Scattering Theory of Nonlinear Thermoelectricity in Quantum Coherent Conductors

0.46 MB·English

by  Jonathan Meair

Scattering Theory of Nonlinear Thermoelectricity in Quantum Coherent Conductors
Jonathan Meair1 and Philippe Jacquod1,2
1Physics Department, University of Arizona, Tucson, AZ 85721, USA
2College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA
Weconstructascatteringtheoryofweaklynonlinearthermoelectrictransportthroughsub-micron
scale conductors. The theory incorporates the leading nonlinear contributions in temperature and
voltage biases to the charge and heat currents. Because of the finite capacitances of sub-micron
scale conductingcircuits, fundamentalconservation laws suchas gauge invarianceand current con-
3
1 servation require special care to be preserved. We do this by extending the approach of Christen
0 and Bu¨ttiker [Europhys. Lett. 35, 523 (1996)] to coupled charge and heat transport. In this way
2 wewriterelationsconnectingnonlineartransportcoefficientsinamannersimilartoMott’srelation
between the linear thermopower and the linear conductance. We derive sum rules that nonlinear
n
transport coefficients must satisfy to preserve gauge invariance and current conservation. We illus-
a
trate our theory by calculating the efficiency of heat engines and the coefficient of performance of
J
thermoelectric refrigerators based on quantum point contacts and resonant tunneling barriers. We
4
identify in particular rectification effects that increase deviceperformance.
2

PACSnumbers: 73.23.-b,73.63.-b,85.50.Fi
]
l
l
a
h I. INTRODUCTION tricity. Ref.[41]calculatedthethermopowertoquadratic
- orderintemperaturegradientsandRefs.[42–44]focused
s
on the nonlinear Peltier coefficient. These works con-
e Today’s energy challenges are calling for novel tech-
m nologies to produce, store and use energy1. These chal- sidered bulk systems with large capacitances, where ac-
cordingly the build-up of local electrostatic potentials
. lenges have among others renewed the interest in the
at old problem of thermoelectricity2. Research in ther- can be neglected. This neglect is no longer justified in
m moelectricity is predominantly directed towards finding smallconstrictions,wheresuchpotentialshavetobeself-
- new materials with better thermoelectric properties3–5, consistently connected to the applied biases in order to
d preserve gauge invariance45. Refs. [46,47] discussed non-
however a recently proposed alternative approach has
n linearthermoelectricity insuchconstrictionsbut didnot
been to try and tailor low-dimensional micro- and nano-
o structured materials6, which may exhibit the sharp os- introduce localpotentials, therebytaking the risk ofvio-
c
lating gauge invariance. Refs. [14,34,48] introduced self-
[ cillations in the electronic density of states necessary to
trigger large thermoelectric effects7. With the ongoing consistentlydeterminedhomogeneousHartreepotentials
2 to preserve gauge invariance.
trendtowardscircuitminiaturizationinelectronics,heat
v
3 evacuation becomes a major problem and it is accord- In this manuscript we generalize the scattering theory
8 inglyimportanttoexplorehowheatcurrentscanbecon- ofweaklynonlineartransportofRef.[45]tocoupledelec-
5 trolled at sub-micron scales. Thus, there is ample mo- tric and heat transport. We go beyond Ref. [48] which
5 tivation for investigatingthermoelectricity in micro- and considered only electric currents in the presence of volt-
1. nanodevices8. To name but few examples of recent in- ageandtemperaturebiases,byadditionallytreatingheat
1 vestigations, sub-micron electrically controlled heat rec- currents. One of our motivations is to provide criteria
2 tifiers9, thermally driven electric rectifiers10,11, thermo- for enhanced thermoelectric efficiency in the nonlinear
1 electric refrigerators12–14 and heat converters15–17 have regime. The standardly used figure of merit ZT is con-
:
v been proposed, and thermoelectric effects in molecular structed out of linear thermoelectric coefficients only2,
i transporthavebeeninvestigated18–24. Onamorefunda- thus while any measure of thermodynamic efficiency in
X
mental level, thermoelectric effects provide information the linear regime can only be a function of ZT, it is
ar on the nature of the underlying quantum fluid, for in- expectablethatZT losesitspredictivepowerinthenon-
stance, breakdowns of the Wiedemann-Franz law25 may linear regime. It is often the case, and we illustrate this
reflectnonFermiliquidbehaviors26–28,quantuminterfer- below, that the maximal efficiency predicted by ZT is
ence effects29,30 or electric and heat transport mediated attained outside the linear regime. In this article we
by both electron and hole quasiparticles in Andreev sys- accordingly focus our attention on thermodynamic effi-
tems31–33. ciencies and coefficients of performance as indicators of
The field of thermoelectricity has until now over- thermoelectric performance. In particular we investigate
looked the fact that thermoelectric devices often oper- howtheybehaveuponincreasingsourcesofnonlinearities
ate in the nonlinear regime of transport (See for in- such as temperature and voltage biases and capacitive
stance Refs. [18,34,35]). Rectification effects occur in couplings. A key aspect of our theory is that in the non-
that regime36–40, and their influence on thermoelectric- linearregimeoftransport,the stateofthe systemisself-
ity is not yet understood. There are only few works we consistently determined by the conditions under which
know about which have discussed nonlinear thermoelec- transport proceeds. We will take this self-consistency
2
intoaccountbyintroducingpiecewiseconstantlocalelec- hisapproachandconsiderheatandelectrictransportme-
trostatic potentials. We will go beyond Refs. [14,34,48] diated only by electron quasiparticles and extend it to
and consider more than one such potential. In the scat- the nonlinear regime. Considering a coherent conductor
tering theory, gauge invariance and current conservation connected to i = 1,2,...M external reservoirs via reflec-
arenaturallypreservedinthelinearregimebytheunitar- tionless leads (see Fig. 1), the electric (I ) and heat (J )
i i
ityofthescatteringmatrix. Inthenonlinearregime,they currents through contact i are given by
arereflectedbysumrulesconnectingnonlineartransport
2e
coefficients. Such sum rules were derived in Ref. [45] for I = dεf (ε)A [ε,U(~r)] , (1a)
i j ij
purely electric transport and below we give a complete h
j Z
X
list of sum rules ensuring gauge invariance and current
2
conservation for weakly nonlinear thermoelectric trans- Ji = dε(ε eVi)fj(ε)Aij[ε,U(~r)] . (1b)
h −
port. j Z
X
The paper is organized as follows. In Section II, we
We took the convention that both electric and heat cur-
construct a scattering theory of weakly nonlinear ther-
rents are positive when they enter the conductor from
moelectric transport valid to quadratic order in temper-
a reservoir. Thus, for example, the conductor is cooled
ature and voltage biases. We express linear and nonlin-
if J < 0 and heated otherwise. In Eqs. (1), ε
ear transport coefficients in terms of the scattering ma- i i ≡
E E is the energy of an electron counted from the
trix and the local potentials generated in the system by FerP−mi Fenergy, f (ε) = 1+exp[(ε eV )/k T ] −1 is
the biases. In Section III, we discuss these local poten- j { − j B j }
theFermifunctioninreservoirj,maintainedatavoltage
tials and recall how they can be determined in practice.
V andatemperatureT ,A [ε,U(~r)]=N [ε,U(~r)]δ
In Section IV we write down sum rules that have to be j j ij i i,j−
[ε,U(~r)]isgivenbythenumberN oftransportchan-
satisfied for the theory to preserve charge and current nTiejl in lead i and the scattering miatrix s via =
conservation as well as gauge invariance. In Section V Tij
Tr[s† s ] where s is the sub-block of s connecting ter-
webrieflydiscussperformancemetricsforthermoelectric ij ij ij
devices. In Section VI we apply our theory to two par- minals j to i and U(~r) is the local potential in the con-
ticular devices, a quantum point contact and a resonant ductor. In this manuscript, we take Ni to be indepen-
tunneling double-barrier. We find in particular that the dentofenergyandbiases. The electrostaticpotentialin-
bias range of operation of these devices is such that our ducedwithintheconductor(seeFig.1)isitselfdependent
weakly nonlinear approximation gives a rather accurate on the voltages and temperatures in the reservoirs, i.e.,
description of the exact thermodynamic efficiency. Con- U(~r, Vk , Tk ). Expanding the currents in Eqs. (1) to
{ } { }
clusions are presented in Section VII. quadraticorderin temperature differences, θj Tj T0,
≡ −
and voltage biases one obtains
I = (G V +B θ )
II. SCATTERING APPROACH TO i ij j ij j
THERMOELECTRICITY Xj
+ [G V V +B θ θ +Y V θ ] , (2a)
ijk j k ijk j k ijk j k
j,k
X
µ3
J = (Γ V +Ξ θ )
T3 µ1 i ij j ij j
I3,J3 I1,J1 Xj
T1 + [Γ V V +Ξ θ θ +Ψ V θ ]. (2b)
ijk j k ijk j k ijk j k
U(~r)
j,k
X
The linear response coefficients were calculated in
I2,J2
T2 µ2 Vg Ref. [50]. They are given by
2e2
G = dε( ∂ f)A (ε), (3a)
ij ε ij
FIG. 1: Sketch of a multi-terminal coherent conductor ca- h −
Z
pacitivelycoupledtoanexternalgateandconnectedtothree 2e
external Fermi liquid reservoirs. The electronic Fermi distri- Bij = dε( ∂εf)εAij(ε), (3b)
hT −
butionsofthereservoirsaredepicted. Inthenonlinearregime, 0 Z
2e
aninhomogeneouselectrostaticpotentialU(~r)developsinside Γ = dε( ∂ f)εA (ε), (3c)
ij ε ij
the conductor, as it is induced by the applied voltages and h −
Z
temperatures at thereservoirs. The arrows indicate our con- 2
ventionthatelectricandheatcurrentsarepositivewhenthey Ξij = dε( ∂εf)ε2 Aij(ε), (3d)
hT −
flow into theconductor. 0 Z
where f(ε) = 1+exp[ε/k T ] −1. The temperature
B 0
{ }
Inlinearresponse,thescatteringtheorytothermoelec- T = 0 is in principle arbitrary, however, better conver-
0
tric transport was worked out by Butcher50. We follow gen6ce of the expansion of Eqs. (2) is obtained when T
0
3
is taken as the temperature in one of the reservoirs or sponse coefficients must hold,
the average of the reservoir temperatures. In particular,
Γ +G (δ δ /2) (πk T )2
the sum rules given below in Section IV guarantee that ijk ik i,j − j,k = B 0 ∂ lnG , (7a)
ε ijk
G 3e
all currents I and J vanish at equilibrium, regardless ijk
i i
of the choice of T0. Eqs. (3b) and (3c) show that the Ψijk +Bik(δi,j −δj,k) = (πkBT0)2 ∂ lnY , (7b)
reciprocity relation Γij = BijT0 is automatically satis- Yijk 3e ε ijk
fied (see Ref. [51] for a detailed discussion of Onsager
B Ξ δ
ijk ij j,k
relations in the linear regime). =e∂ ln Ξ ,
ε ijk
Ξ Ξ δ /(2T ) − 2T
These linear response coefficients are determined by ijk − ij j,k 0 (cid:20) 0 (cid:21)
(7c)
the equilibrium state of the system for which we set
U(eq)(~r)=0. However,in the nonlinearregime,we must where all derivatives on the right-hand side are eval-
consider how the biases affect the localpotential. We do uated at the Fermi energy, ε = 0. These relations
that by introducing characteristic potentials45,48 are the nonlinear counterpart to the Mott relation25
S = (π2k2T /3e)∂ lnGbetweenthethermopowerS =
− B 0 ε
U(~r)= [u (~r)V +z (~r)θ ], (4) B /G and the logarithmic derivative of the conduc-
k k k k 12 12
Xk t−anceG≡G12 intwo-terminalconductors52. Thereisno
counterpartto the Onsagerreciprocity relationsfor non-
which describe how U(~r) depends on the biases to lead- linearcoefficients, because,as canbe seenfromEqs.(5),
ing order. Truncating the expansion of U(~r) to linear they all have different energy integrands. The absence
order in the biases is sufficient to correctly capture cur- of such relations may be attributed to the breaking of
rents to quadratic order. Gauge invariance requires that microscopic reversibility in the nonlinear regime, where
kuk(~r)=145,howeverthereisnosuchrelationinvolv- localpotentials areself-consistentlyrelatedto transport,
ing zk(~r). The nonlinear response coefficients are then and where their dependence on temperature [the coeffi-
P
cients z in Eq. (4)] is not necessarily related to their
k
e3 dependence on voltages [the coefficients u in Eq. (4)].
G = dε( ∂ f)α (ε), (5a) k
ijk ε ijk
h −
Z
e
Bijk = hT2 dε(−∂εf)εβijk(ε), (5b) III. CHARACTERISTIC POTENTIALS
0 Z
2e2
Y = dε( ∂ f)γ (ε), (5c) Our main task is next to determine the characteristic
ijk ε ijk
hT −
0 Z potentials defined in Eq. (4). We briefly summarize how
e2 this can be done45,48. We consider first how additional
Γ = dε( ∂ f)εα (ε)
ijk h − ε ijk chargesareinjectedintotheconductorasitisdrivenout
Z
of equilibrium. Biased reservoirs inject a net addition
1
Gik δi,j δj,k , (5d) of charge, called the bare charge, δq(b). The presence
− − 2
(cid:18) (cid:19) ofδq(b) inducesanelectrostaticpotentialshiftwithinthe
1 Ξ
Ξ = dε( ∂ f)ε2 β (ε)+ ijδ ,(5e) conductorwhich,initsturn,generatesascreeningcharge
ijk hT02 Z − ε ijk 2T0 j,k δq(s). The net injected charge is the sum of the two,
2e δq(~r) = δq(b)(~r) + δq(s)(~r). To leading order the bare
Ψ = dε( ∂ f)εγ (ε)
ijk hT − ε ijk chargecanbeexpressedasalinearfunctionofthebiases,
0 Z
Bik(δi,j δj,k), (5f) δq(b)(~r)= DV(~r)V + Dθ(~r)θ , (8)
− − j j j j
j j
where we defined X X
where DV(~r) = e2 dε( ∂ f) νp(~r,ε) and Dθ(~r) =
αijk(ε) = drδeAδUij((~rε))[2uk(~r)−δj,k] , (6a) ien tdhεe(−un∂jiεtfs)oνfjec(a~rp,aε)cRitaarnec−ceshaεarsgeininjRjeecf.tiv[4it5i]e)sd(eetxepjrmreisnseedd
Z
δA (ε) byRthe energy dependent particle45 and entropic48 injec-
β (ε) = dr ij [2eT z (~r) εδ ] , (6b)
ijk eδU(~r) 0 k − j,k tivities
Z
γijk(ε) = Z dr(cid:18)δeAδUij((~rε))[eT0zk(~r)−εδj,k] νjp(~r,ε) = −41πi k Tr"s†kjeδδUsk(j~r) − eδδUs†k(j~r)skj# , (9a)
δA (ε) X
+eδUik(~r) εuj(~r)(cid:19). (6c) νje(~r,ε) = (cid:18)Tε0(cid:19)νjp(~r,ε), (9b)
Several of the nonlinear response coefficients of Eqs. (5) wherethe traceruns overchannelindices. Eq.(9a)gives
differ primarily by a factor of ε inside the energy inte- the partial density of states at ~r corresponding to injec-
gral. Using a Sommerfeld expansion,we find that at low tion from lead j, which is multiplied by a single-particle
temperaturethefollowingrelationsbetweennonlinearre- entropy factor ε/T in Eq. (9b)48.
0
4
Next, the screening charge δq(s)(~r) is related to U(~r) In particular, two of these sum rules correspond to elec-
via the Lindhard polarization function. The relation triccurrentconservation, G = B =0whiletwo
i ij i ij
is in principle nonlocal, however the nonlocal correc- others reflect gauge invariance, G = Γ =0.
j ij j ij
P P
tions are quantumin essence andthus negligible in good The unitarity of the scattering matrix further implies
P P
metals with high enough electronic density. We thus α = β = γ = (α +α ) =
i ijk i ijk i ijk j ijk ikj
make the quasiclassical approximation of a local Lind- γ = 0, which results in the following sum rules
hard function and write δq(s)(~r) = Π(~r)U(~r), with Pbetjweijekn thePnonlinear rePsponse coeffiPcients,
Π(~r) = DV(~r) = D(~r). From the−definition of DV P
j j j
G = B = Y =0, (13a)
wehaveD(~r)=e2 dε( ∂ f)ν(~r,ε)withthe localden- ijk ijk ijk
sity of sPtates ν(~r,ε) =− ενp(~r,ε). Note that at zero Xi Xi Xi
R j j
temperature,D(~r)isthedensityofstates(inunitsofca- (Gijk +Gikj)= (Γijk +Γikj)=0, (13b)
P
pacitance)attheFermienergy. Puttingallthistogether, j j
X X
the net total injected charge is, to leading order in the
Y = Ψ =0, (13c)
ijk ijk
biases,
j j
X X
δq(~r)= DV(~r)V + Dθ(~r)θ D(~r)U(~r). (10) Ξijk =0, (13d)
j j j j −
i
j j X
X X
G = Γ , B = Ψ . (13e)
jk ijk jk ijk
The conductor may also be capacitively coupled to the − −
i i
reservoirs and external metallic gates. One way to take X X
Two of these sum rules were derived in Ref. [45],
these couplings into account is to discretize the conduc-
G = 0, implied by current conservation and
torinto differentregionslabelledr =1,2,...,eachwith a i ijk
(G +G ) = 0, resulting from gauge invariance.
constant potential Ur. The procedure brings about ad- APllj othijekr sumikrjules are written here for the first time.
ditional relations P
Eqs. (13a) reflect current conservation while Eqs. (13b)
and(13c)correspondtogaugeinvariance. Thesumrules
δq = C (U V ), (11)
r rk r k
− of Eq. (13e) imply J +I(lin)V = 0, where I(lin) is
Xk i i i i i
the linear response current through lead i. Thus in our
with the sum running over all capacitively coupled com- approach energy is Pconserved to quadratic order in the
ponents. Thisequationexpresseschangesinlocalcharge biases.
distributions due to Coulomb interactions. The latter Finally we note the following, additional double-sum
are represented here by geometric capacitances between rules α = β =0 which imply
jk ijk jk ijk
different components of the system.
Once the microstructure considered is consistently P PGijk = Γijk =0, (14a)
defined, in that all electric components (leads, gates, j,k j,k
X X
etc.) that are coupled to the conductor are included
B = Ξ =0. (14b)
ijk ijk
so that electric field lines do not leave the system,
j,k j,k
then the total charge is conserved by Gauss’ theorem. X X
Eqs. (4), (10) and (11) then determine the character- While Eqs. (14a) are guaranteed to hold by Eqs. (13b),
istic potentials in Eqs. (6). The nonlinear transport Eqs. (14b) are new. Eqs. (12), (13c), (14a) and (14b)
coefficients are finally calculated by using the identity ensureinparticularthattheelectricandheatcurrentsof
drδ (...) = e∂ (...) to compute the functional Eqs. (2) vanish at equilibrium.
U(~r) E
−
derivatives in Eqs. (6). Two examples of thermoelec-
R
tric microdevices will be discussed in Section VI, which
V. PERFORMANCE METRICS FOR
illustratehowtheprocedureisappliedinpracticebydis-
THERMOELECTRICITY
cretizingthemicrostructuresintoregionsofuniformlocal
potential.
For two-terminal setups treated in linear response,
Eqs. (1) reduce to
IV. CONSERVATIONS AND SUM RULES
I G B V
= , (15)
J Γ Ξ ∆T
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
Particle conservation is expressed mathematically by
where V and ∆T are the voltage and temperature dif-
the unitarity of the scattering matrix, A =
i ij ferencesbetweenthereservoirs. Alternatively,thislinear
A =0. This translates into the following sum rules
j ij P relation is often rewritten in terms of responses to an
for the linear transport coefficients,
P electric current and a temperature difference,
G = B = Γ = Ξ =0. (12) V R S I
ij ij ij ij = . (16)
J π κ ∆T
iorj iorj iorj iorj p
X X X X (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
5
Eqs.(15) and(16) define the linearresponse coefficients, VI. EXAMPLES
the electric conductance, G, and resistance, R, the See-
beck, S, and Peltier, πp, coefficients as well as the ther- A. Resonant Tunneling Barrier
mal conductances Ξ (at zero voltage, V = 0) and κ (at
zero current, I = 0). We can express the relationships
We illustrate our theory, first by calculating the ther-
between these linear transport coefficients as
modynamic efficiency of thermoelectric devices basedon
a coherent single-level double-barrier system - a reso-
nant tunneling barrier (RTB). Nonlinear electric trans-
R S 1/G B/G
= − . (17) port through a RTB was investigated in Ref. [45], where
π κ Γ/G Ξ BΓ/G
p
(cid:18) (cid:19) (cid:18) − (cid:19) rectification effects were stressed in asymmetric systems
withdifferenttunnelingprobabilitiesthroughtheleftand
Dimensional analysis shows that a single dimensionless rightbarriers. Electrictransportgeneratedbysimultane-
quantity can be constructedfromthe linear responseco- ously applied voltage and temperature biases was inves-
efficientsandthetemperature. Thisisthefigureofmerit tigated in Ref. [48], focusing on the thermopower. This
ZT = (GS2/κ)T . In the linear regime, thermodynamic systemisespeciallyinterestinginthattheweaklynonlin-
0
efficiencies are then only functions of ZT and the tem- eartheorycanbecomparedtofullynonlinearsolutions45.
peraturedifference,forinstanceaheatengineworkingin
the linear regime has maximal efficiency49 ΓL ΓR
IV ∆T √ZT +1 1 µL
η =max = − , (18)
max V
(cid:18) J (cid:19) (cid:18) T0 (cid:19)√ZT +1+1 EF
Er
It is importantto keepinmind thatthis relationis valid µR
only in the linear regime, and that nonlinear effects may TL TR
invalidate it. Linear response may even predict a maxi- eU
malefficiencytooccuroutsideitsrangeofvalidity(attoo
largeabiasvoltage). Adescriptionofthermodynamicef- C
ficiency based only on ZT is therefore inappropriate in V
g
the nonlinear regime. However, the efficiency is generi-
callygivenbyη =IV/J ,whereJ istheheatcurrent
hot hot
flowing from the hot reservoir. While J = J in FIG.2: Sketchofaresonanttunnelingbarrier(RTB)formed
hot cold
−
linear response, J = J in the nonlinear regime. betweentwoFermiliquidreservoirsbytwotunnelbarriersin
hot cold
Note that with this d6efin−ition, it makes sense to substi- serieswithtransparenciesΓLandΓR. Theydefineapotential
well with a single resonant level capacitively coupled to an
tute T T in Eq. (18), in which case one obtains
0 hot
→ external gate.
Carnot efficiency for ZT .
→∞
The efficiency of thermoelectric refrigeration is mea- ThesystemweconsiderissketchedinFig.2. Twotun-
sured by the coefficient of performance (COP) which is nelbarrierswithtunnelratesΓ defineapotentialwell
L,R
the ratio of the heat extracted from the cold reservoir with a single resonantlevel, at energy E above the bot-
r
to the work done2, COP = Jcold/IV. This refrigeration tomofthewell. Leftandright,thewelliscoupledtotwo
“efficiency”islimitedbyCOP<Tcold/∆T. Inthelinear FermiliquidreservoirsattemperaturesTL,R andelectro-
regime, its maximum is also given by ZT as chemicalpotentialsµ . Thewelliscapacitivelycoupled
L,R
to anexternalgate with voltageV which, together with
g
the temperatures and electrochemical potentials in the
J T √ZT +1 1
COP =max = 0 − , reservoirs determine the potential U (assumed spatially
max V
IV ∆T √ZT +1+1
(cid:18) (cid:19) (cid:18) (cid:19) homogeneous) in the well. This models a quantum dot
(19)
in the Coulomb blockade regime as long as neither the
where this time it makes sense to substitute T T .
0 → cold temperatures nor the voltage biases exceed the single-
Notethatthelinearresponsescatteringtheoryleavesthe
particle level spacing in the dot. The equilibrium energy
choice of T0 arbitrary, up to the fact that a well chosen dependent particle injectivity at lead j =L, R is45
T (close to both T and T ) minimizes nonlinear
0 cold hot
corrections. νp(ε)= 1 Γj , (20)
j π(ε+E E )2+Γ2
It is not at all guaranteed that ZT remains a valid F − r
figure of merit once nonlinear effects kick in. Below we and the transmission probability is45
discuss two examples of mesoscopic thermoelectric en-
4Γ Γ
gines and compare their true efficiency to that predicted (ε)= L R , (21)
by linear response. T (ε+EF Er)2+Γ2
−
6
whereΓ=Γ +Γ isthetotaldecayrateoftheresonant
L R
0.06
leveland we defined E =(µ +µ )/2. Inthe following
F L R
we will set E = 0, as the zero of the energy scale and 0.6
F
measure all other energies in units of Γ=const.
Becauseweconsiderasinglehomogeneouspotentialin
0.04
the well, U(~r) = U, the injected charge loses its spatial
P0.4
dependence, δq(~r) δq. Consequently, Eq. (10) can be η O
rewritten as → C
δq = DjVVj + Djθθj −DU. (22) 0.02 0.2
j j
X X
Thewelliscapacitivelycoupledtoanexternalgate,thus
Eq. (11) tells us that the additionalcharge in the poten- 0 0
0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5
tial well is also given by eV / Γ eV / Γ
δq =C(U V ). (23)
g FIG. 4: Left panel: Thermodynamic efficiency for a RTB-
−
based heat engine with (Er −EF)/Γ = 1, µL −µR = eV,
Using Eqs. (22) and (23) we obtain the characteristic kBT0/Γ = 1, TR −TL = 0.8 T0 and different asymmetries
potentials as (ΓL −ΓR)/Γ = 0.95 (red), 0.5 (green), 0 (blue), -0.5 (vio-
let), -0.95 (cyan). Right panel: COP for thesame RTBas in
DjV C Djθ the left panel with TR−TL = 0.1 T0 and working at larger
u = , u = , z = , (24)
j C+D g C+D j C+D bias, thus as a refrigerator. In both cases there is no capaci-
tivecouplingbetweentheRTBandtheexternalgate,C =0.
where the charge and entropic injectivities The solid lines are the weakly nonlinear approximate solu-
are given by DV = e2 dε( ∂ f) νp(ε) and tions and the dotted lines are the fully nonlinear solutions
j − ε j
Dθ = e dε( ∂ f) (ε/T )νp(ε), with the particle (see text). The dashed black line is the linear response solu-
injjectivity νp o−f Eεq. (20). 0 TRhjis fully determines the tion and thesolid black line is themaximum efficiency/COP
R j predicted from linear response.
nonlinear coefficients in Eqs. (5) so that currents can be
calculated using Eqs. (2).
For the RTB the potential U can alternatively be de- charge is replaced by the energy integral
termined nonperturbatively. Eq. (22) for the injected
δq =e dε νp(ε eU)f (ε) ν(ε)f(ε) , (25)
2  j − j − 
Z j
X
 
Γ ] 1 ] 1 with ν(ε) = jνjp(ε). One determines U numerically
h h from Eqs. (23) and (25). The fully nonlinear currents
/ R / R are then obtaPined from Eqs. (1). The result is exact,
ΓL0 ΓL up to the approximation of a single homogeneous local
Γ2|e| Γ[ 2 0 poWtenetfiiarlstinshthowe winellF.ig. 3 the electric and heat currents
I [ -1 J ftoorptrweoseRntTIB’as.ndDiJffeirnenotrdteermtpoervaistuuarellsyhoapvteimbeizeenncohnolsienn-
ear effects. The curves are clearly nonlinear and display
-2 -1 rectification in both the electric (left) and heat (right)
-1 -0.5eV0 / Γ 0.5 1 -1 -0.5 eV0 / Γ 0.5 1 currents. The rectification in the heat current is in gen-
eral more pronounced, due to the additional voltage in
the integrand of Eq. (1b). Most remarkably, perhaps,
FIG. 3: Electric (left) and heat (right) currents from a hot
we see that the behavior obtained via the weakly non-
reservoirtoaRTBwith(Er−EF)/Γ=1,µL−µR =eV and linear approximation (solid lines) does not differ much
TR−TL =0.3T0. The base temperatures are kBT0/Γ=0.3
from the fully nonlinear result (dotted lines), even for
(left) and kBT0/Γ = 1 (right). The different curves corre-
spondtodifferenttunnelbarrierasymmetries(ΓL−ΓR)/Γ= the strongly nonlinear heat current. The pronounced
0.95 (red), 0.5 (green), 0 (blue), -0.5 (violet), -0.95 (cyan). rectification effects will turn out to boost or hinder the
Here,we switch off thecapacitivecoupling between theRTB efficiency of RTB-based devices depending on the asym-
and the external gate, C = 0. The solid lines are the ap- metry (Γ Γ )/Γ.
L R
−
proximate solutions in our weakly nonlinear approach and We next investigate the efficiency of RTB-based ther-
the dotted lines are the fully nonlinear solutions. The black moelectric devices. Fig. 4 shows the thermodynamic ef-
dashedlineistheheatcurrentpredictedfromlinearresponse. ficiency, η = IV/J , of a heat engine (left panel) and
hot
7
the coefficicent of performance, COP = J /IV, of a 0.15 a 2 b
cold
refrigerator in the linear (dashed black line) and in the
1.5
nonlinear approximation(coloredsolid and dotted lines) ax 0.1 max
for different asymmetries (Γ Γ )/Γ. In linear re- m P 1
L − R η O
sponse, I ∝ ΓLΓR/Γ and J ∝ ΓLΓR, so that both η 0.05 C0.5
and COP are independent of (Γ Γ )/Γ. For the cho-
L R
−
senparameters,themaximalefficiencyisatfinitevoltage 0 0
-10 -5 0 5 10 -10 -5 0 5 10
bias, yet linear response gives a good approximation to (E - E ) / Γ (E - E ) / Γ
r F r F
η in the symmetric case, ΓL = ΓR. This is so, because c 0.8 d
in that case, DV,θ = DV,θ and the screening potential 0.06
L R 0.6
given by Eq. (22) identically vanishes in the absence of
P
gate coupling, C = 0. Introducing an asymmetry in the η 0.04 O0.4
RTB changes the rules of the game, as rectification in C
the electric and heat currents kick in. As the asym- 0.02 0.2
metry increases we can see a clear deviation from lin-
ear response, which emphasizes the importance of treat- 00 0.2 0.4 0.6 00 0.5 1 1.5
eV / Γ eV / Γ
ing screening effects when characterizing thermoelectric
performances of such microdevices. The efficiency η in-
creases or decreases, depending on whether rectification FIG. 5: a) Maximal thermodynamic efficiency, ηmax =
play with or against the bias voltage. This depends on maxVη, for a RTB heat engine and b) maximal coefficient
the signofthe bias voltageandthe directionofrectifica- of performance, COPmax = maxVCOP, for a RTB refriger-
ator as a function of the position of the resonant level. c)
tion. FromFig.3weseethatatpositivevoltage,samples
Efficiency and d) coefficient of performance vs bias voltage
withlargenegativevaluesfortheasymmetry(ΓL−ΓR)/Γ for RTB-based devices with (Er −EF)/Γ = 1. In all cases,
exhibitrectificationeffectswheresimultaneouslytheelec- kBT0/Γ=1,(ΓL−ΓR)/Γ=0.95andthecouplingtothegate
triccurrentisenhancedwhiletheheatcurrentisreduced voltagevariesasCΓ/e2 =0(blue),5·10−6 (green),and10−5
compared to their values in the linear regime. Both ef- (red) with a gate voltage given by eVg/Γ = 3000. Panels a)
fectsconspiretoincreasetheefficiencyinthatcase(cyan and c) are for TR−TL =0.8T0 and panels b) and d) are for
curves in the left panel of Fig. 4). The coefficient of TR−TL=0.1T0. Thesolidcoloredlinesaretheapproximate
performance, on the other hand, is always significantly solutions in our weakly nonlinear approach and the dotted
colored linesare thefully nonlinear numericalsolutions. The
smaller than predicted by linear response, and rectifica-
dashedblacklineisthelinearresponsesolutionandthesolid
tion plays in the opposite direction in the refrigerator
blacklineisthemaximumefficiency/COPpredictedfromlin-
case because COP = J /IV while η = IV/J . An-
cold hot ear response.
other remarkable feature is that both performance met-
rics are well predicted by the weakly nonlinear theory
(compare solid with dotted lines) with larger deviations,
before and despite the rather large biases involved, the
as expected, in the refrigerator case at larger biases.
weaklynonlineartheorygivesaratherfaithfulprediction
So far the capacitive coupling to the externalgate has of the true fully nonlinear behavior.
been set equal to zero, C = 0. Fig. 5 finally shows the From the data shown in this section we conclude that
dependence of η and COP on C. Efficiencies and max- nonlinear effects can significantly increase the efficiency
imal efficiencies for a heat engine are shown in the left of RTB-based thermoelectric engines, first via rectifica-
panels,whilecoefficientsofperformanceandmaximalco- tion effects in both the electric and heat currents, and
efficientsofperformanceforarefrigeratorareontheright second via capacitive couplings which lift the resonant
panels. Forthe chosensetofparameters,linearresponse level.
overestimates η and COP when there is no coupling to
the gate. In both panels c) and d), the capacitive cou-
pling is seen to increase η and COP. Quite remarkably B. Quantum Point Contact
the efficiency ranges from well below the linear predic-
tion for CΓ/e2 = 0 to well above it for CΓ/e2 = 10−5, Nonlinear electric transport through a quantum point
strikingly illustrating the dependence of η on nonlinear contact (QPC) has been discussed theoretically in
effects. The same calculation for RTB’s with all other Ref. [45] and experimentally in Ref. [53]. Linear ther-
asymmetriesreportedinFig.4showsystematicincreases moelectric transport through a QPC has been discussed
in η above its value at CΓ/e2 =0 as the capacitive cou- by Proetto54, assuming a saddle potential whose trans-
pling is turned on. The magnitude of the increase is in mission spectrum is known exactly55–57. Oscillations in
all cases similar to that in Fig. 5, which we attribute to thethermopowerwereinparticularreportedinRef.[54],
similar increasesin U with C, regardlessof the asymme- which we reproduce in the inset to the right panel of
try. Panelsa)andb)alsoshowthatbetter performances Fig. 6. Curiously enough, the efficiency of QPC-based
are obtained for a resonant level lying a distance of a thermoelectric devices has not been calculated as of yet,
few times Γ above or below E . We finally note that, as even in the linear regime, and we briefly comment first
F
8
10 10
1
8 2G [2e / h]0.5 8 S [k / e]B2435 Vg
1
0 0
L0 6 0 0(.5E 1- eU1.5) / 2h_ω2.5 6 0 1(E 2- eU3) / 4h_ω5 C0
GT F 0 x T F 0 x UL UR
Z
/
Ξ 4 4
V
g
2 2
C C
0 0
0 (E1 -2 eU3 ) /4 _h ω5 6 0 (E1 -2 eU3 ) /4 _h ω5 6 FIG. 7: Sketch of a quantum point contact (QPC) with the
F 0 x F 0 x local potential in the conductor discretized into a left and a
rightregion. TheQPCiselectrostaticallydefinedbyexternal
FIG. 6: Left panel: dimensionless ratio Ξ/GTL0 for a QPC gates. The capacitive couplings considered in our theoretical
as defined in the text. This ratio is equal to one when the treatment are also indicated.
Wiedemann-Franz law is valid. Right panel: dimensionless
figure of merit ZT. Left inset: conductance steps for the
QPC. The arrows indicate the parameters corresponding to QPC.We consider a geometricallysymmetric QPC with
Figs.8and9. Rightinset: thermopowerS ofaQPC.Param- only a few open transmission channels, which we model
eters are ωy/ωx = 3 and T = 0.10 (blue), 0.25 (green), 0.50 by a saddle potential55
(red), 1.00 (violet).
1 1
eU(x,y)=eU + mω2y2 mω2x2, (26)
0 2 y − 2 x
on the figure of merit for a QPC. We show in Fig. 6
with the electron mass m and the electric potential U
0
data for the ratio Ξ/GTL with the Lorenz number
0 at the center of the QPC. The transmission probability
L0 = π2(kB/e)2/3 as well as ZT = (GS2/κ)T0. When for each transmission channel is given by56,57
theWiedemann-Franzlawholds,onehasΞ/GTL =125.
0
This is the case whenthe QPC is wellopened andtrans- −1
(ε)= 1+e−πǫn(ε) , (27)
mits several conductance channels. The Wiedemann- Tn
Franz law breaks down when the QPC is closing, and h i
with
the sharp increase displayed by Ξ/GTL in that regime
0
suggests a reduction in ZT. The right panel of Fig. 6
1
howeverindicatesanequallysharpincreaseofZT reach- ǫn(ε)=2 ε+EF ~ωy n+ eU0 ~ωx, (28)
− 2 −
ing almost10,whichis understoodonce onerealizes(see (cid:20) (cid:18) (cid:19) (cid:21).
inset)thattheincreaseinΞ/GTL isaccompaniedbyan
0 for n = 0,1,2,... The density of states for each chan-
increase of S54, with the net result that ZT increases as
nel,ν (ε),hasbeencalculatedinRef.[58],usingaWKB
n
the QPC closes. This results in a large efficiency for an
approach. The result exhibits a nonphysical logarith-
almost completely closed QPC operating as a heat en-
mic singularity close to conductance steps, but retains
gine. This however occurs when the power of the device
its validity away from them. We found that electric and
is exponentially small. Some aspects of nonlinear ther-
heat currents lie on smooth curves except in the imme-
moelectrictransportthroughaQPChavebeenaddressed
diate vicinity of the steps. The results presented be-
in Ref. [14]. We here extend these investigations.
low accordingly use a numerical interpolation to inves-
The geometry of the QPC is sketched in Fig. 7. Two tigate the behavior of QPC-based devices close to con-
lateral gates constrain the motion of electrons in a two- ductance steps,whose exactformdoes notmatter,given
dimensional electron gas to a narrow channel. We take the very small range where the expression of Ref. [58]
these two gates as symmetric and at the same potential breaks down. Other than this interpolation, we use the
and thus consider them together as a single gate. The density of states calculated in Ref. [58].
widthofthe channelistunedbythe electricpotentialon The QPC has been discretized into two separate re-
the gates. We spatially split the QPC into a left and a gions, and we express the particle injectivity into region
rightregion,eachwithitsownlocalpotential. Thesetwo r =L,R from lead j =L, R as45
regions are capacitively coupled to one another (capaci-
1
tanceC0)andtothegate. Forphysicallyrealisticparam- νp (ε)= ν (ε) [ (ε)+2[1 (ε)]δ ] , (29)
eters,anasymmetryinthecapacitivecouplingofthetwo rj 2 n Tn −Tn rj
n
regions to the gate has only a small effect on transport, X
and we therefore consider a symmetric coupling with ca- where the prefactor of 1/2 takes into account that only
pacitance C for both the left and right regions of the electrons moving toward the QPC undergo transmission
9
0.03 2 0.05 5
0.04 4
1.5
0.02
0.03 3
P P
η O1 η O
C C
0.02 2
0.01
0.5
0.01 1
00 e0.0V2 / h_ ω0.04 0.06 00 eV0.2 / _hω 0.4 00 eV0.0 5/ _h ω 0.1 00 eV /0 .2h_ ω 0.4
x x x x
FIG. 8: Left panel: Efficiency of a QPC-based heat en- FIG. 9: Left panel: Efficiency of a QPC-based heat en-
gine. Right panel: Coefficient of performance for a QPC- gine. Right panel: Coefficient of performance for a QPC-
based refrigerator. In both instances the QPC is set at based refrigerator. In both instances the QPC is set at
(inEsFet−ofeFUi0g)./6~,ωωxy=/ω1x.5=co3r,reeVspg/o~nωdixn=g t3o00th0,ekrBedT0a/r~roωwx =in0t.h3e, (inEsFet−oefUF0i)g/.~6ω,xω=y/ω0.x5=co3rr,eesVpgo/n~dωinxg=to3t0h0e0,bklaBcTk0a/r~rωowx =in0t.h3e,
TCR~ω−x/TeL2==00.1(bTlu0e,)e,V5·=10−E5F(−greµeRn)=, aµnLd−10µ−R4,(rCe0d)=. 0 and TCR~ω−x/TeL2==00.1(bTlu0e,)e,V2·=10−E5F(−greµeRn)=, aµnLd−4·µ1R0,−C50(r=ed)0. and
or reflection. Eq. (10) now becomes which accordingly may carry two different local poten-
tials. The data we present all have C = 0, as we found
0
δq DV V + Dθ θ DU , (30) that this capacitive coupling does not change the results
r ≃ rj j rj j − r
j j significantly. Note that for the QPC, only the weakly
X X
nonlinear results are shown.
where r = L,R labels the left/right regions of the QPC.
In Fig. 8 we plot both the efficiency of a QPC-based
As in Eq. (23), the injected charge can alternatively be
heatengine(leftpanel)andthecoefficientofperformance
determined by
of a QPC-based refrigerator (right panel) as a function
of the applied bias voltage when the capacitive coupling
δq =C(U V )+C (U U ), (31)
r r− g 0 r − r¯ C to the gate is varied. Both sets of data correspond
toafirsttransmissionchannelcloseto halftransmission,
where r¯ = L(R) when r = R(L). Eqs. (30) and (31)
1/2, corresponding to the red arrow in the inset
determine the characteristic potentials as T ≃
of the left panel of Fig. 6. Fig. 8 shows that in the ab-
sence of capacitive coupling, the maximal efficiency of
u = (C+C0+D)DrVj +C0DrV¯j, u = C (32) the heat engine is slightly lower than one would predict
rj rg
(C+D)(C+2C0+D) C+D from linear response, in agreement with Ref. [14]. The
(C+C +D)Dθ +C Dθ trendishoweverreversedonce the capacitivecoupling C
zrj = 0 rj 0 r¯j, zrg =0, (33) is turned on. Increasing C increases η and even extends
(C+D)(C+2C +D)
0
the bias range of operation of the heat engine towards
which, again,we use to calculate nonlinear transportco- higher bias. Turning our attention next to the refrigera-
efficients in Eqs. (5) and the currents in Eqs. (2). Note tor case (right panel in Fig. 8) we note that the discrep-
that the characteristic potentials now have two labels, ancybetweenlinearpredictionandnonlinearcalculation
one indicating the regions where they are measured, the is stronger, essentially because the refrigerator works at
other one the lead or gate from which they originate. higherbiasvoltage,thereforelargernonlineareffectsmay
We presentour results forthe efficiency ofQPC-based beexpected. IncreasingC increasestheCOPforrefriger-
devices. For all presented data, we set the chemical po- ation just as it increased the power generation efficiency
tentialoftheleftterminalatE andbiastherightreser- for the QPC when operating as a heat engine. In both
F
voir to lower electrochemical potential, µ = E eV. instancesweseethatthenonlinearefficiencysignificantly
R F
This corresponds approximately to the situation co−nsid- exceeds the linear prediction at large enough capacitive
ered by Whitney14 (see his Fig.2d), though he chose a coupling C.
Heaviside function for the QPC transmission. Another Fig.6showsadramaticincreaseofZT whentheQPC
discrepancy between that work and ours is that we dis- is almost totally closed and its conductance is exponen-
cretize the geometry into two distinct spatial regions, tially small. We investigate whether this region presents
10
opportunitiesforthermoelectricity,inparticularwhether pointed outthat the predictive powerof ZT is no longer
thealreadylargelinearefficiencycanbefurtherincreased guaranteed when nonlinear effects kick in, and directly
by nonlinear effects. In Fig. 9 we plot data similar to investigatedthethermodynamicefficiencyandcoefficient
those presented in Fig. 8 but obtained at smaller E , of performance, including nonlinear contributions to the
F
corresponding to the black arrow in the inset of the left heatandelectriccurrents. Weinvestigateddevicesbased
panelofFig.6. Weseethatnonlinearitiesinfluencether- on two microconstrictions,a quantum point contact and
modynamicperformancesonlyweaklyinthatregime,es- aresonanttunneling barrier. Inboth instanceswe found
sentially because currents are exponentially small. situations where the true efficiency significantly differs
from that predicted by linear response, as well as cases
wherethermodynamicefficienciesstronglydependonthe
VII. CONCLUSIONS strength of these nonlinearities – not only on the magni-
tude of the applied temperature and voltage biases, but
We have presented a weakly nonlinear theory of cou- also on the capacitive coupling to external electrostati-
pledheatandelectrictransportinmicro-andnanoscopic cally charge regions. Most importantly, we found that
devices. On those scales it is of central importance that the weakly nonlinear theory faithfully captures the true
local electrostatic potentials are correctly taken care of, nonlinearbehaviorofmicro-andnanoscopicdevices. We
in order to preserve fundamental conservation laws such believethis opensthewayto moregeneralinvestigations
as gauge invariance and current conservation. We de- ofthermoelectricityinquantummicrodeviceswhichuntil
scribed how this can be done and listed sum rules en- now have often been limited to numerical investigations.
suring that these conservation laws are preserved. Most Acknowledgments. This work was supported by the
works on thermoelectricity use the linear figure of merit NSF under grant PHY-1001017. We thank Andrew Jor-
ZT as a metric for thermodynamic performance – the dan for drawing our attention to Ref. [48]. While this
larger ZT, the better the efficiency of a heat engine workwasbeingcompleted,wereceivedadraftofRef.[59]
and the coefficient of performance of a refrigerator. We which discusses related issues in nonlinear transport.
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