Upload
vothuy
View
225
Download
0
Embed Size (px)
Citation preview
This article was downloaded by:[Purdue University][Purdue University]
On: 3 April 2007Access Details: [subscription number 768485394]Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Numerical Heat Transfer, Part B:FundamentalsAn International Journal of Computation andMethodologyPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713723316
The Atomistic Green's Function Method: An EfficientSimulation Approach for Nanoscale Phonon Transport
To cite this Article: , 'The Atomistic Green's Function Method: An Efficient SimulationApproach for Nanoscale Phonon Transport', Numerical Heat Transfer, Part B:Fundamentals, 51:3, 333 - 349To link to this article: DOI: 10.1080/10407790601144755URL: http://dx.doi.org/10.1080/10407790601144755
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf
This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction,re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expresslyforbidden.
The publisher does not give any warranty express or implied or make any representation that the contents will becomplete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should beindependently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with orarising out of the use of this material.
© Taylor and Francis 2007
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
THE ATOMISTIC GREEN’S FUNCTION METHOD: ANEFFICIENT SIMULATION APPROACH FOR NANOSCALEPHONON TRANSPORT
W. Zhang and T. S. FisherSchool of Mechancial Engineering and Birck Nanotechnology Center,Purdue University, West Lafayette, Indiana, USA
N. MingoNASA-Ames Center for Nanotechnology, 229-1, Moffett Field, California, USA
This article presents a general formulation of an atomistic Green’s function (AGF) method.
The atomistic Green’s function approach combines atomic-scale fidelity with asymptotic
treatment of large-scale (bulk) features, such that the method is particularly well suited
to address an emerging class of multiscale transport problems. A detailed mathematical
derivation of the phonon transmission function is provided in terms of Green’s functions
and, using the transmission function, the heat flux integral is written in Landauer form.
Within this theoretical framework, the required inputs to calculate heat flux are equilibrium
atomic locations and an appropriate interatomic potential. Relevant algorithmic and
implementation details are discussed. Several examples including a homogeneous atomic
chain and two heterogeneous atomic chains are included to illustrate the applications of this
methodology.
INTRODUCTION
A growing interest exists in mesoscopic phonon transport, in which devicedimensions becomes comparable to the typical wavelength of a phonon, and thewave nature of phonons becomes prominent. Meanwhile, the shrinking feature sizesof modern electronic and molecular devices are quickly approaching nanometerscales, and phonon transport is often restricted by the heterogeneous boundariesand interfaces embedded in devices. In this quasi-ballistic transport regime, theatomistic Green’s function (AGF) method is efficient at handling interface andboundary scattering. In this article, we present a detailed mathematical derivationand a practical algorithm of the atomistic Green’s function method, as well as exam-ples of the numerical implementation of the method.
Quantized thermal transport has been investigated with a variety of methods ofvarious degrees of complexity. Rego and Kirczenow [1] demonstrated theoreticallythat at low temperatures (<1 K) the thermal conductance of a one-dimensional quan-tum wire is quantized, and the fundamental quantum of thermal conductance is
Received 13 March 2006; accepted 3 November 2006.
Address correspondence to T. S. Fisher, School of Mechancial Engineering, Purdue University,
1205 West State Street, West Lafayette, IN 47907-2088, USA. E-mail: [email protected]
333
Numerical Heat Transfer, Part B, 51: 333–349, 2007
Copyright # Taylor & Francis Group, LLC
ISSN: 1040-7790 print=1521-0626 online
DOI: 10.1080/10407790601144755
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
p2k2BT=3h, regardless of material properties. Similar work on an atomic chain using
a Keldysh formalism was presented by Ozpineci and Ciraci [2]. Hyldgaard [3] appliedthe lattice dynamics method, developed earlier by Pettersson and Mahan [4] andby Stoner and Maris [5], to a one-dimensional lattice model with single-barrierand double-barrier structural configurations. Hyldgaard’s investigation shows theeffect of phonon Fabry-Perot resonances on thermal conductance.
At low temperatures, the phonon distribution is shifted to the long-wavelengthportion of the spectrum; therefore acoustic theory suffices to study transportphenomena. The transmission coefficient for vibrational waves traveling across anabrupt junction between two thin elastic plates can be calculated by the thin-plateelasticity theory described by Cross and Lifshitz [6]. Surface roughness effects
NOMENCLATURE
A matrix for convenience, defined in
Eq. (15)
D one-dimension phonon density of
states, s=m
E energy associated with degrees of
freedom, J
g uncoupled green’s function matrix,
defined in Eqs. (9) and (10)
G green’s function matrix, defined in
Eq. (11)
h Planck constant (¼6.63� 10�34 m2
kg=s)
h reduced Planck constant (¼h=2p)
H harmonic matrix, defined in Eq. (1)
i unitary imaginary number
I identity matrix
J energy flux, W
kB Boltzmann constant
(¼1.38� 10� 23 m2 kg=s2 K)
L bond length, m
M atomic mass, kg
n matrix dimension
N phonon occupation number
S source matrix, defined in Eq. (12)
t time, s
T temperature, K
u vibrational degree of freedom on
displacement, m
u column vector consisting of
vibrational degrees of freedom
U interatomic potential, J
C matrix for convenience, denned in
Eq. (16)
d a small number corresponding to
phonon energy dissipation in contacts
E a quantity defined as x2
r thermal conductance, W=K
R self-energy matrix, defined in Eq. (11)
s matrix representing interactions
among different atom groups
/ complex wave function
U column vector representing vibrational
degrees of freedom in either contact
v column vector representing the change
to the original vector UR after contacts
and the device are connected
w column vector representing vibrational
degrees of freedom in the connected
device
x angular frequency, rad=s
Subscripts and Superscripts
c contact region
cd connection between contact region
and device region
d device region including LD, RD,
and D
D device region not directly bonding with
either contact
l local density of states
LC left contact region
LCB left contact bulk region
LD left Device region
m degree of freedom running index
p matrix row index, or the index of a
degree of freedom
q matrix column index, or the index of a
degree of freedom
R disconnected state
RC right contact region
RCB right Contact bulk region
RD right device region
s submatrix� complex conjugate of a matrixy conjugate transpose of a matrix
334 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
phonon scattering on thermal conductance have been calculated with a full elasticitytheory [7–9] in an attempt to explain why the ratio of thermal conductance to tem-perature decreases at extremely low temperatures [10]. A scattering-matrix methodwith a continuum elastic wave model was developed by Li et al. [11] to evaluatethe thermal conductance of a dielectric quantum wire. This method was laterextended to investigate the effects of discontinuities [12] and defects [13] in crystallattices. The limitations of the acoustic type of approach are its inability to handleexact atomic structures and its neglect of other phonon branches.
Many other theoretical and numerical tools have been developed to simulatemicro=nanoscale phonon transport and include molecular dynamics (MD) [14, 15]and the phonon Boltzmann transport equation (BTE) [16, 17]. Compared to thesemethods, the atomistic Green’s function method is strictly valid at low temperatures.It accounts for boundary and interface scattering efficiently and offers great flexi-bility in handling complicated geometries.
In a typical device-contact setup, contacts (or thermal reservoirs) play signifi-cant roles. Phonon distributions in contacts can significantly change the phonontransport characteristics of the device [18]. A well-designed device-contact geometricconnection for ideal coupling was used in pioneering low-temperature thermal con-ductance measurements by Schwab et al. [10] in which the thermal quantum conduc-tance was experimentally verified. The AGF method uses self-energy matrices torepresent the effect of bulk contacts on the device, thus simplifying the complexitiesof multiscale transport.
Interface and boundary scattering processes are becoming increasingly impor-tant in practical devices. Two primary theories have been employed to explain themechanism of the thermal boundary resistance. One is the acoustic mismatch model(AMM) by Little [19], and the other is the diffuse mismatch model (DMM) bySwartz and Pohl [20]. Both models neglect atomic details of actual interfaces, andthus offer limited accuracy in nanoscale interface resistance predictions [21]. TheAGF method can predict thermal boundary resistance between heterogeneous mate-rials with full consideration of the interfacial atomic structures.
Inspired by the success of the Green’s function method in nanoscale electrontransport simulations [22–24], we have investigated ballistic phonon transport in sev-eral practical configurations using the atomistic Green’s function method [25–27]. Inspite of a few previous publications, we believe that an article dedicated to AGFtheory and algorithm details will facilitate the adoption of this method by a broadercommunity.
In the first part of this article, a mathematical derivation starting from the lat-tice dynamic equation is described, and the phonon transmission function isexpressed in terms of Green’s functions. Relevant details of numerical implemen-tation are discussed in the subsequent section. Several demonstrative atomic chainexamples are presented to show the applicability of the method.
MATHEMATICAL FORMULATION
General Problem Description
The common atomic structure of interest can be generalized as a devicebetween two contacts, as shown in Figure 1. ‘‘Contact1’’ (including atom groups
ATOMISTIC GREEN’S FUNCTION METHOD 335
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
LCB and LC) and ‘‘Contact2’’ (including atom groups RCB and RC) are two semi-infinite thermal reservoirs at constant temperatures T1 and T2, respectively. The‘‘Device’’ includes atom groups LD, D, and RD, and its geometry is arbitrary(e.g., an atomic chain, a nanowire, or a thin film). The types of connections betweenthe device and contacts are arbitrary also (e.g., point contact, limited contact, or pla-nar contact). Atom group LC includes atoms in ‘‘Contact1’’ that bond with‘‘Device’’ atoms. Atoms in group LCB do not have any bonds with ‘‘Device’’ atoms.Therefore, the dynamical properties of these two groups (LCB and LC) will be dif-ferent in a general heterogeneous system. Similar definitions are extended to atomgroups RC and RCB. Atom groups LD and RD include ‘‘Device’’ atoms that bondwith ‘‘Contact1’’ and ‘‘Contact2’’ atoms, respectively. Atoms in group D have nobonds with either contact.
Harmonic Matrix
The AGF method is founded on a harmonic matrix for the system of interest.Prior work [26] has shown that anharmonic scattering can generally be neglected atroom temperature if the characteristic length of the device is less than 20 nm. There-fore, a harmonic matrix can be used to represent interactions among differentdegrees of freedom. The mathematical definition of the harmonic matrix is1
H ¼ fHpqg ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MpMq
p � q2Uqup quq
if p 6¼ q
�P
m 6¼ q
q2Uquq qum
if p ¼ q
8><>: ð1Þ
where up and uq refer to any two atomic vibrational degrees of freedom (i.e., displace-ments), respectively. U represents the total interatomic potential. Mp and Mq areatomic masses associated with degrees of freedom uP and uq, respectively. Thedynamical equation for the system of interest can be written as [28]
ðx2I�HÞ~uu ¼ 0 ð2Þ
1In this article, all summations are explicit, i.e., no Einstein summation rule is used.
Figure 1. Schematic diagram for a general contact-device-contact setup.
336 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
where ~uu is a column vector consisting of vibrational degrees of freedom. We note thatin some complicated situations, the assembled harmonic matrix H can contain com-plex entries, but it is always a Hermitian matrix and thus has only real eigenvalues.
Green’s Function Matrices
In this article, we use matrices s1 and s2 to represent interactions between twodifferent atom groups. A column vector w represents vibrational degrees of freedomin the device, and U, another column vector, represents vibrational degrees of free-dom in either contact. The number of degrees of freedom in the contact approachesinfinity, and we therefore use a sufficiently large number nc to approximate the num-ber of degrees of freedom in the contact. The number of degrees of freedom in thedevice is finite and termed as nd. The number of degrees of freedom in interfaceregions (LC, LD, RC, and RD) is ncd. Based on Eq. (2), the dynamical equationsof the disconnected contacts are
½x2I�H1�UR1 ¼ 0 ð3Þ
½x2I�H2�UR2 ¼ 0 ð4Þ
where H1 and H2 are harmonic matrices of the two contacts. The superscript R refersto the disconnected state. Using v as the change to the original contact vector (UR)after the contact and device are connected, the actual contact vector is U ¼ URþ v.The dynamical equation of the connected contacts and device is then
x2I�H1 �sy1 0�s1 x2I�Hd �s2
0 �sy2 x2I�H2
264
375 UR
1 þ v1
wUR
2 þ v2
8<:
9=; ¼ 0 ð5Þ
where s1 (or s2) is the connection matrix between the left (or right) contact and thedevice. The solutions to Eq. (5) can be written as [24]
v1 ¼ g1sy1 w ð6Þ
v2 ¼ g2sy2 w ð7Þ
w ¼ GS ð8Þ
The matrices used in Eqs. (6), (7), and (8) are defined as
g1 ¼ limd!0½ðx2 þ diÞI�H1��1 (uncoupled Green’s function) ð9Þ
g2 ¼ limd!0½ðx2 þ diÞI�H2��1 (uncoupled Green’s function) ð10Þ
ATOMISTIC GREEN’S FUNCTION METHOD 337
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7 G ¼ x2I�Hd � s1g1sy1|fflfflffl{zfflfflffl}
R1
� s2g2sy2|fflfflffl{zfflfflffl}
R2
264
375�1
(Green’s function) ð11Þ
S ¼ s1UR1|ffl{zffl}
S1
þ s2UR2|ffl{zffl}
S2
ð12Þ
where i is the unitary imaginary number. The importance of the perturbation d isaddressed in the following section. These solutions can be easily verified by back-substituting them into Eq. (5).
S1Sy2 ¼ s1U
R1 URy
2 sy2 ¼ 0 ð13Þ
S2Sy1 ¼ s2U
R2 URy
1 sy1 ¼ 0 ð14Þ
because UR1 and UR
2 are degree of freedom vectors of the two disconnected contacts withnull inner products. Several matrices used later for convenience are also defined as
A ¼ i½G�Gy� ¼ i½g1 � gy1 �|fflfflfflfflfflffl{zfflfflfflfflfflffl}
A1
þ i½g2 � gy2 �|fflfflfflfflfflffl{zfflfflfflfflfflffl}
A2
¼ GC1Gy|fflfflfflffl{zfflfflfflffl}A1
þGC2Gy|fflfflfflffl{zfflfflfflffl}A2
ð15Þ
C ¼ s1A1sy1|fflfflffl{zfflfflffl}
C1
þ s2A2sy2|fflfflffl{zfflfflffl}
C2
ð16Þ
and the proof of Eqs. (15) and (16) can be found in [24].
Energy Flux Between any Two Degrees of Freedom
The energy associated with any degree of freedom consists of kinetic andpotential energies,
Ep ¼1
4
Xq
ðu�pkpquq þ u�qkqpupÞ þMp
2_uu�p � _uup ð17Þ
where
kpq ¼ Hpq
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiMpMq
pð18Þ
The time derivative of this energy (Ep) is
dEp
dt¼ 1
4
Xq
ð _uu�pkpquq þ u�pkpq _uuq þ _uu�qkqpup þ u�qkqp _uupÞ þMp
2ð€uu�p _uup þ _uu�p€uupÞ ð19Þ
338 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
Using Newton’s second law ðMp€uup ¼ �Rqkpquq and Mp€uu�p ¼ �Rqkqpu�qÞ, Eq. (19) canbe expressed as
dEp
dt¼ 1
4
Xq
ðu�pkpq _uuq þ _uu�qkqpup � u�qkqp _uup � _uu�pkpquqÞ ð20Þ
The expressions up ¼ /pe�iwt=ffiffiffiffiffiffiffiMp
pand uq ¼ /qe�iwt=
ffiffiffiffiffiffiffiMq
p(/p and /q with no time
dependence) simflify Eq. (20) to
dEp
dt¼ x
2i
Xq
/�pHpq/q � /�qHqp/p
h i�X
q
Jpq ð21Þ
Equation (21) takes a typical form of an energy conservation equation, and accord-ingly, it is natural to define the energy flux between any two degrees of freedom (up
and uq) as
Jpq ¼x2i
/�pHpq/q � /�qH�qp/p
h ið22Þ
Normalization
Using Eq. (17) and up ¼ /pe�iwt=ffiffiffiffiffiffiffiMp
p, the normalization condition for
phonons can be expressed as
�hx ¼X
p
Ep
¼X
p
1
4
Xq
ðu�pkpquq þ u�qkqpupÞ þMp
2_uu�p � _uup
" #
¼X
p
x2j/pj2
ð23Þ
Therefore
Xp
j/pj2 ¼�h
x; orX
p
��� /pffiffiffiffiffiffiffiffiffi�h=x
p ���2 ¼ 1 ð24Þ
The Expression for Total Energy Flux
Total energy flux is the summation of fluxes between individual degrees of free-dom following Eq. (22). J1, the heat flux between ‘‘Contactl’’ and ‘‘Device,’’ isexpressed as
J1 ¼x Trace wys1U1 � U
y1 sy1 w
h i2i
¼x Trace wys1U
R1 � URy
1 sy1 wh i
2i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Inflow
�x Trace vy1 sy1 w� wys1v1
h i2i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
out flow
ð25Þ
ATOMISTIC GREEN’S FUNCTION METHOD 339
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
where we have used the fact that the trace of any matrix product is independent ofthe order of multiplication. Using Eqs. (6), (7), (8), (12), (13), and (14), the inflowterm can be written as
Inflow ¼x Trace SyGyS1 � S
y1 GS
h i2i
¼x Trace S1S
y1 Gy � S1S
y1 G
h i2i
¼x Trace S1S
y1 A
h i2
ð26Þ
With the normalization condition in Eq. (24), we can express the number of phononsin terms of matrix A1, which relates to the Green’s function matrix [see Eq. (15)]
xURy1 UR
1 )Z
�h
2pN1ðeÞA1 de ð27Þ
where e ¼ x2. Therefore
xS1Sy1 ¼ xs1U
R1 URy
1 sy1 )Z
�h
2pN1ðeÞs1A1s
y1 de ¼
Z�h
2pN1ðeÞC1 de ð28Þ
where N1(e) is the number of phonons in ‘‘Contact 1’’ at the eigenstate e ¼ x2. Com-bining Eqs. (26) and (28), we have
Inflow ¼Z
�hx2p
N1ðxÞTraceðC1AÞ dx ð29Þ
The outflow term in Eq. (25) can be evaluated using Eq. (8):
Outflow ¼ x Trace½wys1gy1 sy1 w� wys1g1s
y1 w�
2i
¼ x Trace½SyGyGSðs1gy1 sy1 � s1g1s
y1 Þ�
2i
¼ x Trace½GSSyGyC1�2i
ð30Þ
Combining Eqs. (12), (13), and (14), we find
xSSy ¼ xS1S1 þ xS2S2
)Z
�h
2pN1ðeÞs1A1s
y1 deþ
Z�h
2pN2ðeÞs2A2s
y2 de
¼Z
�h
2pN1ðeÞC1 deþ
Z�h
2pN2ðeÞC2 de ð31Þ
340 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
Therefore, the outflow term, becomes
Outflow ¼Z
�hx2p
Trace½GðN1ðxÞC1 þN2ðxÞC2ÞGyC1� dx
¼Z
�hx2p
Trace½N1ðxÞA1C1 þN2ðxÞA2C1� dx ð32Þ
The total energy flux between ‘‘Contact 1’’ and ‘‘Device’’ is then
J1 ¼Z
�hx2pfTrace½C1A�N1ðxÞ � Trace½N1ðxÞA1C1 þN2ðxÞA2C1�g dx
¼Z
�hx2p
Trace½C1A2�½N1ðxÞ �N2ðxÞ� dx
¼Z
�hx2p
Trace½C1GC2Gy�½N1ðxÞ �N2ðxÞ� dx ð33Þ
In the same manner, the total energy flux between ‘‘Contact 2’’ and ‘‘Device’’ isexpressed as
J2 ¼Z
�hx2p
Trace½C2A1�½N2ðxÞ �N1ðxÞ� dx
¼Z
�hx2p
Trace½C2GC1Gy�½N2ðxÞ �N1ðxÞ� dx ð34Þ
At steady state, J1 equals �J2 because
Trace½C1A� ¼ Trace½C1GyCG� ¼ Trace½CGyC1G� ¼ Trace½CA1�Trace½C1A� C1A1� ¼ Trace½CA1 � C1A1�
Trace½C1A2� ¼ Trace½C2A1� ð35Þ
Finally, the heat flux can be written in Landauer form [29] as
J ¼Z
�hx2p
NðxÞ½N1ðxÞ �N2ðxÞ� dx ð36Þ
where the transmission function is
NðxÞ ¼ Trace½C1GC2Gy� ¼ Trace½C2GC1Gy� ð37Þ
and thermal conductance ðrÞ is the ratio of heat flux J to the temperature difference,
r ¼ J
DTð38Þ
NUMERICAL IMPLEMENTATION
An algorithmic flow chart for the atomistic Green’s function method isshown in Figure 2. The first step involves constructing the harmonic matrix
ATOMISTIC GREEN’S FUNCTION METHOD 341
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
based on Eq. (1). The evaluation process can be either numerical (for a com-plicated potential) or analytical (for a simple potential), provided that atomiclocations and interatomic potential parameters are known. The second stepinvolves calculating the uncoupled Green’s functions g1 and g2. The size ofg1 or g2 (nc� nc) is typically substantial because of their physical sizes. How-ever, in the transmission calculation, g1 always appears in the form s1g1s
y1 ,
and s1 has only a finite number of nonzero elements. These nonzero elementscomprise a submatrix ss
1 that physically represents interactions between LC andLD. Therefore ss
1 has a finite size of ncd� ncd, as does ss2. Consequently, the
significant part of g1 is its submatrix gs1 ðncd � ncdÞ that corresponds to atoms
in LC. The significant part of g2 is also its submatrix gs2 ðncd � ncdÞ that
corresponds to atoms in RC.Therefore, computational savings can be achieved by considering only these
parts of g1 and g2. The decimation technique [30] is typically employed to calculategs
1 and gs2 because of its computational efficiency and low memory requirements. If
the contact is homogeneous, i.e., if LCB and LC have identical dynamical properties,the decimation technique can be used directly. Otherwise, an additional calculation isneeded as discussed later.
Another issue that deserves discussion is the selection of d, which is a smallnumber corresponding to phonon energy dissipation in contacts, whose role is
Figure 2. Algorithmic flow chart for the atomistic Green’s function method.
342 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
elaborated in [24]. We recommend that d take the following general form:
d ¼ f ðxÞx2 ð39Þ
where f ðxÞ is a monotonically decreasing function. We select f ðxÞ ¼ 0:001ð1� x=xmaxÞ, where xmax is the maximum phonon vibrational frequency. The choiceof d affects the energy resolution in the uncoupled Green’s function and subsequentcalculations. A smaller d value gives better energy resolution but requires longercomputational times.
The last numerical concern involves the significantly large number of matrixinversions in the calculation of Green’s functions. A fast algorithm can be appliedbecause all harmonic matrices have banded structures, and banded-matrix inversionis not as expensive numerically as direct inversion [31]. We also note that only a sub-set of entries in the Green’s function G requires evaluation because only nonzero ele-ments in C1 and C2 are significant in the final transmission formula [see Eq. (37)].Additional computational saving can be achieved by evaluating only the significantpart of G. However, we have observed that in some rare cases, the matrix conditionnumber of ½x2I�Hd � R1 � R2� is sufficiently large, in which case special care needsto be exerted to obtain the significant part of G.
RESULTS AND DISCUSSION
Homogeneous Atomic Chain
A homogeneous atomic chain is employed to demonstrate the AGF methodand is shown in Figure 3a with one degree of freedom per atom. We have selectedthree atoms to constitute the ‘‘Device’’ region, and only nearest-neighbor atomicinteractions are considered. The chosen bond strength (i.e., the spring constant) is32 N=m, approximately equal to the bond strength in silicon. The atomic massand spacing are 4.6� 10�26 kg and 5.5 A, respectively. In this case, the dynamicalproperties of LCB and LC are the same, so the decimation technique can be used
Figure 3. Homogeneous and heterogeneous atomic chain examples. Filled and open circles represent
different types of atoms. The dotted, dashed, and solid lines between atoms represent different bonds.
ATOMISTIC GREEN’S FUNCTION METHOD 343
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
directly to obtain uncoupled Green’s functions. Analytical solutions for theuncoupled Green’s functions gs
1 and gs2 in this simple case can be found in the [32].
The atomistic Green’s function method does not require a priori knowledge ofphonon dispersion curves or density of states. Instead, it can be used to calculate thelocal density of states on each atom in the system. In the case of a one-dimensionalatomic chain, the local density of states matrix (LDOS: Dl) is associated directly withthe local Green’s function (G):
Dl ¼iðG�GyÞx
pLð40Þ
where L is the bond length. The exact local density of states on the ith degree of free-dom will correspond to the ith diagonal element of Dl . The global density of statesfunction is the same as the local density of states in homogeneous materials, and it isshown in Figure 4 together with the theoretical phonon density of states in a homo-geneous atomic chain [33]. A cutoff frequency exists and sets an upper frequency.
The full-spectrum transmission function of the homogeneous chain is calculatedusing the atomistic Green’s function method, and the result is shown in Figure 5. Thetransmission is unity at all frequencies below the cutoff frequency (a consequence ofthe harmonic assumption), and the transmission becomes zero at frequencies abovethe cutoff frequency because the phonon density of states is zero at these frequencies.
Heterogeneous Atomic Chain
Simulation of a heterogeneous atomic chain provides more physical insights ascompared to a homogeneous atomic chain. A heterogeneous atomic chain configuration
Figure 4. Comparison of density of states functions in a homogeneous atomic chain calculated by the
AGF method and an analytical method [33].
344 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
is shown in Figure 3b, with one degree of freedom per atom. One additional step isneeded to calculate gs
1 and gs2:
gs1 ¼ ½ðx2 þ diÞI�HLC � sLC;LCBgLCBsLCB;LC ��1 ð41Þ
gs2 ¼ ½ðx2 þ diÞI�HRC � sRC;RCBgRCBsRCB;RC ��1 ð42Þ
where gLCB and gRCB are uncoupled Green’s functions for LCB and RCB, respectively,and they are obtained from the decimation technique. HLC and HRC are harmonicmatrices for LC and RC, respectively. sLC,LCB and sLCB,LC are matrices that connect
LC and LCB (sLC;LCB ¼ syLCB;LC), and sRC,RCB and sRCB,RC are matrices that connect
RC and RCB (sRC;RCB ¼ syRCB;RC).
The ratios between spring constants and atomic masses dictates the harmonicmatrix and thus the thermal conductance. Therefore, the effect of doubling thespring contact is equal to the effect of reducing the the atomic mass by half, exceptfor the interfacial atoms. To simplify the discussion, only the atomic masses of‘‘Device’’ atoms are changed in the two heterogeneous cases, which conceptuallycan be thought of as isotopes. The properties of ‘‘Contact’’ atoms remain the sameas those in the homogeneous chain case. The atomic masses are 9.2� 10�26 kg and2.3� 10�26 kg in the first and second heterogeneous cases, respectively. The bondstrength between a ‘‘Device’’ atom and a ‘‘Contact’’ atom, though unchanged inthese two specific cases, takes the mean value of ‘‘Device’’ atoms and ‘‘Contact’’atoms if necessary.
Figure 5. Comparison of transmission functions for homogeneous and heterogeneous atomic chains. The
mass of a ‘‘Device’’ atom in the homogeneous case is 4.6� 10�26 kg. The masses of ‘‘Device’’ atoms in the
two heterogeneous cases are 9.2 and 2.3� 10�26 kg, respectively.
ATOMISTIC GREEN’S FUNCTION METHOD 345
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
The local phonon density of states in any heterogeneous chain differs from theglobal density of states in a bulk solid. For the first heterogeneous case (deviceatomic mass ¼ 9.2� 10�26 kg), the LDOS functions of atoms LC, LD, and D areshown in Figure 6 along with the bulk density of states of ‘‘Device’’ atoms. TheLDOS functions of atoms RC and RD are the same as those of LC and LD, becauseof the symmetric configuration, and thus they are not shown. The LDOS functionsexhibit pronounced differences with the bulk density of states resulted from thecoupling between the device and the contacts.
The transmission functions in the two heterogeneous cases are compared to thatof the homogeneous atomic chain in Figure 5. At extremely low frequencies, the hetero-geneous chain exhibits the same transmission as the homogeneous chain because thelong wavelengths of low-frequency phonons limit the influence of the boundary scatter-ing. However, at higher frequencies, phonons are scattered by heterogeneous interfaces,causing decreased transmission. Several Fabry-Perot peaks, where the transmissionfunction reaches unity, are observed due to the resonant scattering of phonons.
The thermal conductances of the homogeneous and heterogeneous cases are com-pared in Figure 7. Due to interface scattering, the heterogeneous atomic chain exhibits asmaller conductance than the homogeneous atomic chain over all temperature ranges.However, at low temperatures, minimal differences exist among conductances of homo-geneous and heterogeneous cases because of the convergence of transmission functionsat low frequencies. The conductances of both homogeneous and heterogeneous systemsexhibit a linear temperature dependence at low temperatures.
Lastly, we nondimensionalize thermal conductance as the ratio of the thermalconductance (r) to the quantum conductance (p2k2
BT=3h). The quantum conduc-tance is the conductance on a homogeneous atomic chain with infinitely large bond
Figure 6. The local density of states functions at various locations, compared to the density of states
function of the homogeneous chain made of ‘‘Device’’ atoms. The mass of ‘‘Device’’ atoms in the homo-
geneous case is 4.6� 10�26 kg. The masses of ‘‘Device’’ atoms in the two heterogeneous cases are 9.2 and
2.3� 10�26 kg, respectively.
346 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
strength or infinitesimal atomic masses. Thus, the nondimensional conductance indi-cates how well each test case represents an ideal phonon conductor. The lengthdependencies of the nondimensional thermal conductances at different temperatures
Figure 8. Length dependence of nondimensional thermal conductances at different temperatures. The case
of zero number of device atoms represents the conductance of a homogeneous chain with ‘‘Contact’’
atomic properties.
Figure 7. Comparison of thermal conductances for homogeneous and heterogeneous atomic chains. The
mass of a ‘‘Device’’ atom in the homogeneous case is 4.6� 10�26 kg. The masses of ‘‘Device’’ atoms in the
two heterogeneous cases are 9.2 and 2.3� 10�26 kg, respectively.
ATOMISTIC GREEN’S FUNCTION METHOD 347
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
in heterogeneous case 1 are plotted in Figure 8. The length of the device region isrepresented by the number of atoms. The case of zero number of device atoms repre-sents a homogeneous atomic chain with ‘‘Contact’’ atom properties (atomic mass is4.6� 10�26 kg). The general trend of the nondimensional homogeneous conductanceis that it decreases with increasing temperature, because higher temperatures shift thephonon spectrum to higher energies and reduce the integral in Eq. (36). The nondi-mensional heterogeneous conductance has a similar temperature dependence as thatof the homogeneous chains. It also decreases as the length increases but levels out toan asymptotic value. In the ballistic transport regime, the length dependence of con-ductance is attributed mainly to the coupling and decoupling of phonon wave func-tions. If the ‘‘Device’’ is extremely short, phonons can easily propagate from one sideto another.
CONCLUSIONS
A full derivation of the atomistic Green’s function method and algorithmicdetails have been presented in this article. Numerical issues in the implementationof the AGF method have been addressed. Several examples illustrated the calcu-lation of transmission functions, local and global density of states functions, andthermal conductances. The unique features of mesoscopic transport, such asFabry-Perot peaks, are reproduced by this method. The ballistic thermal conduc-tances for homogeneous and heterogeneous atomic chains, as well as their lengthdependencies, have been discussed. The AGF method can be further extended tosimulate phonon transport in more sophisticated atomic structures. We expect thatthe presentation of the AGF method herein will make the method more accessible toresearchers and will therefore motivate extended work on this useful approach.
REFERENCES
1. L. G. C. Rego and G. Kirczenow, Quantized Thermal Conductance of Dieletric QuantumWires, Phys. Rev. Lett., vol. 81, p. 232, 1998.
2. A. Ozpineci and S. Ciraci, Quantum Effects of Thermal Conductance through AtomicChains, Phys. Rev. B, vol. 63, p. 125415, 2001.
3. P. Hyldgaard, Resonant Thermal Transport in Semiconductor Barrier Structures, Phys.Rev. B, vol. 69, p. 193305, 2004.
4. S. Pettersson and G. Mahan, Theory of the Thermal Boundary Resistance between Dis-similar Lattices, Phys. Review B, vol. 42, p. 7386, 1990.
5. R, Stoner and H. Maris, Kapitza Conductance and Heat Flow between Solids at Tem-perature from 50 to 300 K, Phys. Rev. B, vol. 48, p. 16373, 1993.
6. M. Cross and R. Lifshitz, Elastic Wave Transimission at an Abrupt Junction in a ThinPlate with Application to Heat Transport and Vibrations in Mesoscopic Systems, Phys.Rev. B, vol. 64, p. 85324, 2001.
7. D. Santamore and M. Cross, Effect of Surface Roughness on the Universal Thermal Con-ductance, Phys. Rev. B, vol. 63, p. 184306, 2001.
8. D. Santamore and M. Cross, Effect of Phonon Scattering by Surface Roughness on theUniversal Thermal Conductance, Phys. Rev. Lett., vol. 87, p. 115502, 2001.
9. D. Santamore and M. Cross, Surface Scattering Analysis of Phonon Transport in theQuantum Limit Using an Elastic Model, Phys. Rev. B, vol. 66, p. 144302, 2002.
348 W. ZHANG ET AL.
Dow
nloa
ded
By: [
Purd
ue U
nive
rsity
] At:
18:5
2 3
April
200
7
10. K. Schwab, E. Henriksen, J. Worlock, and M. Roukes, Measurement of the Quantum ofThermal Conductance, Nature, vol. 404, p. 974, 2000.
11. W.-X. Li, K.-Q. Chen, W. Duan, J. Wu, and B.-L. Gu, Phonon Transport and ThermalConductivity in Dielectric Quantum Wire, J. Phys. D: Appl. Phys., vol. 36, p. 3027, 2003.
12. W.-Q. Huang, K.-Q. Chen, Z. Shuai, L. Wang, and W. Hu, Discontinuity Effect on thePhonon Transmission and Thermal Conductance in a Dielectric Quantum Waveguide,Phys. Lett. A, vol. 336, p. 245, 2005.
13. K.-Q. Chen, W.-X. Li, W. Duan, Z. Shuai, and B.-L. Gu, Effect of Defects on the Ther-mal Conductivity in a Nanowire, Phys. Rev. B, vol. 72, p. 45422, 2005.
14. D. Poulikakos, S. Arcidiacono, and S. Maruyama, Molecular Dynamics Simulation inNanoscale Heat Transfer: A Review, Microscale Thermophys. Eng., vol. 7, p. 181, 2003.
15. Z. Zhong, X. Wang, and J. Xu, Equilibrium Molecular Dynamics Study of Phonon Ther-mal Transport in Nanomaterials, Numer. Heat Transfer B, vol. 46, pp. 429–446, 2004.
16. S. Mazumdar and A. Majumdar, Monte Carlo Study of Phonon Transport in Solid ThinFilms including Dispersion and Polarization, J. Heat Transfer, vol. 123, p. 749, 2001.
17. S. Srinivasan, R. Miller, and E. Marotta, Parallel Computation of the Boltzmann Trans-port Equation for Microscale Heat Transfer in Multilayered Thin Films, Numer. HeatTransfer B, vol. 46, pp. 31–58, 2004.
18. D. Angelescu, M. Cross, and M. Roukes, Heat Transport in Mesoscopic Systems, Super-lattices Microstruct., vol. 23, p. 673, 1998.
19. W. Little, The Transport of Heat between Dissimilar Solids at Low Temperature, Can. J.Phys., vol. 37, p. 334, 1959.
20. E. Swartz and R. Pohl, Thermal Boundary Resistance, Rev. Mod. Phys., vol. 61, p. 605,1989.
21. R. Stevens, A. Smith, and P. Norris, Measurement of Thermal Boundary Conductance ofa Series of Metal-Dielectric Interfaces by the Transient Thermoreflectance Technique, J.Heat Transfer, vol. 127, p. 315, 2005.
22. D. Ferry and S. Goodnick, Transport in Nanostructures, pp. 156–180, Cambridge Univer-sity Press, New York, 1997.
23. S. Datta, Electronic Transport in Mesoscopic Systems, pp. 117–163, Cambridge UniversityPress, New York, 1995.
24. S. Datta, Quantum Transport: Atom to Transistor, pp. 223–233, Cambridge UniversityPress, New York, 2005.
25. W. Zhang, N. Mingo, and T. Fisher, Simulation of Phonon Interfacial Transport inStrained Silicon-Germanium Heterostructures, J. Heat Transfer, in press, 2007.
26. N. Mingo and L. Yang, Phonon Transport in Nanowires Coated with an AmorphousMaterial: An Atomistic Green’s Function Approach, Phys. Rev. B, vol. 68, p. 245406, 2003.
27. N. Mingo and L. Yang, Erratum: Phonon Transport in Nanowires Coated with anAmorphous Material: An Atomistic Green’s Function Approach [Phys. Rev. B 68,245406 (2003)], Phys. Rev. B, vol. 70, p. 249901, 2004.
28. N. Ashcroft and N. Mermin, Solid State Physics, p. 439, Saunders College Publishers,Philadelphia, 1976.
29. A. Buldum, S. Ciraci, and C Fong, Quantum Heat Transfer through an Atomic Wire, J.Phys. Condensed Matters, vol. 12, p. 3349, 2000.
30. F. Guinea, C. Tejedor, F. Flores, and E. Louis, Effective Two-Dimensional Hamiltonianat Surfaces, Phys. Rev. B, vol. 28, p. 4397, 1983.
31. G. Golub and C. V. Loan, Matrix Computations, 3rd ed., p. 152, The John HopkinsUniversity Press, Baltimore, MD, 1996.
32. M. Lannoo and P. Friedel, Atomic and Electronic Structure of Surfaces, pp. 42–48,Springer-Verlag, Berlin, 1991.
33. C. Kittel, Introduction to Solid State Physics, 8th ed., p. 108, Wiley, New York, 2005.
ATOMISTIC GREEN’S FUNCTION METHOD 349