction between electron and hole.
; n=1,2….
; me,h*-electron and hole effective masses
Absorption spectrum revealing excitonic levels
Summary:
1. Optical transition in direct and indirect gap materials.Role ofphonons at high and low T in case of indirect transitions
2. Energy dependence of absorption coefficient. Concept of joint density of states.
3. Moss-Burstein effect in doped matearials
4. Excitonic effects. Energy spectrum of bound e-h state.
Ref. Charles Kittel, Introduction to Solid State Physics, 4th Edition p 364, 614
Previous Lecture on Topic 10, Optical transitions
Direct transitionIndirect transition
Topic11: Doping of Semiconductors. Donors and Acceptors
1. Control of conductivity. Central of microelectronics
2. Analogy with hydrogen atom
3. Derivation of expressions for binding energy and Bohrradius
4. For GaAs EB~5.8 meV, aB~99 A
5 Consistent with approximation of macroscopical dielectricconstant and effective mass
He
n Rm
mE
=
2
* 1
ε Be
n am
mr
=
*
ε
Group IV semiconductors Si, Ge, (C)
Donors- group V P, As, SbAcceptors- group III B, Ga, Al
Group III-V semiconductors GaAs, InSb
Donors –group VI (on As site) S, Se, Tegroup IV (on Ga site) Si
Acceptors- group IV (on As site) C, Sigroup II ( on Ga site) Zn, Be, Cd
Doping of semiconductors by donors and acceptors
Binding energy of electron(hole) on donor (acceptor)
Use analogy with hydrogen atom
Hydrogen Rydberg
Electron Bohr radius in Hydrogen
eVem
R en 6.13
)4(2 20
2
4
==πε
He
een R
m
m
nn
emE ×
−=−= 2
*
220
22
4* 11
)4(2 επεε
me – free electron mass; e-electron charge;
Aem
nr
en 53.04 02
22
== πε(for n=1)
For semiconductors one needs to take into account electron motion in crystal, rather than in vacuum. Need to consider effective electron mass
and dielectric constant
*em
ε
(for n=1)
02*
22
4πεεem
nr
en
=
Donor bound states Donor Bohr radius
For GaAs ee mm 067.0* =
12=ε
meVRm
memE H
e
ee 8.51
)4(2 2
*
20
2
4*
1 −=×
−=−=
επεε
Donor binding energy (n=1)
Donor Bohr radius
Aem
nr
en 984 02*
22
== πεε
The bound state wavefunction of electron on donor extends over many atomic diameters=> approximation of effective mass and dielectric constant is valid
Band gaps at room temperature
Ge 0.66 meVSi 1.1 eVGaAs 1.4 eV
Donor binding energiesGe 10 meVSi 45 meVGaAs 5.7 meV
Acceptor binding energiesGe 11 meVSi 46 meVGaAs 27 meV
Binding energies
1. Very much less than band gap
2. Of order kBT at 300 K
High degree of ionisation at room T => Effective doping allows control of conductivity
Make sure you understand (know)
1. Concepts of donor and acceptor2. Example of donors (acceptors )3. Hydrogen model for the energy levels of electron (hole)on donor (acceptor)4. Why this model works well5. Binding energies in comparison to thermal energies are small
J.R.Hook and H.E.Hall, Second edition P136-138
Topic 12: Semiconductor statistics, Intrinsic and Extrinsic Semiconductors
1. Intrinsic semiconductor. Fermi level close tocentre of gap.
2. Extrinsic conduction, result of doping.
3.Extrinsic, saturation and exhaustion regimes.Variation of carrier concentration with temperature.
4. Compensation. Compensation ratio. Number of ionised impurities at low and high temperature
Fermi distribution
The probability of occupation of a state with energy E is given by Fermi distribution
If there is no doping where is the position ofFermi level EF in band gap?
At final T since populations in bands small, EF must remain close tothe middle of the band gap
EF-Fermi level. By definition
TVF N
FE ,)(
∂∂= ; F- Free energy
Consider the case of T=0
Calculation of electron(hole) density as a function of T
)exp(2
22/3
2
*
Tk
EE
h
Tkmn
B
gFBee
−
= π
Electron density
2/3
2
*22
=
h
TkmN Be
c
π -effective density of states
(Eq. 1)
Intrinsic regime.
Band to band excitation of electrons and holes.Electron ne and hole np concentrations are equal
Here Nc, v –effective densities of states in conduction and valence bands respectively.
Example:At 300 K nintrinsic=2x1019 m-3 (Ge)
= 1x1016 m-3 (Si)
)2
exp(Tk
ENNnn
B
gvcpe
−==
Extrinsic conduction as a resultof doping
Consider material doped only with donors
At very low temperatures DB ETk <<ED- donor binding energy
most donors are not ionised
Fermi Level is close to 2/DgF EEE −≈
)2
exp()( 2/1
Tk
ENNn
B
DDce
−=Precise concentration of electrons
Compensation
For ND>NA,
At high T (kT>>ED) all impurities are ionised:ne = ND-NA
Both donors and acceptors are present
ND,A-concentration of donors (D) or acceptors (A)
At low T (kT<<ED):
ND>NA => n-type semiconductorND<NA=> p-type semiconductor
DgF EEE −≈)exp(
Tk
ENn
B
Dce
−≈
Temperature dependence of carrier numberin doped material
Temperature dependence of Fermi level and carrier number in doped material
Summary
1. Concept of Fermi level.
2. Position of Fermi level in doped and undoped material.
3. Temperature behaviour of Fermi level
4. Derivation of expression for electronconcentration using Fermi distribution
5. Compensation in material doped with donors and acceptors. Electron concentration at low and high T. Number of ionised impurities at low and high T.
J.R.Hook and H.E.Hall, Second edition p139-147
Topic 13: Hall effect
1. Motion of charge carrier in crossed electric and magnetic fields (d.c. ω=0, ω≠0 corresponds to cyclotronresonance )
2. Solve equation of motion
3. Hall coefficient RH=-1/(ne), expressions for mobility, conductivity
4. Hall field exactly balances the Lorentz force
5. Method to determine sign of charge carriers in semiconductor and metals
6. RH negative for e.g. simple alkali metals. Can be positive for divalent metals due to overlapping bands
)()(* BVEeV
dt
Vdm
ee ×+−=+
τ
Geometry of Hall effect
The Lorentz force acting on electrons
The Lorentz force is compensated by electric force
BVe ×−
Consider geometry of Hall bar
Assume that electrons are the main carriers
HEe−BandV are electron velocity and external magnetic field
HE is generated Hall electric field
Motion of charge carriers in constantelectric and magnetic fields
General equation of motion
)()(* BVEeV
dt
Vdm
ee ×+−=+
τ*em Electron effective mass
eτ Electron scattering time
);;( zyx VVVV =
Electron velocity
Calculation of Hall field
0=yV Since current cannot flow out of the bar
Electric field yE (Hall field) has value, since it exactly Counter balances the Loretz force due to B-field
Hall coefficient
Define Hall coefficientBj
ER
x
yH =
Here electric current in X direction (ne-electron concentration )
xe
eexex E
m
eneVnj ⋅=−= *
2τ
( )enmBEen
mEeBR
eexee
exeH
1
/
/*2
*
−=−=ττ
Electron concentration can be extracted from Hall effect
Electron mobility *e
e
m
eτμ =
Conductivity μσ ene=
Measuring both σ and ne we can obtain μ
If both electron and holes are present in the system, it is morecomplicated
Write equations of motion for both electrons and holes
If the mobility of minority carriers is high these carries may determine the sign of the Hall coefficient
( )( )eh
ehH npe
npR
μμμμ
+−=
22
Final answer:Here n, p are electron and hole concentrations
RH negative for e.g. simple alkali metals. Can be
positive for divalent metals due to overlapping bands
Summary
1. Hall experiment is used to determine carrier concentrationand mobility
2. Sign of Hall coefficient is determined by the sign of the main carriers
3. If both electrons and holes are present the mobility of minority carriers may define the sign of Hall
coefficient
J.R.Hook and H.E.Hall, Second edition p152-154
Topic 14 Cyclotron Resonance
1. Method to determine effective mass in semiconductors
2. Electron moves in circular orbits in constant
magnetic field B. The precession angular
frequency *ce
eB
mω =
3. a.c electric field ~ exp( ); ( ~ )cE i tω ω ω perpendicular B results in strong acceleration of electrons. => effect of cyclotron resonance
4. Condition for observation of cyclotron resonance 1cω τ >
τ − electron scattering time.
5. To observe cyclotron resonance one uses low temperature and pure samples to maximise τ , and higher magnetic fields to maximise ωc
6. Quantum mechanical description of cyclotron
resonance. Electron spectrum consists of Landau levels. Transition between Landau levels due to
photon absorption of energy cω
Electron motion in magnetic field B.
Simple treatment
*
* 2
e
e
dm e B
dt
m r e rB
υ υ
ω ω
= − ×
=
υ
-electron velocity r- radius of circular orbit
*em - electron effective mass
Electron moves in circular orbits about B with frequency of angular motion given by
*ce
eB
mω =
Electron motion in magnetic field B||Z and electric field
E
=(Ex , Ey)=(E cos(ωt), E sin(ωt)) in the xy plane.
*( )x x
x ye e
d eE B
dt m
υ υ υτ
+ = − +
*( )y y
y xe e
d eE B
dt m
υ υυ
τ+ = − −
,x yυ υ - Projections of electron velocity on X and Y τ − electron scattering time.
Introduce complex velocity
x yu iυ υ= +
*( exp( ) )
e e
du u eE i t iuB
dt mω
τ+ = − −
Solution u=u0 exp(iωt).
0 * ( / )e c e
ieEu
m iω ω τ=
− −
Cyclotron frequency *ce
eB
mω =
Maximum amplitude of electron velocity*
0 0u u at
~ cω ω
Radiation of frequency ~ cω ω will be absorbed.
Method to determine effective mass m* of either electron or hole.
Quantitative example
*/c eeB mω =
100
2c GHz
ωπ
=,
m*=0.07 me (me-free electron mass)
*
0.247e cmB T
e
ω= =
To observe cyclotron resonance one needs that a particle completes several orbits between collisions
1cω τ >
τ ∼1.5 ∗10−12 sec
The higher magnetic field, the higher the experimental frequency and thus the shorter scattering times that can be tolerated To observe cyclotron resonance one maximises scattering time by using low temperatures and investigating pure samples
It is possible to use cyclotron resonance to determine the sign whether charge carriers are electrons or holes. In the above sign convention for electron
*ce
eB
mω =
is positive=>
the electric field used is σ+ circularly polarised
E
=(Ex , Ey)=(E cos(ωt), E sin(ωt)) **********************************************
For holes *ce
eB
mω = −
is negative, and thus the
radiation is σ− circularly polarised
E
=(Ex , Ey)=(E cos(ωt), -E sin(ωt))
Quantum mechanical treatment of cyclotron resonance. In external magnetic field B||Z electron motion is quantised in xy plane. Quantised Landau levels are formed.
2
*( 1/ 2)
2z
c
kE n
mω= + +
kz – momentum in Z direction
Cyclotron energy *ce
eB
mω =
Cyclotron resonance corresponds to transition from one Landau level to the next one accompanied by
absorption of photon of energy *ce
eB
mω =
Quantised cyclotron orbits in k-space
Cyclotron resonance in Si
Few resonances are observed in Si because dispersion is anysotropic
Summary
1. Cyclotron resonance can be used to determine effective mass and sign of carriers 2. To observe cyclotron resonance scattering time should be long enough to allow electron to complete few circular orbits 3. In quantum mechanical treatment cyclotron resonance corresponds to transition between Landau levels Reference: Hook, Hall, Second Edition, pages 155-159
Topic 15: Mobility and Scattering Processes in Semiconductors. 1. Two dominant scattering mechanisms of electrons
affecting conductivity: scattering with ionised impurities (or missing atoms –vacancies and other structural defects) and scattering with phonons.
2. Mobility *e
e
m
τμ =, τ−scattering time.
Mattheisen’s rule: 1 1 1
( ) ( )ph impT Tτ τ τ= +
,
τph , τimp –scattering times with phonons and impurities, respectively
1 1 1imp phμ μ μ− − −= +
3 / 2 3 / 2~ ( ) ~ ; ~ ( ) ~imp imp ph phT T T Tμ τ μ τ −
for the mobilities determined by ionised impurity and phonon scattering respectively
3. Comparison with metals. In semiconductors impurity scattering is temperature dependent since electron velocity is temperature dependent.
In metals impurity scattering is temperature independent.
Scattering by ionised impurities in semiconductors: Defect (impurity) scattering is temperature dependent, since mean velocity of electrons is temperature dependent. In case of nondegenerate gas electrons obey Boltzman distribution. Average kinetic energy of electrons ( v -electron velocity)
* 2 3
2 2e Bm v k T
=
Scattering by charged impurities significant
when thermal energy ( ~ Bk T ) is of the order of
Coulomb potential (
1~
r )
:
1~Bk T
r
*Scattering cross-sectional area of impurity
22
1~ ~A r
T
*Mean free path of electron 21
~ ~l TA ;
*Electron scattering time
23 / 2
1/ 2( ) ~ ~imp
l TT T
v Tτ =
*Mobility due to impurity scattering
2/3~)(~ TTimpimp τμ
Analogy with Rutherford scattering of alpha particles: mean free path associated with such scattering ~ to the square of electron energy
2~l v => scattering time 3( ) ~ph
lT v
vτ =
Scattering with phonons in semiconductors: Mobility due to phonon scattering
3 / 2~ ( ) ~ph ph T Tμ τ −
Predominant scattering with longitudinal acoustic phonons (LA), which produce series of compressions and dilations and modulate energies of conduction and valence bands locally. The LA phonons which scatter electron must
have wavelengths phλ at least as large as the
electron wavelength eλ :
2 2~
ph e
π πλ λ
Scattering of electron with energy Bk T by phonon involves a major change of its momentum but not its energy, since the energy
of phonon with momentum
2 2~
ph e
π πλ λ is small
compare to Bk T
Free mean path between collisions 1
~lT
=>3/ 2( ) ~ ( ) ~ph ph
lT T T
vμ τ −=
Comparison to metals: In metals at low temperature conductivity and thus mobility is independent of T
At low T scattering with defects/impurities is dominant. The scattering rate is temperature independent since the electron velocity given by Fermi energy is temperature independent.
Summary
1. Two dominant scattering mechanisms of electrons: scattering with ionised impurities (or missing atoms –vacancies and other structural defects) and scattering with phonons.
2. Temperature dependence of the mobilities determined by ionised impurity and phonon scattering respectively in the limit of low and high T
3. Comparison to metals. In metals at low temperature conductivity and thus mobility is independent of T
Reference: J.S. Blackmore, Solid State Physics, Second Edition, pages 330-339
Topic 16: Plasma reflectivity, Plasma Oscillations
1. Collective response of electron gas to electromagnetic field 2. Dielectric constant in a material depends of the frequency of external
radiation ω and wavevector k
3. Oscillation of electron density at plasma angular frequency
0
22
εω
m
nep = occurs in the ultraviolet for typical metals. n-electron
density; m-electron effective mass;ε0-permittivity of free space 4. Dielectric function 22 /1)( ωωωε p−= ; Dispersion of plasmon
frequency as a function of k-vector derived
from222 ),( ckk =ωεω . c-speed of light in free space.
5. For pωω < , ε is real and negative, and hence k is imaginary. Wave is damped in metal, which has a very high reflectivity
6. For pωω > ε is real and positive , and k is real. wave propagates.
7. Colour of metals e.g. lithium, potassium, silver, copper gold 8. Quantised oscillations of plasma: plasmons. Observed in electron
energy loss experiments
Transverse electromagnetic field in a plasma Collective response of electron gas to an electromagnetic field Dielectric function in a metal ),( kωε depends on the wavevector and frequency of external radiation. Plasma frequency determines reflectivity of metal up to high photon energy (~3 eV)
Derivation of ε(ω) in presence of high density of carriers (electrons)
Motion of free electron in electromagnetic field E~exp(iωt)
Frequency dependence of dielectric function
Electromagnetic field in metal.
However colour of some metals arises from interband absorption. For Cu, light in with green and blue wavelengths is absorbed and red light is reflected. The same for Au. When semiconductors are heavily doped the plasma effect become significant
Longitudinal Optical modes in a Plasma. Plasmons
Don’t mix up with transverse modes!!
Plasmons are quanta of longitudinal plasma oscillations
Energy of plasmon phω
To observe, one measures energy loss spectra for electron reflected from metallic films
Summary.
1. Transverse optical modes in a plasma. 2. Reflection and propagation of transverse mode
3. Role of plasma frequency
4. Longitudinal modes in a plasma. Plasmons.
5. Experiments to reveal plasmons
Reference: Charles Kittel, Introduction to Solid State Physics, 4th Edition, p 269-278
Topic 17 Amorphous Semiconductors
1. No long range order, but local bonding still exists
2. Band edges strongly broadened
3. Electrons scattered by disorder-localised states near band edges
4. Defect states in gap due to dangling bonds
5. Dangling bond states can be removed by
hydrogen. Allows material to be doped. Permits applications
6. Growth is fast nonequilibrium process-atoms don’t
have time to reach equilibrium positions
7. Contrast with growth of crystals which is a “slow” equilibrium process
Amorphous semiconductors Local bonding present as in crystalline materials No long range order. Removes translational symmetry. Prepared as thin films by evaporation or sputtering Atom do not have time to achive crystalline order before positions become fixed on surface
Atoms are knocked out of target by high energy ions. Are then deposited on substrate
As a result of lack of translational symmetry momentum selection rule (∆∆∆∆ k=0) is relaxed Absorption edges broadened but bands and band gaps still exists- determined in part by local bonding Electrons strongly scattered by disorder-> localised states near band edges because disorder is very strong In addition there are many unsatisfied dangling bonds � large density of states in the middle of gap
Density of defect states is very high, up to 1025m-3
Compensates donor or acceptor doping and therefore doping is not effective However if amorphous Si is grown from SiH4 , dangling bonds are satisfied by H (“neutralises” defect states) and doping is effective Application – large area of displays, solar cells Very cheap to produce TFT – thin film transistors in laptops + solar cells
Topic 18 Quantum wells
1. Potential wells created by growth of thin layers of crystalline semiconductors
2. Quantum confinement of electron and hole states
3. New bad gap result
4. Many applications as light emitters
Quantum wells (crystalline) Thin layer of crystalline semiconductor of thickness < de Broglie wavelength of electrons (200 A, 50-100 A typical) leads to quantatisation of motion in Z direction. The layer consists of 20-40 atomic layers only In 3D electron kinetic energy
)(2
222
2
zyx kkkm
hE ++=
In case of electron motion in a thin layer z-motion is quantised=>
)()(2
22
2
zEkkm
hE nyx ++=
Motion in XY plane of the thin layer is unaffected
Alterating layer of smaller (GaAs) and larger (Al x Ga1-xAs) band gap materials. Creates potential well for electrons and holes.
Typical Quantum well
Potential well formed for electrons and holes
Eg –energy gap of bulk GaAs material. In case of quantum well above onset of optical absorption occurs at Eg
’, because the band gap is increased due to quantised motion in Z direction
Applications Laser, Light emitting diodes (LED), detectors Design of heterostructures based on quantum wells determines wavelength of emitted light: 1.5 µm for fibre optics communications 0.6 µm for DVDs 0.4 µm for Blue-Ray etc, etc……