Efficiency fluctuations in quantum thermoelectric devices

0.27 MB·English

by  Massimiliano Esposito

Efficiency fluctuations in quantum thermoelectric devices
Massimiliano Esposito
Complex Systems and Statistical Mechanics, Physics and Material
Science Research Unit, University of Luxembourg, Luxembourg
Maicol A. Ochoa and Michael Galperin
Department of Chemistry & Biochemistry, University of California San Diego, La Jolla CA 92093, USA
5
(Dated: March 19, 2015)
1
0 We present a method, based on characterizing efficiency fluctuations, to asses the performance
2 of nanoscale thermoelectric junctions. This method accounts for effects typically arising in small
junctions,namely,stochasticityinthejunction’sperformance,quantumeffects,andnonequilibrium
r
a features preventing a linear response analysis. It is based on a nonequilibrium Green’s function
M (NEGF)approach,whichweusetoderivethefullcountingstatistics (FCS)forheatandwork,and
whichinturnallows ustocalculate thestatistical propertiesofefficiencyfluctuations. Wesimulate

8 thelatterforavarietyof simplemodels whereourmethodisexact. Byanalyzingthediscrepancies
1 withthesemi-classicalpredictionofaquantummasterequation(QME)approach,weemphasizethe
quantum nature of efficiency fluctuations for realistic junction parameters. We finally propose an

] approximateGaussianmethodtoexpressefficiencyfluctuationsintermsofnonequilibriumcurrents
l
l and noises which are experimentally measurable in molecular junctions.
a
h
PACSnumbers: 05.70.Ln85.65.+h85.80.Fi84.60.Rb
-
s
e
m I. INTRODUCTION Due to the smallsize ofthe system,thermalfluctuations
will play a much more import role than in bulk samples,
.
t
a The development of thermoelectric materials is at the resulting in a high variability in the junction’s perfor-
m mance. This variability requires a statistical characteri-
forefrontoftheresearchrelatedtoenergyconversionand
zation of the energy conversion which can be performed
- storage. While researchonthermoelectricityinbulkma-
d usingthemethodsofstochasticthermodynamics[44–47].
terials goes back to the middle of the last century [1],
n Such studies have been recently done for small classical
o measurements at the nanoscale (and in particular, stud-
energyconverters [48–53]. Themainideaistodefinethe
c iesofthermoelectricityinmolecularjunctions)wereonly
efficiency along a single realization of the operating de-
[ reported recently [2]. The small size of the junctions
vice and to develop techniques to study its fluctuations.
gives rise to new physical phenomena, not accessible at
2 Experimental studies of efficiency fluctuation have been
v the macroscopic level, and which are considered promis-
very recently performed in Ref. [54].
2 ing for reaching more effective energy conversion. The
3 thermoelectric properties of nanoscale junctions havein- The third centralfeature of smallthermoelectric junc-
2 deed received a lot of attention in the last years, both tions which needs to be accounted for are quantum ef-
3
experimentally [3–14] and theoretically [15–29]. fects. Indeed, quantum coherences can significantly af-
0
. Experimentalstudiesonthermoelectricityinnanoscale fect charge and energy transfers in molecular junctions
1 junctions makeuse ofthe macroscopictheoryofthermo- as discussed theoretically in Refs [55–60] and shown ex-
0 electricitytoassesthejunction’sperformance. Thelatter perimentally in Refs. [61–66].
5
1 is characterized by the figure of merit, a quantity exclu- In this paper we provide a general method to study
: sively defined in terms of linear response transport coef- the performance of nanoscale thermoelectric junctions
v
ficientsandthusill-definedoutofnonequilibrium. While based on efficiency fluctuations. This methods accounts
i
X the linear theory is reasonable in bulk material, it fails for the three key features characterizingsmall junctions,
r in small thermoelectric junctions which can operate in namely, the variability in performance due to fluctua-
a the nonlinear regime (for instance in the resonant tun- tions, operation modes arbitrary far from equilibrium,
neling regime). This fact motivated a number of studies and quantum effects. It is based on the joint energy and
to consider the macroscopic efficiency of the junction as particlefull counting statistics(FCS) whichwe calculate
an alternative to the figure of merit to characterizes the withinthenonequilibriumGreen’sfunctions(NEGF)for-
performanceofthejunction[30–43]. Themacroscopicef- malism. While the quantum FCS of particle currents
ficiency is the traditional thermodynamic efficiency of a (e.g. electrons) in junctions is well developed [67–78],
heatengine defined as the fractionofaveragepowerout- that of energy was mostly limited to the quantum mas-
put extracted from the heat arising from the hot source. terequation(QME)approach [71,79–84]withitsknown
Itiswelldefinedfarfromequilibriumandupperbounded limitations [76, 85, 86]. By numerically calculating effi-
by the Carnot efficiency. ciency fluctuations for a set of simple models and com-
Thenonequilibriumfeaturesofthejunctionarenotthe paring our NEGF results with those obtained using a
only characteristic to be accounted for at the nanoscale. QME approach, we identify the regimes where efficiency
2
fluctuations display truly quantum features. Moreover, (a)
we propose an approximate Gaussian scheme enabling /2
to estimate efficiency fluctuations solely based on exper-
imentally measurable quantities in molecular junctions
[87–92], namely the nonequilibrium energy and matter
currents and noises.
+
Thestructureofthepaperisthefollowing. Afterintro-
- /2
ducing the FCS of energy, work, and heat within NEGF
in Section II, we consider efficiency fluctuations in Sec-
tionIII.InSectionIV,wenumericallyevaluateefficiency
(b)
fluctuationsforvariousmodels,compareourresultswith
theQMEapproachanddescribetheapproximatescheme
to estimate efficiency fluctuations experimentally. We L R
summarize our findings in section V.
T T
L R
M
II. FCS OF PARTICLE AND ENERGY FLUXES
The particle FCS for a single level strongly coupled to
Fermi reservoirswas derived in Ref. [69] and generalized
to amultilevelinteractingsysteminRef.[75]. Later,the
methodology was applied to describe inelastic transport
in junctions in Ref. [77], where the role of quantum co-
FIG.1: (Coloronline)(a)Sketchofthecountingfieldλdress-
herence on the FCS was discussed.
ing of the Keldysh contour forward (−) and backward (+)
Here we extend the methodology to count particles
branches. (b) Sketch of the a nano-thermoelectric junction
andenergyinasystemstronglycoupledtoitsreservoirs.
consisting of a molecule M embedded between two contacts
Similar to the particle FCS [71], the treatment starts L and R with TL <TR and µL >µR.
by dressingthe evolutionoperator,Uˆ(t,t′) with particle,
γP, andenergy, γE, counting fields atinterface K ofthe
K K
junction <(>)
andG isthelesser(greater)projectionsoftheGreen
λ
function obtained from a counting field dressed version
Uˆγ(t,t′)=e−iγKPNˆKe−iγKEHˆKUˆ(t,t′)e+iγKPNˆKe+iγKEHˆK of the Dyson equation (see e.g. Ref. [77] for details).
(1)
While the expression for the derivatives of the adia-
Note that [Nˆ ;Hˆ ]=0. The counting fields depend on
K K baticpotentialinthecountingfieldscanbeeasilyformu-
the Keldysh contour branch (see Fig. 1a)
lated in terms of the field-dressed Green functions and
self-energies(see Eq.(3) aboveor Ref. [77] for a detailed
γ = +λPK/2 at − γE = +λEK/2 at − (2) discussion), the corresponding expression for the adia-
K (−λPK/2 at + K (−λEK/2 at + batic potential itself is more complicated. An explicit
expression for the adiabatic potential within the NEGF
Here−and+arethetimeorderedandanti-timeordered based particle FCS for a single noninteracting level was
branches of the contour, respectively. derived in Ref. [69]. Exact results for the particle FCS
Following the procedure outlined in Refs. [69, 77], at for one-dimensional tight-binding junction models were
steady state we get the following expression for deriva- presented in Ref [72]. Here we consider the case of a
tivesofthecumulantgeneratingfunction,S =−i(t −t ) multilevel noninteracting system. In particular,we show
f i
U (here U is the adiabatic potential), in the counting that for a single level coupled to its reservoirs (and pos-
fields λM (M =P,E) sibly also to other levels with the latter not coupled to
K
reservoirs)orforamultilevelsystemcoupledtoreservoirs
∂ dE
U(λP,λE)=− O Iλ(E) (3) throughsingle molecularorbitals,the explicit expression
∂λM K K 2π M K fortheadiabaticpotentialinthepresenceofbothparticle
K Z
and energy counting fields is (see Appendix for details)
where O =1(E) for M =P(E), and
M
dE
Iλ(E)≡Tr Σ<(E)ei(λPK+EλEK)G>(E) (4) U({λ})=i ln 1+T(E)
K K λ 2π
Z (cid:18)
n−G<λ(E)Σ>K(E)e−i(λPK+EλEK) × fL(E)[1−fR(E)][e+i(λPL−λPR+E(λEL−λER))−1] (5)
o n
is the energy resolved dressed particle current at inter- +fR(E)[1−fL(E)][e−i(λPL−λPR+E(λEL−λER))−1]
face K, Tr{...} is the trace over the system subspace,
o(cid:19)
3
where W˙ =(µ −µ )IP, and the average heat taken from the
L R
hotreservoirwhichfuels the device, Q˙ =−(IE−µ IP),
T(E)≡Tr{ΓL(E)Gr(E)ΓR(E)Ga(E)} (6) namely η¯ = W˙ /Q˙. It is upper bounded by the CRarnot
efficiency η¯ ≤ 1−T /T . The fluctuating efficiency on
is the Landauer transmission coefficient at energy E. L R
theotherhandisdefinedastheratiobetweenthefluctu-
Here Gr(a)(E) are the retarded (advanced) projections
ation power w/t and heat flow q/t measured at the level
of the system Green function in absence of the counting
of a single experiential realization of duration t, namely
fields, and Γ (E) is the electron dissipation matrix at
K η = w/q. Efficiency fluctuations are not bounded and
energyE due to coupling to contactK (K =L,R). The
arecharacterizedbytherateJ(η)atwhichtheprobabil-
size of the matrix is that of the molecular subspace of
ity to observe a given efficiency η decays during a long
the problem. Below we consider systems for which ex-
measurement realization [48, 49]
pression (5) is satisfied. In these systems, the electron
dissipationratematricesarealwaysdiagonalinthe local t→∞
P(η) = exp{−J(η)t}. (13)
basis. We denote by Γ and Γ the parameters charac-
L R
terizing the electron escape rates into the left and right This rate is called the large deviation function (LDF) of
contact, respectively. efficiency. Itcanbederivedfromthe heatandworkFCS
The particle and energy average currents and noises obtained from the energy and heat FCS (5) as follows.
can be directly obtained from the adiabatic potential U, The heat entering the system from the hot (cold) reser-
Eq. (5), as voir is given by the right (left) energy current minus µ
R
(µ ) times the right (left) particle current. At steady-
IKM =−∂λMU|{λ}=0 (7) staLte, the particle and energy currents are equal (but
K
SM1M2 =i∂ ∂ U| , (8) with opposite signs) at the two interfaces. Therefore, by
K1K2 λMK11 λMK22 {λ}=0 the first law of thermodynamics, the work generated by
the particles moving across the system is equal to the
where K = L,R and M = P,E. Explicitly, the average
sum of the heat from the left and right reservoir which
currents read
is thus µ −µ multiplied by the right particle current.
R L
IM ≡IM =−IM (9) ThismeansthatifλQ countstheheatfromthehotreser-
L R
voirandifλ countsthework,wegetthattheheatand
dE W
= O T(E)[f (E)−f (E)] work FCS reads
M L R
2π
Z dE
U =i ln 1+T(E) (14)
while the noises read 2π
Z (cid:18)
SM1M2 ≡SLML1M2 =SRMR1M2 =−SLMR1M2 =−SRML1M2 × fL(E)[1−fR(E)][e−i([E−µR]λQ−[µL−µR]λW)−1]
(10)
n
=SM1M2 +SM1M2, +fR(E)[1−fL(E)][e+i([E−µR]λQ−[µL−µR]λW)−1] .
shot therm
o(cid:19)
where the shot and the thermal (equilibrium) noise re- Introducingtheslightlymodifiedversionoftheadiabatic
spectively read potential, φ≡−iU, and redefining the counting fields as
γ ≡iλ and λ≡iλ , we get that
W Q
dE
SM1M2 = O O (11)
shot 2π M1 M2 dE
Z φ(γ,λ)= ln 1+T(E)
×T(E) (1−T(E))[fL(E)−fR(E)]2 Z 2π (cid:18)
SM1M2 = dE O O (12) × fL(E)[1−fR(E)][e−([E−µR]λ−[µL−µR]γ)−1] (15)
therm 2π M1 M2 n
KX=L,RZ +f (E)[1−f (E)][e+([E−µR]λ−[µL−µR]γ)−1] .
R L
×T(E)f (E)[1−f (E)].
K K o(cid:19)
Note that the fluctuation theorem symmetry holds
Expressions (7)-(12) are exact for any non-interacting
system bi-linearly coupled to two contacts. 1 1 1
φ γ,λ =φ − −γ, − −λ (16)
T T T
(cid:18) (cid:19) (cid:18) L R L (cid:19)
III. EFFICIENCY FLUCTUATION as can be verified using the property
In order to operate as a thermoelectric junction the fR(E)[1−fL(E)]e(cid:16)[E−µR](T1R−T1L−λ)−[µL−µR](−T1L−γ)(cid:17)
asmndallRquwaitnhtuTms<ystTemaisndemµbed>deµdbe(steweeeFnigt.w1obl)e.adTshLe =fR(E)[1−fL(E)]e−E−TLµL+E−TRµR e−([E−µR]λ−[µL−µR]γ)
L R L R (17)
macroscopicefficiencyofsuchajunctionisdefinedasthe
ratiobetweentheaveragepowergeneratedbythedevice, ≡f (E)[1−f (E)]e−([E−µR]λ−[µL−µR]γ).
L R
4
The efficiency LDF is finally obtain by setting λ = ηγ
-7
10
and minimizing φ relative to the field γ, namely [48, 49]
J(η)=−minφ(γ,ηγ). (18)
γ 3
Theconvexityof(15)togetherwiththefluctuationtheo-
remsymmetry (16)has been used inclassicalsystems to )
2
prove two important results. First, the single minimum J(
in J(η) (i.e. the most probable efficiency) corresponds
to the macroscopic efficiency η¯, second, the single maxi-
1
mum in J(η) (i.e. the least likely efficiency) corresponds L t R
1 2
to the Carnot efficiency 1−T /T [48, 49]. By showing
L R
thatthefluctuationtheoremsymmetry(16)holdsforthe
adiabaticpotentialofquantumjunctions, wethus gener- -2 -1 0 1 2
alized these remarkable results to the quantum realm.
In the limit of weak system-lead coupling, Γ ≡ Γ +
L
Γ →0, Eq. (15) reduces to the QME approach predic-
R
tion [71, 93] FIG. 2: (Color online) Efficiency LDF for a two-level bridge
calculated within the NEGF (solid line, blue) and the QME
φ(γ,λ)= (19) (dashed line, red) approaches. See text for parameters.
Γ (E ) Γ (E ) 2
− s s + s s +ΓL(E )ΓR(E )
2 2 s s s s
s (cid:18) (cid:20)(cid:18) (cid:19) Unless specified otherwise, the parameters of the cal-
X
× fL(Es)[1−fR(Es)][e−([Es−µR]λ−[µL−µR]γ)−1] culations are TL = 100 K, TR = 600 K, µL = 0.02 eV
andµ =0. Weusethewidebandapproximationwhich
R
n 1/2 assumes that the electron escape rates Γ and Γ are
+f (E )[1−f (E )][e+([Es−µR]λ−[µL−µR]γ)−1] . L R
R s L s energy-independent constants. The NEGF calculations
o(cid:21) (cid:19) were performed on an energy grid spanning the region
Here ... is the sum over the eigenorbitals of the from −1 to 1 eV with step 10−5 eV.
s
system with eigenenergies E , and Γ (E ) ≡ ΓL(E )+ We start by considering the two-level bridge model
s s s s s
ΓR(EP) is the total escape rate from the eigenorbital s depicted in inset in Fig. 2 when the system is weakly
s s
evaluatedatenergyofthetheorbital. Thequasi-classical coupled to the contacts. The position of the levels is
nature ofthis resultis manifestsince Eq.(19) disregards ε = ε = 0.1 eV, the electron hopping parameter is
1 2
the reservoir induced correlations between the eigenor- t=0.05eV,andtheelectronescaperatesareΓ =Γ =
L R
bitals of the system. This form of adiabatic potential 2·10−4 eV. As expected, in this regime both the NEGF
was used in Refs. [48, 49] together with (18) to calculate andtheQMEpredictionsfortheefficiencyfluctuationco-
efficiency fluctuations in a photoelectric device. incide(comparethe solidandthe dashedlines inFig.2).
Large values of J(η) indicates unlikely efficiency fluctu-
ations while the minimum is the most likely efficiency
IV. NUMERICAL EXAMPLES η¯correspondingto the macroscopicefficiency considered
intraditionalthermodynamics. Althoughhardlyseenon
We now compare the efficiency fluctuations (18) pre- this figure, the most unlikely efficiency is located at the
dicted using the NEGF heat and work FCS (15) with Carnotefficiency 1−TL/TR ≈0.83. The probabilitydis-
the QME prediction (19). Since we exclusively consider tribution in this regime is thus quite narrowly centered
non-interacting models, we emphasize that the NEGF around the most likely efficiency.
treatmentisexactwhiletheQMEapproachisanapprox- We consider two types of junctions, a two-level bridge
imate approach only valid in the weak coupling limit to (topinsetinFig.3)andasingleleveljunctioncoupledto
the contact and which neglects coherences between sys- an isolated orbital (bottom inset in Fig. 3). Both junc-
tem eigenstates (we use the rotating wave approxima- tionsareinregimeswherethesystemisstronglycoupled
tion to guarantee positivity). The discrepanciesbetween to the contacts. The latter is the simplest model often
these two approachescanthus be attributed to broaden- used to describe destructive interference effect in trans-
ing effects induced by strong coupling and to eigenbasis portthroughajunction(seee.g. Ref.[76]). Theposition
coherences. of the levels is ε = ε = 0.12 eV, the electron hopping
1 2
The calculations are performed by numerically evalu- parameter is t = 0.05 eV, and the electron escape rates
ating the adiabatic potential φ(γ,ηγ) (using Eq.(15) for are Γ = Γ = 0.1 eV. Figure 3a shows that QME re-
L R
the NEGF and Eq.(19) for the QME) and numerically sults of the two models are identical. This result stems
minimizing it as a function of the counting field γ for a from the fact that in the rotating wave approximation,
fixed value of the efficiency η according to Eq.(18). the QME neglects coherences in the system eigenbasis
5
-5
10
-5
10
(a)
15
10
) )
10
( (
J J
5
5
L t R
1 2 (a)
QME QME
-5
10
-5
2 10 t 3 s
L R
t 1 t 2 t 4
10 L 1 R
) ) 6
( (
J J
5
2 (b)
(b)
NEGF NEGF
-2 -1 0 1 2 -2 -1 0 1 2
FIG. 3: (Color online) Efficiency LDF for a two-level bridge FIG. 4: (Color online) Efficiency LDF for a donor-bridge-
(top inset; solid line, blue) and a single level coupled to an acceptor junction calculated within the (a) QME and (b)
isolated orbital (bottom inset; dashed line, red), calculated NEGF approaches. Results are shown for constructive in-
withinthe(a)QMEand(b)NEGFapproaches. Thevertical terference (s=t; solid line, blue), single path (s=0; dashed
dashed line shows the Carnot efficiency. See text for param- line, green), and destructive interference (s = −0.8t; dotted
eters. line, red). The vertical dashed line shows the Carnot effi-
ciency. Otherparameters are as in Fig. 3.
[76, 86]. Fig. 3b shows the exact efficiency fluctuations
for the two models. The interference effects responsible efficiency fluctuations. In other words, the performance
forthediscrepancybetweenthetwocurvesdonotsignif- ofthe junction increasesbut atthe costofbecoming less
icantly alter the qualitative shape of the efficiency LDF. reproducible.
However, when comparing Figs. 3a and b, we note that As a final example we consider a single level junc-
the broadeningeffects resultingfromthe strongcoupling tion (see inset in Fig. 5). Within the QME approach,
to the contacts clearlytend to increasethe magnitude of the efficiency does not fluctuate in this model because
the efficiency fluctuations and also intensifies the asym- heat and work are directly proportional to each other,
metry of the fluctuations around the most likely value. a condition known as tight coupling [50]. However, the
We note that even the most likely value is affected. The NEGF approachbreaks the tight coupling condition due
least likely value is nevertheless still exactly located at tothehybridizationofthemolecularlevelwiththestates
the Carnot efficiency. in the contacts. The position of the level is taken as
We now turn to the donor-bridge-acceptor (DBA) ε = 0.1 eV and Figure 5 shows the results of calcula-
junction depicted in the inset of Fig. 4. This setup en- tions for several strengths of the system-reservoir cou-
ables to study the effect of intra-molecular interference pling: ΓL = ΓR = 0.1 eV (solid line), 0.05 eV (dashed
on efficiency fluctuations. We see that the trend pre- line), 0.01 eV (dash-dotted line), and 0.001 eV (dotted
dicted by the QME, when moving from constructive to line). AsΓ→0(weakcouplinglimit)thedistributionbe-
destructive interference (solid to dashed to dotted line), comesverynarrowandcenteredaroundthe macroscopic
is the oppositeofthe realtrendobtainedusing the exact efficiency (µL−µR)/(ε−µR).
NEGF.Itisinterestingto observethatdestructiveinter- WenowdiscusswaystorelatetheefficiencyLDFtoex-
ference tend to increase the most likely efficiency but at perimentally measurable characteristics of the junction.
thesametimesignificantlyincreasethemagnitudeofthe Forthe setupsketchedinFig.1b, the averagepowerand
6
-6
10
-5
10
L R
10
) )
1
( (
J J
5
-2 -1 0 1 2 -1 -0.5 0 0.5 1
FIG. 5: (Color online) Efficiency LDF calculated within the FIG. 6: (Color online) Efficiency LDFfor a single level junc-
NEGF for a single level junction. The results are shown tion(seeinsetinFig.5)calculatedforexperimentallyrelevant
forseverallevel-contactscouplingstrengthsrangingfrom the parameters. The predictions of theexact NEGF calculations
strongest(solidline,blue)totheweakest(dottedline,black). (solid line, blue) are compared to the linear response pre-
TheverticaldashedlineshowstheCarnotefficiency. Seetext dictions (29) (dashed line, black), and to the Gaussian ap-
for parameters. proximation predictions (33) (dotted line, red). See text for
parameters.
the heat flux from the hot reservoir are
Thus following Ref. [48], in the linear response regime
W˙ =∆µIP (20) the efficiency LDF can be expressed in terms of these
measurable quantities as
Q˙ =− IE −µ IP , (21)
R
η(κ∆T +GST ∆µ)+GS∆T∆µ+G∆µ2 2
R
(cid:0) (cid:1) J(η)= ,
where IP and IE are defined in Eq. (9) and ∆µ ≡µL− (cid:2)4[η2κTLTR+2ηGSTLTR∆µ+GTL∆µ2(cid:3)]
µR. In the linear response regime (obtained by lineariz- (29)
ing the Fermi distributions in 1/TL(R) and µL(R)/TL(R) where ∆T =TR−TL.
around equilibrium µL = µR = EF and TL = TR = T), We now attempt to estimate the efficiency LDF be-
we get that yondthelinearregime,solelyintermsoftheparticleand
energynonequilibriumcurrentsandnoises,Eqs.(9)-(12).
W˙ ≈G∆µ2+L∆µ∆β (22) Notethatinmolecularjunctions,theparticleandenergy
Q˙ ≈R∆µ+F ∆β (23) currents as well as the particle noise are experimentally
measurable[87–89,91]andtheenergynoisewillsoonbe-
where ∆β =1/T −1/T and come measurable [94–96]. To do so, we approximate the
L R
cumulantgeneratingfunction(15)byaquadraticexpan-
dE 1 sion in counting fields γ and λ around point γ = λ = 0.
G=− T(E)f′(E) (24)
This is a Gaussian assumption which leads to
2π T
L
Z
L= dE T(E)f′(E)(E−µ ) (25) φ(γ,ηγ)≈aγ2+bγ, (30)
R
2π
Z where we used the fact that φ(0,0)=0, and defined
dE E−µ
R= T(E)f′(E) R (26)
Z 2π TL a≡η2SEE + (µL−µR[1−η])2SPP (31)
F =− dE T(E)f′(E)(E−µ )2. (27) 2 2
2π R −η(µ −µ [1−η])SPE
Z L R
Here f′(E) = [d/dx1/(ex + 1)] and R = b≡−ηIE +(µL−µR[1−η])IP, (32)
x=(E−EF)/T
L/T . The coefficients in (24) are related to experimen-
L which are solely expressed in terms of the measurable
tally measurable quantities. Indeed, G is the electrical
nonequilibrium particle and energy fluxes, Eq. (9), and
conductance, and if κ denotes the heat conductance and
ofthenonequilibriumnoisecharacteristicsofthejunction
S the Seebeck coefficient, we have that
(10). Within this Gaussian approximation,we find that
F L b2
κ= ; S = . (28) J(η)= . (33)
TLTR GTLTR 4a
7
We have thus shown that efficiency fluctuations are dynamicsinRef. [48],themostlikelyefficiencycoincides
experimentally measurable close to equilibrium (29) and withthemacroscopicefficiency,whiletheleastlikelyeffi-
in the Gaussian approximation (33). We now verify ciencycorrespondstotheCarnotefficiency. Weusedsim-
the validity of these approximations in Fig. 6 where the ple models with realistic molecular junction parameters
exact efficiency LDF, Eqs. (15) and (18), is compared to compare our NEGF based method to the commonly
to the linear response, Eq. (29), and the Gaussian ap- used QME approach. We showed that the latter may
proximation, Eq. (33), result for an experimentally rel- failqualitativelyforstrongsystem-reservoircouplingdue
evant set of parameters: T = 300 K, T = 350 K, to its inability to properly account for quantum coher-
L R
µ = 0.002 eV, µ = 0 [3]. We also set ε = 0.1 eV and ences in the system. We finally proposed a method to
L R
Γ =Γ =0.1eV.Weseethatneartheminimumcorre- estimate efficiency fluctuations using the experimentally
L R
sponding to the macroscopic efficiency, the three curves measurable particle and energy nonequilibrium currents
coincide (thus justifying the use of linear response to es- and noises. Linear response and Gaussian approxima-
timates of averagequantities). At the same time, the ef- tions were proposed as ways to construct efficiency fluc-
ficiencyfluctuationsarepoorlycapturedbythelinearre- tuations from experimental measurements. We showed
sponseapproximation(dashedline)butreproducedquite that while linear response approach, often used in the
well by the Gaussian approximation (dotted line). experimentalliteratureto discuss thermoelectricproper-
ties of junctions, captures the macroscopic efficiency, it
fails to account for the efficiency fluctuations. At the
V. CONCLUSION same time, the Gaussian approximation was shown to
work very well within experimentally relevant range of
We studied the thermoelectric properties of nanoscale parameters.
junctions. Sincestochasticityandquantumcoherenceare
expected to be important at small scale, we proposed to
characterize the performance of these devices by study-
Acknowledgments
ing efficiency fluctuations rather than the widely used
figure of merit which is intrinsically restricted to the lin-
earregime. Weprovidedasystematicproceduretostudy M.E. is supported by the National Research Fund,
efficiencyfluctuationswhichaccountsforallquantumef- LuxembourgintheframeofprojectFNR/A11/02. M.G.
fects and is based on the work and heat FCS obtained gratefully acknowledges support by the Department of
within the NEGF formalism. As predicted for classical Energy (Early Career Award, de-sc0006422).
Appendix A: Cumulant generating function of a multilevel non-interacting system
Here we derive the generalform of the adiabatic potential U(λ) for a noninteracting n-levelsystem. For simplicity,
we consider the specific case of one particle counting field λ in the left molecule-contact interface. Multiple counting
fieldsand/orenergyFCSareformulatedsimilarly. Wefirstfindexpressionforthepotentialderivativeinthecounting
field, Eqs. (3) and (7), and then integrate it in the field to get the potential itself.
WestartbywritingthedressedGreenFunctionG(λ)asa2n×2ndimensionalblockmatrixintheKeldyshcontour
Gc G<
G(λ)= λ λ , (A1)
G> Gc˜
(cid:20) λ λ(cid:21)
inverse of which is [69]
−1 −iΓL(fL(E)−1/2)−iΓR(fR(E)−1/2)+IE−HM ieiλΓLfL(E)+iΓRfR(E)
G (λ)=(cid:20) −ie−iλΓL(1−fL(E))−iΓR(1−fR(E)) −iΓL(fL(E)−1/2)−iΓR(fR(E)−1/2)−IE+HM(cid:21).
(A2)
We will use Jacobi’s formula for the derivative of the determinant of a matrix that in our case reads
d d
det(G−1(λ))=Tr adj G−1(λ) G−1(λ) , (A3)
dλ dλ
(cid:26) (cid:27)
(cid:0) (cid:1)
whereadj(M)denotestheadjugatematrixofamatrixM (adj(M)M =Idet(M)=Madj(M)). Forourconsideration,
it will be important to work with special submatrices of G−1. For an n×n matrix M, we define M(j|i) to be the
(n−1)×(n−1) matrix that is obtained from M by removing the jth row and the ith column. In this notation
8
the (i,j)-matrix element for the adjugate of M can be expressed as adj(M) = (−1)i+jdet(M(j|i)). Also below
ij
M[j ...j |i ...i ] will denote the submatrix of M composed of rows j ...j and columns i ...i .
1 r 1 r 1 r 1 r
The first step in the derivation is to obtain from Eq. (A2)
d 0 −eiλΓ f (E)
G−1(λ)= L L , (A4)
dλ −e−iλΓL(1−fL(E)) 0
(cid:20) (cid:21)
and utilizing Eq. (A3) calculate
1 d 1 d
det(G−1(λ))= Tr adj G−1(λ) G−1(λ)
det(G−1(λ))dλ det(G−1(λ)) dλ
(cid:26) (cid:27)
(cid:0) (cid:1)
adj G−1(λ) 0 −eiλΓ f (E)
=Tr L L
(det(cid:0)(G−1(λ))(cid:1)(cid:20)−e−iλΓL(1−fL(E)) 0 (cid:21))
Gc(E) G<(E) 0 −eiλΓ f (E)
=Tr λ λ L L
(cid:26)(cid:20)G>λ(E) Gcλ˜(E)(cid:21)(cid:20)−e−iλΓL(1−fL(E)) 0 (cid:21)(cid:27)
=Tr G<(E)(−e−iλ)Γ (1−f (E))+G>(E)(−eiλ)Γ f (E)
λ L L λ L L
=iILλ(cid:8)(E). (cid:9) (A5)
Using this last result in Eq.(3) and integrating with respect to the counting field λ leads to
dE det(G−1(λ))
U(λ)=i ln . (A6)
2π det(G−1(0))
Z (cid:20) (cid:21)
where we used the known property U(0)=0. Eq. (A6) is the first important result.
We now have to evaluate the determinants inside the logarithm in Eq. (A6). The determinants can be evaluated
after applying elementary transformations to G−1. First, we notice that we can write
−Σ>(E)+Ga,−1 −i(1−eiλ)Γ f (E)+Σ<(E)
det(G−1(λ))= L L , (A7)
i(1−e−iλ)Γ (1−f (E))+Σ>(E) −Gr,−1−Σ>(E)
(cid:12) L L (cid:12)
(cid:12) (cid:12)
where Gr,−1 =IE−H +i(Γ (cid:12)+Γ )/2 and Ga,−1 =(Gr,−1)†, by adding and subtracting to each(cid:12)submatrix in Eq.
M L(cid:12) R (cid:12)
(A2) appropriate matrices. Then we add to the i-th row, i≤n the (n+i)th row of the matrix. After which, on the
resulting matrix, we add the (n+j)-th column to the j-th column for each j ≤n. This leads to
i(1−e−iλ)Γ (1−f (E))+Ga,−1 −i(1−eiλ)Γ f (E)+i(1−e−iλ)(1−f (E))Γ
det(G−1(λ))= L L L L L L (A8)
i(1−e−iλ)Γ (1−f (E))+Σ>(E) i(1−e−iλ)Γ (1−f (E))−Gr,−1
(cid:12) L L L L (cid:12)
(cid:12) (cid:12)
Setting λ=0 we arri(cid:12)ve at the result for the first of the determinants in Eq. (A6) (cid:12)
(cid:12) (cid:12)
det(G−1(0))=det(Ga,−1)det(−Gr,−1), (A9)
Togetthe seconddeterminantinEq.(A6)wehavetoworkwith the generalformofEq.(A7). Explicitevaluations
lead to an expression that can be grouped in powers of (1−e−iλ) and (1−eiλ) of at most n power. In particular,
noticing that (1−e−iλ)(1−eiλ)=(1−e−iλ)+(1−eiλ) we can write
n
det(G−1(λ))=det(Ga,−1)det(−Gr,−1)+ (1−e−iλ)sa +(1−eiλ)sb , (A10)
s s
s=1
X
where a and b are the coefficients of the polynomial given by [97]
s s
a =(i)s(1−f (E))s (−1)σ(α+n)+σ(β)det(Γ [α|β])det(N(α+n|β)) (A11)
s L L
α∈XQs,nβ∈XQs,n
b =(−i)s(f (E))s (−1)σ(α)+σ(β+n)det(Γ [α|β])det(M(α|β+n)) (A12)
s L L
α∈XQs,nβ∈XQs,n
where N and M are 2n×2n matrices given by
−Σ>(E)+Ga,−1 iΓ f (E) −Σ>(E)+Ga,−1 Σ<(E)
N = R R , M = , (A13)
Σ>(E) −Gr,−1−Σ>(E) −iΓ (1−f (E)) −Gr,−1−Σ>(E)
R R
(cid:20) (cid:21) (cid:20) (cid:21)
9
Q is the set of s-tuples (i ,...,i ) of natural numbers such that 1 ≤ i < i < ··· < i < i ≤ n, σ(α) = α
s,n 1 s 1 2 s−1 s i
for α ∈ Q , and α+n = (α +n,α +n,...,α +n). From equations (A11) and (A12) we have in particular
s,n 1 2 s
a = ((1−f (E))f (E))ndet(Γ Γ ) and b = ((1−f (E))f (E))ndet(Γ Γ ). Eqs. (A10)-(A12) give the Pmost
n L R L R n R L L R
general form for the second determinant in Eq. (A6).
Finally, substituting Eqs. (A9) and (A10) into (A6) we obtain the general form for the adiabatic potential
dE n a (1−e−iλ)s b (1−eiλ)s
s s
U(λ)=i ln 1+ + , (A14)
2π det(Ga,−1)det(−Gr,−1) det(Ga,−1)det(−Gr,−1)
" #
Z s=1
X
with the coeffcients a and b given by Eqs. (A11) and (A12).
s s
We now consider two specific examples where we can recover the expression derived in Ref. [69] for a single level
junction from our general result, Eq. (A14).
1. One level.
By direct computation of Eqs. (A11) and (A12) we find
a =Γ Γ (1−f (E))f (E) b =Γ Γ (1−f (E))f (E).
1 L R L R 1 L R R L
Thus
det(G−1(λ)) Γ Γ
=1+ R L (1−e−iλ)(1−f (E))f (E)+(1−eiλ)(1−f (E))f (E)
det(G−1(0)) Ga,−1(−Gr,−1) L R R L
=1+Gr(E)Γ Ga(E)Γ(cid:0) (eiλ−1)(1−f (E))f (E)+(e−iλ−1)(1−f (E))f ((cid:1)E)
R L L R L R
(cid:0) (cid:1)
which yields the expression for adiabatic potential derived in Ref. [69].
2. n-level system coupled to the contacts through single orbitals.
Inthis caseΓ andΓ aren×n matriceswithallentriesequalzerobutoneelementinthe diagonal. Examples
L R
of systems of this kind are the two level bridge (see inset in Fig. 2) or D-B-A type of the junction (see inset in
Fig. 4). Here we can take [Γ ] =δ δ γ and [Γ ] =δ δ γ , which results in a =b =0 for s>1 and
L ij 1j i1 L R ij nj in R s s
a =i(1−f (E))(−1)1+n+1det(Γ [1|1])det(N(1+n|1))
1 L L
−Σ>(|1)(E)+Ga,−1(|1) iΓ f (E)
=i(1−f (E))(−1)nγ R R
L L −iΓ (1|1)(1−f (E)) −Gr,−1(1|)−Σ>(1|)(E)
R R
(cid:12) (cid:12)
=i(1−fL(E))(−1)nγL((cid:12)(cid:12)(cid:12)−1)n+2n−1iγRfR(E) −Σ−>i(Γn|1(1)(|1E))(1+−Gfa,−(E1()n)|1) (cid:12)(cid:12)(cid:12)−Gr,−1(1|n)−0 Σ>(1|n)(E)
R R
(cid:12) (cid:12)
=(1−fL(E))fR(E)γLγRdet(Ga,−1(n|1))det((cid:12)(cid:12)−Gr,−1(1|n)) (cid:12)(cid:12)(A15)
(cid:12) (cid:12)
where we used Σ>(n|1)(E)=Σ>(1|n)(E)=0. Similarly
b =f (E)(1−f (E))γ γ det(Ga,−1(1|n))det(−Gr,−1(n|1)). (A16)
1 L R L R
From the definition of the adjugate we have (−1)1+ndet(Ga,−1(1|n)) = adj(Ga,−1) ,
n1
(−1)n+1det(−Gr,−1(n|1)) = adj(−Gr,−1) . Also in this particular case adj(Ga,−1) = adj(Gr,−1)
1n 1n 1n
and adj(Ga,−1) = adj(Gr,−1) . Finally, substituting Eqs. (A15) and (A16) into Eq. (A14) and rearranging
n1 n1
terms, we get
dE
U(λ)=i ln 1+Tr{Gr(E)Γ Ga(E)Γ } (A17)
R L
2π
Z (cid:20)
× (eiλ−1)(1−f (E))f (E)+(e−iλ−1)(1−f (E))f (E)
L R L R
(cid:21)
(cid:0) (cid:1)
Eq. (A17) is the Levitov-Lesovikformula for a multi-level system.
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