Large enhancement in thermoelectric efficiency of quantum dot junction due to increase of level degeneracy

0.33 MB·

by  David M T Kuo

Large enhancement in thermoelectric efficiency of quantum dot junction due to
increase of level degeneracy
David M T Kuo1, Chih-Chieh Chen2, and Yia-Chung Chang3,4
1Department of Electrical Engineering and Department of Physics,
National Central University, Chungli, 320 Taiwan
2Department of Physics, Zhejiang University, Hangzhou 310027 China
3Research Center for Applied Sciences, Academic Sinica, Taipei, 11529 Taiwan and
7 4 Department of Physics, National Cheng Kung University, Tainan, 701 Taiwan
1 (Dated: February 21, 2017)
0
2 Itistheoreticallydemonstratedthatthefigureofmerit(ZT)ofquantumdot(QD)junctionscan
be significantly enhanced when the degree of degeneracy of the energy levels involved in electron
b
transport is increased. The theory is based on the the Green-function approach in the Coulomb
e
blockade regime by including all correlation functions resulting from electron-electron interactions
F
associated with the degenerate levels (L). We found that electrical conductance (Ge) as well as
0 electron thermal conductance (κe) are highly dependent on the level degeneracy (L), whereas the
2 Seebeck coefficient (S) is not. Therefore, the large enhancement of ZT is mainly attributed to the
increase of Ge when the phonon thermal conductance (κph) dominates the heat transport of QD
] junction system. In the serially coupled double-QD case, we also obtain a large enhancement of
l
l ZT arising from higher L. Unlike Ge and κe, S is found almost independent on electron inter-dot
a
hoppingstrength.
h
-
s
e I. INTRODUCTION ducedsignificantlydueto theintroductionofQDs. Such
m
a reduction of κ due to phonon scattering with QDs
ph
. in SiGe nanowire filled with QDs was verified theoreti-
t Recently,manyeffortshavebeendevotedtothesearch
a cally in Ref. 12. However, the behaviors of G , S and
ofhigh-efficiencythermoelectric(TE)materials,because e
m
κ in nanowires filled with QDs remain unclear because
of the high demand of energy-saving solid state coolers e
d- and power generators.1,2 TE devices have very good po- of the complicated many-body problem involved. The
full many-body effect on the behaviors of electron ther-
n tential for green energy applications due to their desir-
moelectric coefficients may be analyzed by considering
o able features, including low air pollution, low noise, and
c longoperationtime. However,thereexistscertainbarrier a single QD or double QDs (DQD) embedded in a sin-
[ gle nanowire to reveal the importance of the electron
for TE devices to replace conventional refrigerators and
Coulomb interaction.
3 powergeneratorssince TE materialswith figureofmerit
v (ZT) larger than three are not yet found.1,2 The figure Theoretical studies have indicated that a TE de-
5 of merit, ZT = S2GeT/κ,defined in the linear response vice made of molecular QD junction13−14 can reach the
1
regime is composed of the Seebeck coefficient (S), elec- Carnot efficiency if one can neglect κ . Such a diver-
5 ph
4 trical conductance (Ge), thermal conductance (κ) and genceofZT forQDsisrelatedtothedivergenceofGe/κe,
0 equilibrium temperature (T). κ is the sum of the elec- which violates the Wiedeman-Franz law (WFL).15 The
. tronthermalconductance (κe) andphononthermalcon- violation of WFL is a typical feature for QDs with dis-
1
0 ductance (κph). It has been shown that low-dimensional creteenergylevels.16 Itishardtorealizethermaldevices
7 systems including quantum wells3, quantum wires4 and with Carnot efficiency as considered in Refs. 13 and 14,
1 quantum dots (QDs)5 have very impressive ZT values becauseitisimpossibletoblockadeacousticphononheat
: when compared with bulk materials.3−9 In particular, flow completely in the implementation of solid state TE
v
ZT of PbSeTe QD array (QDA) can reach two,5 which devices1,2. Therefore, finding a way to enhance ZT of
i
X is mainly attributed to the reduction of κph in QDA.2 QD junctions under an achievable κph value is crucial.
r However,QDjunctionswithZT ≥3arenotyetreported Here,wedemonstratethatby increasingthe leveldegen-
a experimentally. There are some technical difficulties in eracyinQDs,itispossibletoenhancethethermoelectric
using the QD junction to achieve ZT ≥3 via the reduc- efficiency significantly given the condition κ /κ ≫ 1.
ph e
tion of phonon thermal conductivity.1,2 The level degeneracy in a QD can be determined by its
More than two decades ago, Hicks and Dresselhauss point-groupsymmetry. For sphericalQDs made of semi-
theoretically predicted that ZT values of BiTe quantum conductorswithzincblende(e.g. III-Vcompounds)ordi-
wells and quantumwires can be largerthanone atroom amond crystal structure (e.g. Si or Ge), the point group
temperature.10,11 In particular, ZT values of nanowires isTd. Thus,theorbitaldegeneracyLcanbedescribedby
(with diameter smaller than 1 nm) may reach 10 based singlet (A ), doublet (E ) or triplet (T ). If the QD en-
1 2 2
on the assumption of very low lattice thermal conduc- ergylevelsarewelldescribedbytheeffective-massmodel
tivity (κ = 1.5Wm−1K−1 for Bi Te ). Recently, there (neglecting the crystal-field effect), then the orbital de-
L 2 3
areconsiderableinterestonZT valuesofnanowiresfilled generacyis determined by the associatedorbitalangular
with QDs,1,2 because it is expected that κ can be re- momentumquantumnumberℓ,andtheleveldegeneracy
ph
2
becomes L = 2ℓ+1. For example, the p-like states in cording to the Meir-Wingreen formula19
a spherical QD are 3-fold degenerate with L = 3 (not
including spin degeneracy). In an QD junction, one can Jn = ie dǫ(ǫ−µα)nΓα[G<(ǫ) (3)
tune the gate voltageto access the level with desired de- α h XZ e j jσ
jσ
generacy. The high level degeneracy (L) is also feasible
+ f (ǫ)(Gr (ǫ)−Ga (ǫ))],
inQDsmadeofmulti-valleysemiconductorssuchasSior α jσ jσ
Ge. Our theoreticalresults may serveas usefulguideline
where n = 0 is for the electrical current and n = 1
for optimizing ZT of semiconductor QD1,2 or molecular
for the heat current. Γα(ǫ) = |V |2δ(ǫ − ǫ ) is
QD systems13, in which a dominating phonon thermal j k k,j k
the tunneling rate for electrons frPom the α-th reservoir
conductivity cannot be avoided.
and entering the j-th energy level of the QD. f (ǫ) =
α
1/{exp[(ǫ−µ )/k T ]+1} denotes the Fermi distribu-
α B α
tion function for the α-th electrode, where µ and T
α α
II. FORMALISM are the chemical potential and the temperature of the α
electrode. e, h, and k denote the electron charge, the
B
Planck’s constant, and the Boltzmann constant, respec-
Here we consider nanoscale semiconductor QDs em-
tively. G<(ǫ), Gr (ǫ), and Ga (ǫ) denote the frequency-
beddedinananowireconnectedwithmetallicelectrodes. jσ jσ jσ
domain representations of the one-particle lessor, re-
An extended Anderson model is employed to simulate a
tarded, and advanced Green’s functions, respectively.
QD junction with degenerate levels.17−19 The Hamilto-
nian of the QD junction system considered is given by
H =H +H , where
0 QD A. Thermoelectric coefficients
H0 = ǫka†k,σak,σ+ ǫkb†k,σbk,σ (1) Thermoelectric coefficients including Ge, S and κe in
X X
k,σ k,σ the linear response regime can be evaluated by using
+ VLd† a + VRd† b +c.c. Eq.(3)withsmall∆V =(µL−µR)/eand∆T =TL−TR.
k,ℓ ℓ,σ k,σ k,ℓ ℓ,σ k,σ
X X Weobtainthefollowingexpressionsofthermoelectricco-
k,ℓ,σ k,ℓ,σ
efficients:
The first two terms of Eq. (1) describe the free electron δJ0
gasinthe leftandrightelectrodes. a† (b† )createsan Ge = ( α )∆T=0 (4)
k,σ k,σ δ∆V
electronofmomentumkandspinσ withenergyǫk inthe δJ0 δJ0
left (right) electrode. VL (VR) describes the coupling S = −( α ) /( α ) (5)
k,ℓ k,ℓ δ∆T ∆V=0 δ∆V ∆T=0
between the the ℓ-th energy level of the QD system and
δJ1 δJ1
left (right) electrode. d† (d ) creates (destroys) an κ = ( α ) +( α ) S (6)
ℓ,σ ℓ,σ e δ∆T ∆V=0 δ∆V ∆T=0
electron in the ℓ-th energy level of the QD.
δJ1
= ( α ) −S2G T
∆V=0 e
δ∆T
HQD = XEℓnℓ,σ+XUℓnℓ,σnℓ,σ¯ (2)
ℓ,σ ℓ where
1
+ 2ℓ,jX,σ,σ′Uℓ,jnℓ,σnj,σ′ (δδ∆JαV0 )∆T=0 = ihe Z dǫΓαj(ǫ)×
X
jσ
where Eℓ is the spin-independent QD energy level, δG<(ǫ) δf (ǫ)
and nℓ,σ = d†ℓ,σdℓ,σ, Uℓ and Uℓ,j describe the intralevel [ δfjσ(ǫ) +(Grjσ(ǫ)−G(jaσ(ǫ))] δ∆αV , (7)
and interlevel Coulomb interactions, respectively. For α
nanoscalesemiconductorQDs,theinterlevelCoulombin-
teractionsaswellasintralevelCoulombinteractionsplay δJ0 ie
asignificantroleontheelectrontransportinsemiconduc- (δ∆αT)∆V=0 = h Z dǫΓαj(ǫ)×
X
tor junctions. It is worth noting that H possesses the jσ
QD
particle-hole symmetry. One can prove it with a simple δG<(ǫ) δf (ǫ)
swap of electron and hole operators (dℓ,σ → c†ℓ,σ). The [ δfjασ(ǫ) +(Grjσ(ǫ)−G(jaσ(ǫ))] δ∆αT , (8)
form of H is changed only by constant terms when
QD
QDenergylevelsaredegenerate. This indicates thatdy-
namicphysicalquantityisunchangedintheholepicture. δJ1 i
( α ) = dǫΓα(ǫ)(ǫ−E )×
To reveal the transport properties of a QD junction δ∆T ∆V=0 h Z j F
X
connected with metallic electrodes, it is convenient to jσ
usetheGreen-functiontechnique. Theelectronandheat [δG<jσ(ǫ) +(Gr (ǫ)−Ga (ǫ))]δfα(ǫ), (9)
currents from reservoir α to the QD are calculated ac- δf (ǫ) jσ jσ δ∆T
α
3
( δJα1 ) = i dǫΓα(ǫ)(ǫ−E )× clude κph, we have adopted the Landauer formula given
δ∆V ∆T=0 h Z j F in Refs. 28 and 30.
X
jσ
δG<(ǫ) δf (ǫ) 1 ¯h3ω2 eh¯ω/kBT
[ δfjσ(ǫ) +(Grjσ(ǫ)−Gajσ(ǫ))] δ∆αV . (10) κph = hZ dωT(ω)phkBT2(eh¯ω/kBT −1)2, (11)
α
δG<jσ(ǫ) is obtained by taking the derivative of the equa- where ω and Tph(ω) are the phonon frequency and
δfα(ǫ) throughput function, respectively. In a perfect wire
tionofmotionwithrespecttothe changeinFermi-Dirac throughput is unity for each open channel, then κ
ph
distribution, fα(ǫ). Here we have assumed the varia- in such a perfect case is given by κ = kB2π2TNph =
tion of the correlation functions with respect to δfα(ǫ) g (T)N , where N is the totalnumphb,0er of ope3nhmodes
is of the second order. Note that we have to take the 0 ph ph
limit ∆V → 0 for the calculation of (δδ∆JαV0 )∆T=0 and and g0(T) = kB23hπ2T = 9.456×10−4T(nW/K2) is called
(δJα1 ) . E is the Fermi energy of electrodes. The the quantum conductance.[34] Experimentally, it was
δ∆V ∆T=0 F found that the linear-T behavior in ballistic regime only
one-particle Green’s functions in Eqs. (7)-(10) are re-
holdsfortemperaturebelow0.8Kforwiresofsize 200nm
latedrecursivelytohigh-orderGreen’sfunctionsandcor-
with N = 4 (which includes one longitudinal, one tor-
relation functions via a hierarchy of equations of mo- ph
sional, and two flexural modes)34. Beyond 0.8K, the
tion(EOM)20. This hierarchyselfterminates atthe 2N-
nonlienar-T behavior was observed due the contribution
particle Green function, where N is the number of levels
ofhigh-energyphononmodesbeingthermallypopulated.
considered in the QD or coupled QDs.
Low-temperaturethermoelectric properties of Kondo in-
To reveal the effect of degenerate levels on the ther-
sulatornanowirewasalsostudiedinRef. 35byusingthe
moelectric efficiency of QD junction system, all needed
Callaway model to describe κ .
Green’s functions and correlation functions arising from ph
Ifthereexistsphononelasticscatteringfromthedisor-
electron-electron interactions in the QDs considered are
dereffectsofnanowiresurface27−33orinterfaceboundary
computed self-consistently following the procedures de-
of QDs embedded in the nanowire,36,37 the throughput
scribed in our previous work.20,21 Our procedure is be-
functionbecomesmorecomplicated.27−33 Ingeneral,the
yond the mean-field theory, which is widely used in
T (ω)dependsonthelength(L˜)anddiameter(D)ofthe
solving the equation of motion in the Green function ph
nanowire, phonon mean free path ℓ (ω), and Debye fre-
calculation.14 For L = 4, our calculation involves solv- 0
quency (ω ). Realistic calculation of T requires heavy
ing one-, two-, ···, up to eight-particle Green functions. D ph
numerical work for treating the detailed phonon disper-
sion curves,29−33,36−37 which is beyond the scope of this
article. However, an empirical expression which works
B. Phonon thermal conductance
well in general for semiconductor nanowires at a wide
range of temperature can be found in Ref. 24, which
ThethermoelectricefficiencyofaQDjunctionembed-
reads
ded in a nanowire is determined by the figure of merit,
ZT = S2GeT/(κe+κph), which involves the κph of the T (ω)= Nph,1(ω) + Nph,2(ω) (12)
QD junction system. The optimization of molecular QD ph 1+L˜/ℓ (ω) 1+L˜/D
0
junctions under the condition of κ /κ ≫ 1 has been
e ph
theoretically investigated in references[13,14]. However, withthefrequency-dependentmeanfreepathℓ0(ω)given
the condition of κ /κ ≫1 is very difficult to realize in by
e ph
practice. Themaingoalofthisstudyistoinvestigatethe
1 δ2 ω
effect of energy level degeneracy on thermoelectric effi- =B ( )2N (ω), (13)
ℓ (ω) D3 ω ph
ciencyundertherealisticconditionwithκ /κ >1. The 0 D
ph e
phononthermal conductance ofnanowireshave been ex- where N (ω) = 4 + A(D)2( ω )2 (for ω < ω ) de-
tensively studied experimentally and theoretically.22−32 ph a ωD D
notes the number of phonon modes. The dimensionless
InRefs.22-24ithasbeenshownexperimentallythatκ
ph parameters are chosen to be A = 2.17 and B = 1.2.
displaysalinearT behaviorfrom20K to300K forsilicon
Notation a denotes the lattice constant of nanowire.
nanowires with diameter 22 nm. The linear T behavior
N (ω)=N (min(ω,v /δ))andN (ω)=N (ω)−
ph,1 ph s ph,2 ph
of κ also holds for T between 100K and 400K for ger-
ph N (ω). v is the sound velocity of nanowire and δ de-
manium nanowires with diameter 19 nm.25 Due to the scprhib,1esthetshicknessoftheroughsurfaceofnanowire24,28
reduction of κ , ZT of silicon nanowires increases sig-
ph In Eq. (12), one essentially replaces the frequency-
nificantly (with ZT = 1 at 200K) in comparison with
dependent mean free path ℓ (ω) by a constant D for the
ZT =0.01 for bulk siliconat roomtemperature.4,26 The 0
high-frequency modes (ω >v /δ).
s
linear T behavior of nanowires is an interesting topic.
For a certain range of temperatures, a simple expres-
Many theoretical efforts have been devoted to clarifying
sionofκ (T)formolecularQDjunctionsystemmaybe
ph
why nanowires with diameters near 20 nm exhibit the used. One can approximately write13
linear T behavior.27−32 For a true one-dimensional sys-
tem, the linear T behavior of κ is expected.33 To in- κ =F g (T), (14)
ph ph s 0
4
where F is a dimensionless correctionfactor used to de- features by ǫ (n =1,···6), which result from the res-
s 3,n
scribe the effect of non-ballistic phonon transport due onantchannelsatE , E +U , E +2U , E +U +2U ,
0 0 I 0 I 0 0 I
to surface roughness and phonon scattering from QDs, E +U +3U ,andE +U +4U ,respectively. (Here,we
0 0 I 0 0 I
which replaces the throughput function T (ω) used in have adopted U =U =20Γ . U denotes the interlevel
ph 0 I 0 I
Eq.(11). ThesimpleexpressionofEq.(14)withF =0.1 CoulombinteractionsfortheL=3case.) Thesechannels
s
will be used to describe κ throughout this article ex- correspond to physical processes of filling the QD with
ph
cept in Fig. 4. In Fig. 4, we compare ZT as a func- onetosixelectrons. Athighertemperatures(k T =5Γ
B 0
tionoftemperature obtainedby usingbothEq.(12)and and k T =10Γ ), the 1st M-shapedspectral feature for
B 0
Eq. (14). It is found that with the simple scaling factor ZT is broadened and enlarged. (The last M-shaped ZT
F we can describe the behavior of κ reasonably well feature for L=3 is not shown in Fig. 1) For L=3, the
s ph
for thin nanowires in the temperature range of interest. other spectral features of ZT (at ǫ ; n = 2,3,4,5) are
3,n
With this simple scaling we can clarify the effect of level suppressed. Thisisattributedtothesignificantreduction
degeneracy (L) on ZT for different magnitudes of κ . of maximum S2 for those channels. Because of electron-
ph
holesymmetry inthe systemHamiltonian,itis expected
thatthespectrumofZT issymmetricalaboutthemiddle
III. RESULTS AND DISCUSSION point of the Coulomb gap (MPCG). For L = 3, MPCG
occurs at eV =100Γ . Therefore,we only need to focus
g 0
Based on Eqs. (4)-(6), we numerically calculate ther- ontheanalysisofZT optimizationnearthefirstspectral
moelectric coefficients including all correlation functions feature in the level-depletion regime, which is defined as
arising from electron Coulomb interactions in the QDs. the regime when the average occupation number of the
Fig. 1 shows ZT of a single QD junction as a function QD summed over spin (Nt) is less than one. In general,
of the QD level E0, which is tuned by gate voltage Vg it occurs at E0 >EF.
according to E = E +50Γ −eV for the case of non
0 F 0 g
degeneracy(L=1)and3-folddegeneracy(L=3). Note
that the role of gate voltage introduced here allow us 1.00 e
to tune the difference between the QD level energy and 0.75 (a) e 3,1 e 1,1 e 3,2 e 3,3 e 3,4 e 3,5
Fmeertmriicaelneturgnyn.elTinhgroruagtheowuitththΓis a=rtiΓcle,=weΓa=doΓptaansdymal-l ZT 00..2550 kBT=1G 0 1,2 e 3,6
L R 0
energyscalesarein termsofΓ . Γ ≈1meV intypically 0.00
QD junctions; thus, reasonabl0e va0lues for U0 and UI in 11..05 (b)ZTmax LL==13 kBT=5G 0
realistic semiconductor QDs are in the range of 20-100 T
Z 0.5
Γ . Fig. 1(a),(b)and(c)arefork T =1Γ , k T =5Γ
0 B 0 B 0 0.0
manudmkZBTT =for1t0hΓe0,3-rfeoslpdeccatsiveeilsy.siIgtniifiscsaenetnhtihgahterththeamnatxhie- T 01..82 (c) ZTmax FEs0==0E.F1+50G 0-eVg kBT=10G 0
corresponding value for the non-degenerate case when Z 0.4
the temperatureishigh. Forexample,the maximumZT 0.0
(labeled by (ZT) ) is enhanced by near two-times for 0 25 50 75 100 125 150
k T = 5Γ and mmaoxre than two-times for k T = 10Γ , eVg/G 0
B 0 B 0
although the enhancement of ZT for L = 3 is small at
FIG.1: FigureofmeritasafunctionofQDenergyleveltuned
k T = 1Γ . We observe several new spectral features
wBith simila0r ZT values at E values spaced apart ap- bygatevoltage(E0=EF +50Γ0−eVg)forleveldegeneracy,
proximately by the charging e0nergy U or U , which is L=1 and3. (a) kBT =1Γ0, (b)kBT =5Γ0, and(c) kBT =
0 I 10Γ0. The correction factor for phonon scattering, Fs =0.1.
caused by the intralevel and interlevel Coulomb interac-
We have adopted the intralevel Coulomb interaction U0 =
tions. For the non-degenerate case (L = 1), thermoelec- 60Γ0 for L = 1 and U0 = UI = 20Γ0 for L = 3. UI denotes
tric coefficients can be calculated in terms of the trans- theinterlevel Coulomb interaction.
mission coefficient T (ǫ), which can be expressed as
LR
To gain better understanding of the enhancement
T (ǫ) 1−N N
LR −σ −σ
4Γ Γ = (ǫ−E )2+Γ¯2 + (ǫ−E −U )2+Γ¯2, (15) mechanism for (ZT)max resulting from increased degen-
L R 0 0 0 eracy,wecalculatetheG ,S,κ andZT oftheQDjunc-
e e
whereΓ¯ =(ΓL+ΓR),andN−σ denotesthesingle-particle tion as functions of the level energy (∆ = E0 −EF) at
occupation number. Eq. (15) illustrates two resonant kBT =5Γ0forL=1andL=3andtheresultsareshown
peaks at ǫ = E0 and ǫ = E0 + U0 with the probabil- inFig.2. InFig. 2(a)themaximumGevalueisenhanced
ity weights of (1−N ) and N , respectively, which withincreasingofdegeneracy,althoughitsdependenceof
−σ −σ
are relatedto the two M-shapedspectralfeatures in ZT L is not linear.This is mainly attributed to complicated
(labeled by ǫ and ǫ ) with the dip position corre- correlation functions arising from the electron Coulomb
1,1 1,2
sponding to the resonance energies. (Here the intralevel interactions in QD. G is much smaller than G = 2e2
e 0 h
Coulomb interaction used is U = 60Γ ) Similarly, for (theelectronquantumconductance)evenforL=3,which
0 0
L=3 at k T =1Γ we label the six M-shaped spectral is mainly attributed to strong electronCoulombinterac-
B 0
5
tions. We note that the Seebeck coefficient is almost Because Eq. (16) does not take into account the in-
independent of L, whereas G and κ are enhanced with terlevel correlation functions arising from U , Eq. (16)
e e I
increasing L. However, since κ /κ ≫ 1, the L de- isnotadequateforillustratingthermoelectriccoefficients
ph e
pendence of κ won’t affect ZT appreciably. Thus, the for the situation ∆/k T ≤ 1. Nevertheless, we see that
e B
enhancement of ZT shown in Fig. 1(b) mainly comes (ZT) of Fig. 2 does not occur in the ∆/k T ≤ 1
max B
fromtheincreaseofG ,notS. Inthe Coulombblockade regime. Therefore, we consider the limit of weak cou-
e
regime, κ and G are highly suppressed. Thus, if one pling between QD and electrodes (Γ = Γ = Γ → 0)
e e L R
cfoarneixnatrmopdluec)e, tahmenecthhaenmismaxitmourmeduvcaeluκephof(ZwTithcaFns =rea0c.h1 in Eqs. (15) and (16) and obtain Ge = kBTGc0oΓsπhL2(P2Lk,∆B1T),
S = −∆/(eT), and κ = 0. The L-dependent behavior
1 for L=1 and around 2 for L=3 as illustrated in Fig. e
of (ZT) is then determined by the simple expression
2 (d). max
(∆/eT)2G ΓπLP T
0 L,1
ZT = , (18)
k Tcosh2( ∆ )k
2)/h 00..23 kBT=5G 0 (a) B 2kBT ph
(eG e00..01 D =E0-EF wbyhiGche reaxtphlaerintshathnaSt tahnedLw-hdyepZenTdaenptprZoaTchisesd0etaesrmEi0ne→d
)/eB --21 (b) EF forL=1. NotethatZT forL=3doesnotapproach
(S k -3 (zseereo tahse∆da→she0d, blienceauosfeFSig.ha2s(ba))fi.nitSeucvhaluaeraetsu∆lt =also0
)h (c)
/0 2 L=1 indicates that the correlationarising fromUI can not be
GB 1 L=3 neglected for QD when E is close to E . Some novel
k 0 F
( e 20 (d) nanoscale TE devices resulting from the inclusion of UI
kZT 1 Fk psh==0F.1sg0(T) wreecrteifitehrse3o8reatnicdalcluyrdreisnctudssioeddesfo.3r9designing electron heat
0
Figure 3 shows G , S, κ and ZT as functions of tem-
0 20 40 60 80 e e
D /G 0 perature with ∆ = 15Γ0 for L = 1,3, and 4. From
the application point of view, the temperature depen-
FIG. 2: (a) Electrical conductance (Ge),(b) Seebeck coeffi- dence of ZT is an important consideration for develop-
cient (S), (c) electron thermal conductance (κe) and (d) fig- ingroomtemperaturepowergeneratorsusedinconsumer
ure of merit (ZT) as a function of QD energy energy level electronics.2G ishighlyenhancedforL=3and4inthe
e
(∆ = E0 −EF) for different orbital degenerated states at whole temperature regime, but the difference of L = 3
kBT =5Γ0. Fs =0.1 was used in thecalculation of ZT. andL=4issmall,indicatingasaturationbehaviorasL
exceeds 3, mainly because of the factor P in Eq. (16).
L,1
The calculation of thermoelectric coefficients for the As seen in Fig. 3(b), S is nearly independent of L for
L = 3 case including all correlation functions arising k T < 7Γ , but becomes weakly dependent on L at
B 0
fromelectronCoulombinteractionsisquite complicated. higher temperature. This implies that the effect of res-
To reveal L-dependent (ZT)max, we consider the trans- onances at ǫL,2 and ǫL,3 can not be ignored for L > 1
missioncoefficientTLR(ǫ)includingonlythecontribution in the high temperature regime (kBT ≥7Γ0). Although
fromtheresonantchannelǫ−E0,whichisapproximately κe is enhanced with increasing L as shown in Fig. 3(c),
given by its effect is insignificant since κ is much smaller than
e
κ . Therefore, the behavior of ZT with respect to k T
4Γ Γ LP ph B
T (ǫ)≈ L R L,1 , (16) is determined by the power factor (PF = S2G T). We
LR (ǫ−E )2+Γ¯2 e
0 foundimpressiveenhancementofZT forL=3and4for
k T ≥4Γ . ComparingFigs. 2(d) and3(d), we see that
where P is the L-dependent probability weight for B 0
L,1
the maximum values of ZT occur near k T =∆/2.4 for
the resonant channel at ǫ = E . We have P = B
0 3,1
the tunneling rate considered (Γ = 1Γ ). Based on such
(1−N )(1−(N +N )+c)(1−(N +N )+c), 0
−σ −σ σ −σ σ
acondition,wecaninfer thatthe maximumZT atroom
where c=hn n i denotes the intraleveltwo particle
ℓ,σ ℓ,−σ
correlation function.38 temperaturekBT =25meV willoccurnear∆=60meV
for Γ = 1meV.
Thermoelectric coefficients determined by the T (ǫ) 0
LR
of Eqs. (15) and (16) can be calculated by G = e2L , In Figures (1)-(3), κph is assumed to obey the simple
e 0
S =−L /(eTL ) and κ = 1(L −L2/L ). L is given expression give in Eq. (14). Here we examine whether
1 0 e T 2 1 0 n thelargeenhancementofZT duetoleveldegeneracywill
by
be destroyed when a more realistic throughput function
2 ∂f(ǫ) given by Eq. (12) is considered. Fig. 4 shows the com-
L = dǫT (ǫ)(ǫ−E )n , (17)
n hZ LR F ∂EF parison of ZT and κph calculated by both Eq. (12) and
Eq. (14). In Fig. 4(a), κ used to obtain the solid
ph
where f(ǫ) = 1/(exp(ǫ−EF)/kBT +1) is the Fermi distri- (L = 3) and dashed curves (L = 1) for ZT are cal-
bution function of electrodes. culated by using Eqs. (11) and (12) with D = 4 nm,
6
2()e/h e00..12 (a) ZT 12 (D=Da1=)54Gnm0 LL==31:: L L==31
G 0.0
()S k/eB ---321 (b) D =15G 0; Fs=0.1 W/K) 11..280 (b) DD==1105 nnmm d=2 nm
G)/hk0B 011...505 (c) k(nph 00..06 D=20 nm
( e 0.02 (d) K) 0.02 (c) D=4 nm: Fs=0.1
k L=1; L=3; L=4 W/ k (L=3): k (L=1)
T 1 n 0.01 e e
Z (h
0 p0.00
1 5 10 15 20 25 30 k 0 25 50 75 100 125 150 175 200 225
kBT/G 0 Temperature (K)
FIG. 3: (a) Electrical conductance (Ge),(b) Seebeck coeffi- FIG. 4: (a) Figure of merit (ZT) and (b,c) phonon ther-
cient(S),(c)electronthermalconductance(κe)and(d)figure mal conductance (κph) as a function of kBT. The length of
ofmerit (ZT)asafunctionof kBT forvariousvaluesoflevel nanowire used is L˜ = 2000 nm. Other physical parameters
degeneracy (L) with ∆ = 15Γ0. Other physical parameters are thesame as those of Fig. 3.
are thesame as those of Fig. 2.
ZT due to increase of level degeneracy still exists in
δ = 2 nm and L˜ = 2µm. Other parameters are given the case of coupled double QDs (DQDs). The Hamil-
byphysicalpropertiesofsiliconsemiconductors.24,28The tonianofaDQDisgivenbyH =H +H +
DQD QD,L QD,R
triangle (L = 3) and square marks (L = 1) for ZT are ULR ℓ,jnL,ℓ,σnR,j,σ′+tLR ℓ,j(d†L,ℓ,σdR,j,σ+h.c).40−42
calculated by using κph based on Eq. (14). The maxi- H P (H ) denotes tPhe Hamiltonian of the left
QD,L QD,R
mum ZT values of solid and dashed lines are near 1.8 (right) QD as defined in Eq. (2). For simplicity, the
and 0.9, respectively. The results of Fig. 4(a) indicate interdot electron hopping strengths (t ) and electron
LR
that the large enhancement of ZT resulting from L is
Coulomb interactions (U ) are assumed uniform. Al-
LR
unchangedevenwhen a more realistic expressionfor κ
ph though electron tunneling currents through DQDs have
(which is nonlinear in temperature) is used. When we beenextensivelystudiedbyseveralauthors,40−42 theop-
compare the spectra of ZT given by the solid curve and
timization of ZT including the effect of all correlation
the curve with triangle marks for the case of L = 3,
functions arisingfromelectronCoulombinteractionshas
the curves with triangle marks have better ZT value at
not been reported. Here, we assume one nondegenerate
low temperature due to the lower value of κ in the
ph energy level for each QD (L = 1). The energy levels of
linear-T expression. Fig. 4(b) shows κ for nanowires
ph left QD andrightQD arethe same (denoted E0). Based
with diameter of 10,15, and 20 nm, which agree well
on Eqs. (4)-(6), G , S, κ and ZT as functions of the
e e
with the experimental results of Ref. 24, lending sup- QD energy level (which is related to the gate voltage
port for the validity of this model. The comparison of by E = E +50Γ −eV ) for K T = 3, 5, and 7 Γ
0 F 0 g B 0
behaviors of κ for a 4 nm nanowire obtained by both
ph are plotted in Fig. 5. There are four peaks labeled by
Eq. (12) (solid curve) and Eq. (14) (triangles) is shown
ǫ (n = 1,2,3,4) in the spectrum of electrical conduc-
1,n
in Fig. 4(c). It is found that the results obtained by
tance (G ). The Seebeck coefficient (S) behaves like the
e
the simple linear-T expressionofEq.(14)arefairlyclose
derivativeof−G andis vanishinglysmallatthe MPCG
e
to that obtained by the realistic expression of Eq. (12)
due to electron-hole symmetry. The maximum S occurs
for temperatures between 50K and 200K. In Fig. 4(c),
near the onset of the first peak in G or the ending of
e
κ shows a nonlinear-T behavior between 1K and 50,
ph the last peak. Both κ and G are symmetrical with re-
e e
whichismainlyattributedtofrequency-dependentmean
spect to MPCG. Similar to the single QD L = 1 case
freepath. Thedashedanddotedlinesshowthebehavior
in Fig. 1(d) the maximum ZT values occur at either the
of electronic thermal conductance (κ ) with respect to
e level-depletionregimeorthe full-chargingregimeasseen
temperature. Note that the G , S and κ in Fig. 4 are
e e in Fig. 5(d). Here, the maximum ZT is close to 1.7 at
calculated according to the simplified method described
k T =3Γ with F =0.1.
B 0 s
in Ref. 38, where we only considered single-particle oc- To further understand the relationship between phys-
cupation numbers and intralevel two-particle correlation ical parameters and thermoelectric coefficients, we con-
functions. The curves with triangle and square marks sider some approximationswhich include only the domi-
shownin Fig.4(a) arealmostidentical to the black solid nantcorrelationfunctionstoderiveT (ǫ)forDQDwith
LR
line and red dashedline of Fig. 3(d) obtained by the full L=1. Simple analytic expressions of G< (ǫ),Gr (ǫ) and
j,σ j,σ
calculation. Ga (ǫ)canbefoundinourpreviouswork42whenweonly
j,σ
Next we examine whether the large enhancement of include intradottwo-particle correlationfunctions in the
7
hn n i denote the two particle correlation functions
ℓ,σ¯ j,σ
2)e/h 00..12 (a) e 1,1 e 1,2 e 1,3 e 1,4 L=1 tainodnsh(ninℓ,cσ¯lnudj,iσnngj,bσ¯oiththientrtahdreoet-panardtiicnlteercdoortretleartmiosn). fNuontce-
( e MDCG that the probability weights satisfy the conservation law
G
0.0
)e 6 (b) mP1,m = 1. UL(R) and UL,R denote the intradot
/B 0 aPndinterdotCoulombinteractions. Whenallcorrelation
k
(S -1-62 functionsofDQDsareincludedasinRefs. 21and40,itis
)h 0.8 (c) kBT=3G 0 difficult to find an analytical expression for TLR(ǫ). The
/0 kT=5G thermoelectric coefficients, G , S, κ , and ZT obtained
GB 0.4 B 0 e e
k( keT 011...005 (d) ZTmax G LE kBtTLR=7GE0G R ZTmax bvaeryeruyosbginotgaoidEneqad.g(rb1ey9em)thaerenetfpuwllolittchtaeldtchuionlsaeFtiiosghn.o,6wi.nnTclihunedFirneiggsu.al5tl,lswcaorhreircienh-
Z 0.5 L R
0.0 lation functions. It should be noted that Eq.(19) works
0 25 50 75 100 125 150 175 200 225 250 275 well only in the limit t /k T ≪ 1. If one would like
eVg/G 0 LR B
to study the spin-dependent thermoelectric coefficients
of DQD in the low temperature regime (k T <t ), all
B LR
FIG. 5: (a) Electrical conductance (Ge), (b) Seebeck coeffi- correlations functions should be included.40
cient (S), (c) electron thermal conductance (κe) and (d) fig-
ure of merit (ZT) as a function of QD energy level tuned InFigs. 5and6,the peakpositionsǫ1,n (n=1,2,3,4)
by gate voltage (E0 = EF +50Γ0 −eVg) in a DQD junc- correspondtothechannelswithprobabilityweightsP1,1,
tion with L = 1 for various temperatures. We have con- P ,P andP ,respectively. Thereexistsaninterest-
1,3 1,6 1,8
sidered the electron hopping strength tLR = 1Γ0, interdot ing behaviorfor S at low temperature (kBT =3Γ0). We
Coulomb interaction ULR = 40Γ0 , intradot Coulomb inter- foundthatSisalinearfunctionofeV nearthemaximum
g
actions UL =UR =100Γ0, ΓL =ΓR =1Γ0 and Fs =0.1, of G . From the κ behavior shown in Figs. 5(c) and
e e
6(c), the electron heat flow is maximized near the mid
point between the first (or last) two resonant channels,
probabilityweights. Here,weincludealltwo-particleand
and it increases with increasing temperature. Eq. (19)
three-particle correlation functions. Then, the following
allows us to obtain an analytic form of thermoelectric
expressionofT (ǫ)is obtainedbysolvingthe hierarchy
LR coefficients, which is very useful for clarifying how ther-
ofequationsofmotion(whichterminatesatthe4-particle
moelectric coefficients are influenced by tunneling rates,
Green function) via a similar procedure as described in
Ref. 42. inter-dot hopping strength, and electron Coulomb inter-
actions. OuranalysisshowsthatS isindependentoft
LR
TLR(ǫ)/(4t2LRΓLΓR)= |µLµRP1−,1t2LR|2 a(2n1d)iatnhda(s2a2)l)inneeaarrdmepaexnimdeunmceZoTf.∆A=saEc0o−nsEeqFu(esnecee,Etqhse.
+ P1,2 trendofZT withrespectto tLR isdeterminedbythatof
|(µL−ULR)(µR−UR)−t2LR|2 Ge for κph/κe ≫1.
+ P1,3 (19)
|(µL−ULR)(µR−ULR)−t2LR|2
++ ||((µµLL−−2UULL)(Rµ)RP(µ1−R,5UP−1L,UR4L)R−−t2LURR|2)−t2LR|2 2()Ge/he0001...0122 (a) e 1,1 e 1,2 e 1,3 e 1,4 L=1
+ P1,6 )/eB 06 (b)
|(µL−UL−ULR)(µR−UR−ULR)−t2LR|2 (Sk-1-26
+ |(µL−UL−ULR)P(µ1,R7 −2ULR)−t2LR|2 )/h000..48 (c) kkBTT==37GG 0; kBT=5G 0
+ |(µL−UL−2ULR)(µPR1,−8 UR−2ULR)−t2LR|2, TkG(keB011...005 (d) ZTmax B 0 ZTmax
Z 0.5
0.0
where µ = ǫ − E + iΓ and µ = ǫ − E + iΓ .
L L L R R R 0 25 50 75 100 125 150 175 200 225 250 275
The probability weights are given by P1,1 = 1−NL,σ¯ − eVg/G 0
N −N +hn n i+hn n i+hn n i−
R,σ¯ R,σ L,σ¯ R,σ¯ L,σ¯ R,σ R,σ¯ R,σ
hnL,σ¯nR,σ¯nR,σi,P1,2 =NR,σ¯−hnL,σ¯nR,σ¯i−hnR,σ¯nR,σi+ FIG. 6: (a) electrical conductance (Ge), (b) Seebeck co-
hnL,σ¯nR,σ¯nR,σi,P1,3 =NR,σ−hnL,σ¯nR,σi−hnR,σ¯nR,σi+ efficient (S), (c) electron thermal conductance (κe) and (d)
hn n n i, P = hn n i − hn n n i, figureofmerit(ZT)asafunctionofQDenergyleveltunedby
L,σ¯ R,σ¯ R,σ 1,4 R,σ¯ R,σ L,σ¯ R,σ¯ R,σ
P1,5 =NL,σ¯−hnL,σ¯nR,σ¯i−hnL,σ¯nR,σi+hnL,σ¯nR,σ¯nR,σi, gatevoltage(E0=EF+50Γ0−eVg)inaDQDjunctionwith
P =hn n i−hn n n i,P =hn n i− L=1forvarioustemperaturescalculatedbyusingEq. (19).
1,6 L,σ¯ R,σ¯ L,σ¯ R,σ¯ R,σ 1,7 L,σ¯ R,σ
hn n n i, and p = hn n n i, where Otherphysical parameters are the same as those of Fig. 5.
L,σ¯ R,σ¯ R,σ 1,8 L,σ¯ R,σ¯ R,σ
8
Next we consider the case with two-fold degenerate From Eqs. (21) and (22), we have G = e2L and
e 0
levels (L = 2) for each QD in the DQD junction. Such S = −∆/(eT). This reveals the behavior of S around
two-folddegeneracycanberealizedinQDswithsuitable the maximum of G at k T = 3Γ in Fig. 6(b) and L-
e B 0
symmetry. For example, the x- and y-like states in a dependent ZT determined by G in Fig. 7(d). Note
max e
disk-shaped QD are degenerate. Due to symmetry, the that if we artificially set F = 0 (i.e. neglecting κ ),
s ph
intradot electron hopping process is prohibited, whereas one can prove that the enhancement of ZT arising
max
the interdot electron hopping strength is nonzero. We from L will disappear due to the L-independence of the
assume t = 1Γ , the same as that of the L = 1 ratio G /κ and L-independence of S. If we choose a
LR 0 e e
case. The intradotCoulombinteractions are takento be much higher value of F (e.g. F = 1) in Eq. (14), we
s s
U =U =U =50Γ , where i,j =1,2 denote the will have the condition κ /κ ≫1. In this situation, L
L,i,j R,i,j I 0 ph e
two degenerate levels within the same QD. The interdot dependence of ZT is fully determined by G and we
max e
CoulombinteractionistakenasU =40Γ . Thecalcu- haveZT linearlyproportionaltoL,sinceP inEq.(18)
L,R 0 L,1
lation of thermoelectric coefficients of DQD with L = 2 is close to 1 under the condition (E −E )/k T ≫ 1,
0 F B
involvessolvingone-,two-,···,uptoeight-particleGreen andtheZT valueswillbe approximatelyproportionalto
functions. Due to the presence of t term, the numer- 1/F . Finally, we would like to point out that QD junc-
LR s
ical procedure is much more complicated than that of a tions embedded in a silicon nanowire can be realized by
single QD with L = 4. Based on Eqs. (4)-(6), we cal- the advanced technique reported in Refs. 45 and 46.
culate the thermoelectrical coefficients of DOD for the
L = 2 as functions of the QD energy level as shown
itnheFliegf.t7h.anTdhseidfiersotffMouDrCrGes)onaarnetacphparnonxeimlsaoteflyGegiv(oenn 2)e/h 00..23 (a)L=2 e 2,1 e 2,2 e 2,3 e 2,4
by ǫ2,1 = E0, ǫ2,2 = E0 +ULR, ǫ2,3 = E0 +ULR +UI, (G e0.1 MDCG
and ǫ = E +2U +U , in which the small t is 0.0
neglec2t,e4d sinc0e tLRL≪R UL,RI. The oscillatory behLaRvior )/eB 06 (b)
of Ge displayed in Fig. 7(a) is similar to the Ge spec- (S k -1-26
tsruareombesnetrsveodf PexbpSeeriQmDen(twalhlyichinhatusnanesliixn-gfolcdurdreegnetnemraeate- )/h0 01..82 (c) kkBTT==35GG 0
GB B 0
excited state) and carbon nanotube QD (which has an ( ke 00..04 kBT=7G 0
eight-fold state).43,44 Although the S spectrum exhibits k 2.4 (d) ZTmax
1.6
morebipolaroscillatorystructures,themaximumSvalue T
Z 0.8
does not increase with increasing L. This feature is the 0.0
sameasthatofasingleQDcase. Fig.7(d)showsanlarge 0 25 50 75 100 125 150 175 200 225 250 275
enhancement of maximum ZT arising from the degener- eVg/G 0
acy effect. In the current case, ZT reaches around
max
2.7. Comparing Fig. 7(d) with Fig. 5(d), we see an en- FIG. 7: (a) Electrical conductance (Ge), (b) Seebeck coeffi-
hancementofmaximumZT fromaround1.7to2.7when cient(S),(c)electronthermalconductance(κe)and(d)figure
ofmerit(ZT)asafunctionofQDenergyleveltunedbygate
Lincreasesfrom1to2forDQDjunctionwhenF =0.1.
s voltage (E0 = EF +50Γ0 −eVg) in a DQD junction with
We expect even larger enhancement to occur for higher
L = 2 for various temperatures. UL,ℓ,j = UR,ℓ,j = 50Γ0 and
level degeneracy. Unfortunately, the computation effort
ULR =40Γ0. Otherphysicalparametersarethesameasthose
for L > 2 for a DQD junction is prohibitively large if of Fig. 5.
all Green functions and correlations functions are to be
included.
ToclarifythebehaviorofZT inthelevel-depletion
max
regime (with N < 1), we can approximately write (by
t IV. CONCLUSION
keeping only the dominant channel)
4Γ Γ t2 L P We have theoretically investigated the effects of level
T (ǫ)≈ L R LR L,1 , (20)
LR |(ǫ−E +iΓ )(ǫ−E +iΓ )−t2 |2 degeneracy on thermoelectric properties of QDs embed-
0 L 0 R LR
dedinathinnanowirejunctionintheCoulombblockade
wherePL,1istheprobabilityweightforDQDinthelevel- regime. All the correlation functions arising from elec-
depletionregime. UndertheassumptionofΓL =ΓR →0 tronCoulombinteractionsforelectronsinthedegenerate
and tLR/kBT ≪1, we have levelsare included in ourcalculation. We found that the
maximumvaluesofZT canbehighlyenhancedwithlevel
2 πΓ t2 LP 1
L = LR L,1 (21) degeneracy under the typical condition with κ much
0 hkBT (4t2LR+Γ2) cosh2(2k∆BT) largerthanκe. When(E0−EF)/kBT ≫1,S isipnhdepen-
dent on L. Therefore, the enhancement of ZT in the
and max
level-depletionregimeismainlyattributedtotheincrease
2 πΓ t2 LP ∆
L = LR L,1 . (22) of Ge. Large enhancement of ZT due to the increase of
1 hkBT (4t2LR+Γ2) cosh2(2k∆BT) level degeneracy is also found in the presence of finite
9
electron hopping in coupled QD system. In our stud- coupled QDs1,2 and molecular QDs.13
ies, we assumed a simple expression κ = F g (T) for Acknowledgments
ph s 0
the phonon thermal conductance. However, it is worth This work was supported under Contract Nos. MOST
pointingoutthatourconclusiononthe effectoflevelde- 103-2112-M-008-009-MY3 and MOST 104-2112-M-001-
generacy on ZT is not limited to the linear T-behavior 009-MY2.
of κ (as illustrated in Fig. 4). The enhancement due E-mail address: [email protected]
ph
to level degeneracy holds as long as κ >κ , regardless E-mail address: [email protected]
ph e
ofthe temperaturedependance ofκ . Thisimplies that
ph
the design principle based increasing level degeneracy is
applicable for a composite materials involving arrays of
1 G.Chen,M.S.Dresselhaus, G.Dresselhaus,J.P.Fleurial 23 D. Li, Y. Y. Wu, R. Fan, P. D. Yang, and A. Majumdar
and T. Caillat, International Materials Reviews, 48, 45 Appl.Phys. Lett. 83, 3186 (2003).
(2003). 24 R. K. Chen, A. I. Hochbaum, P. Murphy,J. Moore, P. D.
2 A.J.Minnich,M.S.Dresselhaus,Z.F.RenandG.Chen, Yang, and A. Majumdar, Phys. Rev. Lett. 101, 105501
Energy Environ Sci, 2, 466 (2009). (2008).
3 R. Venkatasubramanian, E. Siivola,T. Colpitts,B. 25 M. C. Wingert, Z. C.-Y. Chen, E. Dechaumphai,J.
O’Quinn,Nature 413, 597 (2001). Moon,Ji-Hun Kim,J. Xiang, and R. K. Chen, Nano Lett,
4 A.I.Boukai,Y.Bunimovich,J.Tahir-Kheli,J.K.Yu,W. 11, 5507 (2011).
A.GoddardIIIandJ.R.Heath,Nature,451,168(2008). 26 A.I.Hochbaum,R.K.Chen,R.D.Delgado,W.Liang,E.
5 T. C. Harman, P. J. Taylor, M. P. Walsh, B. E. LaForge, C. Garnett, M. Najarian, A. Majumdar and P. D. Yang,
Science 297, 2229 (2002). Nature, 451, 163(2008)
6 Y.M.LinandM.S.Dresselhaus,Phys.Rev.B68,075304 27 N. Mingo and L. Yang,Phys. Rev.B 68, 245406 (2003).
(2003). 28 P. G. Murphy and J. E. Moore, Phys. Rev. B 76, 155313
7 K. F. Hsu,S. Loo,F. Guo,W. Chen,J. S. Dyck,C. Uher, (2007).
T.Hogan,E.K.Polychroniadis,M. G.Kanatzidis,Science 29 D. Donadio and G. Galli, Phys. Rev. Lett. 102, 195901
303, 818 (2004). (2009).
8 A. Majumdar, Science 303, 777 (2004). 30 Gursoy B. Akguc and Jiangbin Gong, Phys. Rev. B, 80,
9 C.R.KaganandC.B.Murray,Naturenanotechnolgy,10, 195408 (2009).
1013 (2015). 31 T. M. Gibbons, By. Kang,S. K. Estreicher and C. Car-
10 L. D. Hicks , and M. S. Dresselhaus, Phys. Rev. B 47, bogno, Phys.Rev.B 84, 035317 (2011).
12727(1993). 32 E. B. Ramayya, L. N. Maurer, A. H. Davoody, and I.
11 L.D.Hicks,andM.S.Dresselhaus,Phys.Rev.B47,16631 Knezevic, Phys.Rev.B 86, 115328 (2012).
(1993). 33 T. Zhu and E. Ertekin.Phys. Rev.B 93, 155414 (2016).
12 D.L.Nika,E.P.Pokatilov,A.A.Balandin,V.M.Fomin, 34 K. Schwab, E. A. Henriksen, J. M. Worlock and M. L.
A. Rastelli, and O. G. Schmidt, Phys. Rev. B 84, 165415 Roukes,Nature, 404, 974 (2000).
(2011). 35 Y.Zhang,M.S.Dresselhaus,Y.Shi,Z.RenandG.Chen,
13 P. Murphy, S. Mukerjee and J. Moore, Phys. Rev. B 78, Nano. Lett. 11, 1166 (2011).
161406(R) (2008). 36 T. Zhu and E. Ertekin.Phys. Rev.B 90, 195209 (2014).
14 P.TrochaandJ.Barnas, Phys.Rev.B85,085408 (2012). 37 T. Zhu and E. Ertekin.Nano Lett. 16. 4763 (2016).
15 R. Q. Wang, L. Sheng, R. Shen, B. G. Wang, and D. Y. 38 D.M.-T. Kuoand Y.C. Chang, Phys.Rev.B 81, 205321
Xing, Phys. Rev.Lett. 105, 057202 (2010). (2010).
16 G. Mahan, B. Sales and J. Sharp, Physics Today, 50, 42 39 B. Sothmann, R. Sanchez and A. N. Jordan, Nanotech-
(1997). nolgy, 26, 032001(2015).
17 H. Haug and A. P. Jauho, Quantum Kinetics in Trans- 40 B. R. Bulka and T. Kostyrko, Phys. Rev. B 70, 205333
port and Optics of Semiconductors (Springer, Heidelberg, (2004).
1996). 41 B. Muralidharan and S. Datta, Phys. Rev. B 76, 035432
18 S. Datta, Electronic Transport in Mesoscopic Systems (2007).
(Cambridge UniversityPress, Cambridge U.K. (1995)). 42 D. M.-T. Kuo, S. Y. Shiau and Y. C. Chang, Phys . Rev.
19 Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, B 84, 245303 (2011).
2512(1992). 43 P. Liljeroth, L. Jdira, K. Overgaag, B. Grandidier, S.
20 C.C.Chen,Y.C.Chang,andD.M.-T.Kuo,Phys.Chem. SpellerandD.Vanmaekelbergh,Phys.Chem.Chem.Phys.
Chem.Phys.17,6606(2015);C.C.Chen,PhDthesis,Uni- 8, 3845 (2006).
versityofIllinoisatUrbana-Champaign(2015)availableat 44 S. Moon, W. Song, J. S. Lee, N. Kim, J. Kim, S. G. Lee
https://www.ideals.illinois.edu/handle/2142/88937. and M. S.Choi, Phys. Rev.Lett. 99, 176804 (2007).
21 C.C.Chen,D.M.-T.KuoandY.C.Chang,Phys.Chem. 45 G. L. Chen, David MT Kuo, W. T. Lai and P. W. Li,
Chem. Phys. 17, 19386 (2015). Nanotechnology 18, 475402 (2007).
22 D. Li, Y. Y. Wu, P. Kim, L. Shi, P. D. Yang, and A. 46 J. W. Lee, J. Lee, S. H. Jung, Y. Jang, B. L. Choi, C.
Majumdar, Appl. Phys.Lett. 83, 2934 (2003). W. Yang, D. Whang and E. K. Lee, Nanotechnology 27,
10
305703 (2016).