Strong bounds on Onsager coefficients and efficiency for three terminal thermoelectric transport in a magnetic field

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by  Kay Brandner

Strong bounds on Onsager coefficients and efficiency
for three terminal thermoelectric transport in a magnetic field
1 2 1
Kay Brandner, Keiji Saito, and Udo Seifert
1II. Institut fu¨r Theoretischen Physik, Universita¨t Stuttgart, 70550 Stuttgart, Germany
2Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Japan 223-8522
(Dated: December 11, 2013)
For thermoelectric transport in the presence of a magnetic field that breaks time-reversal
3 symmetry,astrongboundontheOnsagercoefficientsisderivedwithinageneralset-upusingthree
1 terminals. Asymmetric Onsager coefficients lead to a maximum efficiency substantially smaller
0 thantheCarnotefficiencyreachingonlyηC/4inthelimitofstrongasymmetry. Relatedboundsare
2 derivedfor efficiency at maximum power, which can become larger than theCurzon-Ahlborn value
ηC/2,andfor acooling device. Ourapproach revealsthatinthepresenceofreversiblecurrentsthe
n
standard analysis based on the positivity of entropy production is incomplete without considering
a
therole of current conservation explicitly.
J

3
PACS: numbers: 05.70.Ln, 72.15.Jf

]
h
c The Onsager reciprocity theorem derived more than duction of the attainable efficiency of any thermoelec-
e eighty years ago is arguably the most important con- tric device within the broadand well-establishedclass of
m
straintforthephenomenologicalframeworkoflinearirre- three-terminal models [13–16].
- versiblethermodynamics[1]. Initsoriginalformitstates Thermoelectric transport can conveniently be dis-
t
a that the linear kinetic coefficient Lik(B), which relates cussed within the model sketched in Fig.1. Two particle
t
s the affinity k to the flux Ji is equal to the reciprocal reservoirsofrespectivetemperaturesTL >TR andchem-
t. coefficientLFki( B) providedboth i andk refer to quan- ical potentials µL <µR are connected by a conductor C,
a −
tities with even signature under time-reversal [2]. The whichallowsfortheexchangeofheatandparticles. Con-
m
constant magnetic field B breaks the time-reversal sym- sequently,assoonasthesteadystateisreached,constant
-
d metry. A generalization which includes variables with heat and particle currents, Jq and Jρ, flow between the
n odd signature was given by Casimir in 1948 [3]. Ever left and the right reservoirs. Depending on the sign of
o since, the reciprocal relations have turned out to be ex- these currents,the machineworkseither as a powergen-
c tremely useful for a plethora of applications. A promi- erator or a refrigerator. We now assume that both the
[
1 nalelnotwosnteoitsretahtetuhneifiveadriothuesotrhyeormf toheelermctoriecleecfftericctistysuwchhicahs itceamlppeortaetnutriealdidffieffreernecnece∆∆Tµ≡TLµ−LTR µ>R0<and0tahreecshmemal-l
v the Peltier, the Seebeck or the Thomson effect on an compared to the respective r≡eferen−ce values, which we
2
9 equalfootingthusrevealingtheirinterdependencies[2,4]. chooseto be T TR andµ µR. Inthis linearresponse
≡ ≡
4
It is well known that properly designed energy filters
0
allowasubstantialenhancementoftheefficiencyofther-
1. moelectricdevices[5]. Linkeandco-workers[6,7]showed C
0 that, in principle, even Carnot efficiency is attainable. μ T
3 P P
Despite these promising theoretical results, the actual
1
: performanceofpresentdevices is notoriouslymuchmore
v
modest, whichkeepsthe searchfor better thermoelectric
i
X materials a very active and increasingly important re-
r searchfieldasreviewedin[8–11]. Recently,inanintrigu-
a
ing paper, Benenti et al. pointed out that the presence μ μ
B
ofa magneticfield couldinprinciple enhance the perfor- L R
manceofathermoelectricdevice[12]. Infact,theirquite T T
general thermodynamic analysis invoking only the On- L R
sager reciprocity relations and the positivity of entropy
production suggested the possibility of a device deliv-
ering finite power while still reaching Carnot efficiency.
FIG. 1. Sketch of a thermoelectric device in the presence of
Such a spectacular optiondeservesboth further scrutiny a magnetic field B. The entire dashed box represents the
and a search for a microscopic realization. In this Let-
conductor C, which is connected to two reservoirs. For the
ter, we show that unitarity of the scattering matrix as special case of the three terminal model, the conductor con-
a general physical principle imposes a strong restriction sistsessentiallyofascatteringregionandanadditionalprobe
on the Onsager coefficients that lead to a significant re- terminal.
2
regime,thecurrentsJ andJ arerelatedtotheaffinities them. This set-up constitutes the quite general class of
ρ q
2
∆µ/T and ∆T/T via the phenomenological three-terminal models on which we base our subsequent
ρ q
F ≡ F ≡
equations analysis. We emphasize that the additional terminal P
playsacrucialrolesinceitiswellknownthatinapurely
J =L +L
ρ ρρFρ ρqFq (1) coherent two terminal set-up the off-diagonal Onsager
Jq =Lqρ ρ+Lqq q. coefficients must be even functions of the magnetic field
F F
[17] and hence there are no reversible currents. Such
The constant rate of entropy production accompanying
probeterminals,whosetemperatureandchemicalpoten-
this transport process reads
tial are chosenself-consistently, were originallyproposed
S˙= J + J =L 2+L 2+(L +L ) . (2) byBu¨ttiker[18]andhavebecomeacommontooltosimu-
Fρ ρ Fq q ρρFρ qqFq ρq qρ FρFq late inelastic events inanotherwise conservativesystem.
Clearly,thefeaturesofthismachinearecompletelydeter- Since we now have to deal with three reservoirs, we
mined by the kinetic coefficients L (i,j = ρ,q), which need accordingly four affinities A (µ µ)/T and
ij Fρ ≡ A −
are subject to two fundamental constraints. First, the A (T T)/T2 (A = L,P), where we still use the
second law requires S˙ 0, which is equivalent to the Fteqmp≡eratuAre−andchemicalpotentialoftherightreservoir
≥
conditions as reference values µ and T. For convenience, we collect
the affinities in two vectors FA ( A, A). Analo-
L ,L 0 and L L (L +L )2/4 0. (3) ≡ Fρ Fq
ρρ qq ≥ ρρ qq− ρq qρ ≥ gously we define the current vectors JA = (eJA,JA),
ρ q
Second, Onsager’s theorem implies the symmetry rela- with JA and JA the particle and heat currents flowing
ρ q
tion out of reservoir A and e the electronic unit charge. This
vector notation allows us to write the phenomenological
Lρq(B)=Lqρ( B). (4) equations in the rather compact form
−
Here, we have reintroduced the magnetic field B which JL FL L′ L′
was suppressed so far in order to keep the notationslim. JP =L′ FP with L′ ≡ L′LL L′LP . (6)
In the light of the Onsager reciprocity relation (4), (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) PL PP(cid:19)
the expression (1) for the total entropy production rate Here,theL′ (A,B =L,P)are2 2-matricesofkinetic
suggests a natural splitting of each current Ji into a re- coefficients.ABBy virtue of the addit×ional constraint JP =
versible and an irreversible part defined by 0,wecanmoreovereliminateFP andtherebyreduce(6)
to a system of two equations
L L L +L
rev ij ji irr ij ji
J − and J L + .
i ≡ 2 Fj i ≡ iiFi 2 Fj JL =LFL (7)
(5)
Obviously, Jrev vanishes for B = 0, i.e. for unbroken
i with an effective matrix of kinetic coefficients
time-reversalsymmetry. However,once the latter is bro-
ken, although not contributing to S˙, the reversible cur- L L′ L′ L′−1L′ Lρρ Lρq (8)
rentscan,inprinciple,becomearbitrarilylarge,sincebe- ≡ LL− LP PP PL ≡ Lqρ Lqq
sides(3)nofurthergeneralrelationsbetweenL (B)and (cid:18) (cid:19)
ij
L (B) areknown. Suchunconstraintreversiblecurrents relevant for the net currents from L to R.
ji
couldultimatelygiverisetothepossibilityofdissipation- Explicit expressions for the 2 2 block matrices L′
× AB
lesstransportnoticedbyBenentietal. [12]asmentioned of kinetic coefficients showing up in (6) arise from the
above. multi-terminal Landauer formula [19, 20]
For progress investigating possible further constraints
tmhaedpehmenooremsepneocliofigci.caAl speat-ruapdigdmisacutiscsemdodsoelffaorrmthuestcobne- L′ = e2T ∞ dE F (E) 1 E−eµ 2
AB h E−µ E−µ
ductor C is shown inside the dashed box in Fig. 1. It Z−∞ e e !
consists of a central scattering region, with a constant (cid:16) (cid:17) 2
δ S (E,B) , (9)
AB AB
magnetic field B, and a third electronic reservoir (ter- × −| |
(cid:16) (cid:17)
minal) P, whose temperature and chemical potential are
where
chosen such that no net exchange of particles and heat
with the rest of the system occurs. The scattering re- 2 E µ −1
F(E) 4k T cosh − (10)
gion is connected to all of the reservoirs via perfect, B
≡ k T
one-dimensional, infinitely long leads. In order to keep (cid:20) (cid:18) B (cid:19)(cid:21)
the model as simple as possible, we assume moderately is the negative derivative of the Fermi function, k de-
B
damped [6], non-interacting electrons, which means that notes Boltzmann’s constant and h Planck’s constant.
noinelasticscatteringeventstakeplaceoutsidethereser- S (E,B) = S (E,B) are the matrix elements of the
AB BA
voirsandtheelectronsaretransferredcoherentlybetween 3 3scatter6ingmatrixS(E,B)thatdescribesthepassage
×
3
ofelectronswithenergyE throughthescatteringregion. 1
CurrentconservationrequiresthatS(E,B)isunitaryand
the time reversal invariance of unitary dynamics implies 0.8
the symmetry S(E,B) = S(E, B)t [19]. We emphasize
that in the presence of a magn−etic field, this symmetry ηC 0.6
/
pspeirtmeitthseS(fEac,tBt)hatot btheenLon′ symarmeesttrililc.syCmonmseetqruice,nttlhye,dree-- ∗ηmax 0.4
AB
ducedmatrix(8)willgenericallyacquireanon-vanishing
0.2
asymmetric part as desired.
We now derive a constraint on the asymmetry of the 0
reduced matrix L. First, we define the Hermitian matri- -4 -2 -1 0 1 2 4
ces x
1
K′ (L′+L′t)+i√3(L′ L′t) (11)
≡ −
0.8
and C
η
/ 0.6
K≡(L+Lt)+i√3(L−Lt)≡(cid:18)KKρρ∗ρq KKρqqq(cid:19), (12) ()Pmax 0.4
∗
η
where L′ is the full 4 4 Onsager-matrix introduced in 0.2
(6),whileListheredu×cedmatrix(8). Second,asaconse-
quence of the unitarity of the scattering matrix S(E,B), 0
K′ can be shown to be positive semidefinite on C4 [21]. -4 -2 -1 0 1 2 4
Third, since for any z C2 x
∈
z FIG. 2. Bounds on benchmarks in units of ηC as functions
z†Kz=z′†K′z′ ≥0 with z′ ≡(cid:18)−L′−PP1L′PLz(cid:19), (13) oηfm∗atxh,etahseymlomweertryonpearηa(mPmetaexr)∗x.. TThheesuoplipdercudrivaegsrarmeprsehsoewnst
it then follows that the reduced matrix K must be pos- the bounds following from relation (15). In the limit x →
±∞ both functions asymptotically approach the value 1/4,
itive semidefinite on C2. Therefore the matrix elements
whichisshownbythedash-dottedlines. Forcomparison,the
of K have to obey the inequalities boundsobtainedbyBenentietal. [12]solelyfromthesecond
law(3)havebeenincludedasdashedcurves. Thedottedline
Kρρ,Kqq ≥0 and KρρKqq −KρqKρ∗q ≥0. (14) in the lower panel indicates theCurzon-Ahlborn limit 1/2.
We note that the K (i,j =ρ,q) are rather complicated
ij
functions of the matrix elements of the full Onsager- The new bound (15) has profound consequences for
matrix L′, which would make it a quite challenging task the performance of the model as a thermoelectric heat
to obtain (14) directly from (8) and (9). If we insert the engine. Following the lines of Benenti et al. [12], we
definition (12) of the Kij in terms of the Lij, we imme- introduce the dimensionless parameters
diately get the relation
y L L /DetL and x L /L . (17)
ρq qρ ρq qρ
2 2 ≡ ≡
L L +L L L L 0, (15)
ρρ qq ρq qρ− ρq− qρ ≥
Expressed in terms of x and y, the inequality (15) reads
which constitutes our first main result. We emphasize
that (15) provides a much stronger constraint than (3) h(x) 4y 0 if x<0,
≤ ≤ (18)
since the former can be rewritten as 0 4y h(x) if x>0,
≤ ≤
LρρLqq (Lρq+Lqρ)2/4 3(Lρq Lqρ)2/4. (16) where h(x) 4x/(x 1)2 . The most important bench-
− ≥ − ≡ −
marks for the performance of a heat engine - namely the
Indeed,werecover(3)onlyifLρq isequaltoLqρ. Assoon maximumefficiency ηmax andtheefficiency atmaximum
asthekineticcoefficientscontainfiniteasymmetricparts, power η(Pmax) - admit the analytic expressions [12]
thelefthandsideof(15)mustbestrictlylargerthanzero.
In other words, the reversible currents associated with √y+1 1 xy
the asymmetricpartof L come atthe price ofa stronger ηmax =ηCx√y+1+−1 andη(Pmax)=ηC4+2y. (19)
lowerboundontheOnsagercoefficientsandhenceonthe
entropyproductionrate(2)thanthebaresecondlaw(3) Here,we havedenotedby ηC =1−TR/TL ≈TFqL the
requires. Carnot efficiency, which is the absolute upper bound for
4
1 strongconstraintonthelineartransportcoefficientsthat
can be obtained neither from Onsager’s principle of mi-
0.8 croreversibility nor from thermodynamic arguments in-
voking only the second law. This constraint implies
rC 0.6 strongboundsonbothmaximumefficiencyandefficiency
η
/ at maximum power if this device is operated as a ther-
∗ε 0.4
moelectric heat engine or refrigerator.
0.2 We emphasize that our results rely solely on the uni-
tarity of the scattering matrix. Therefore our new re-
0 lation (15) applies to any specific model which can be
-4 -2 -1 0 1 2 4
formulated within the three terminal set-up, including
x especially the ones based on quantum dots [16] and bal-
listic microjunctions [15], which have been proposed in
FIG. 3. Bounds on the coefficient of performance of a re-
this context recently. It might even apply to a model
frigerator ε in units of ηCr as a function of the asymmetry introduced by Entin-Wohlman and Aharony [14], which
parameter x. The dashed line follows from bare second law,
includesaphononbathlocallyinteractingwithelectrons
thesolid line from the stronger relation (15).
transferred between the reservoirs. Furthermore it is
noteworthy that by following exactly the same lines the
the attainable efficiency following from the second law. constraint (15) can be shown to apply to the three ter-
Both benchmarks become maximal for 4y = h(x). The minal railway switch transport model [13]. This model
resulting bounds, ηm∗ax and η∗(Pmax), which are our sec- may be regarded as a classical analogue of the quan-
ond main result, are plotted in Fig. 2. It shows how tum three terminal model discussed here. Indeed, our
the maximum efficiency decays rapidly as the asymme- considerations are not restricted to the quantum realm,
try parameter x deviates from its symmetric value 1. In since ultimately the unitarity of the scattering matrix is
the limit x , maximumefficiency and efficiency at nothing but the manifestation of the law of current con-
→±∞
maximum power both approach η /4. This result im- servation, which should be considered the fundamental
C
plies essentially that from the perspective of maximally principle underlying our bound.
attainable efficiency the thermodynamic cost of the re- Despite this generality,the question whether a similar
versible currents is larger than the benefit they bring. bound can be found for more complex model classes,
For efficiency at maximum power, the situation is some- includingforexamplealargernumberofprobeterminals
what different. If x is only slightly larger than 1, the or models with genuinely interacting electrons, remains
Curzon-Ahlborn-limit η = η /2 [22–26], reached for open and should constitute an important subject for
CA C
x=1, canbe overcomeina smallrange ofx values with further investigations.
a maximumof 4η /7 at x=4/3. This resultshows that
C
despite the strongbounds onηmax it mayin principle be Acknowledgments: K.B. and U.S. acknowledge sup-
possible to improve the performance of the machine by port from ESF through the EPSD network. K.S. ac-
breaking the time-reversal symmetry slightly. knowledges support from MEXT (23740289).
Finally, we discuss the consequence of the bound (15)
for the modelas a refrigerator. In this casethe mostim-
portant benchmark is the coefficient of performance ε
≡
JL/T LJL [2] defined as the ratio of the heat current
−extqracteFdρfroρmthecoldreservoirandtheabsorbedpower. [1] L. Onsager, Phys. Rev.38, 2265 (1931).
Intermsofthedimensionlessparameters(17)εreads[12] [2] H. B. Callen, Thermodynamics and an Introduction to
Thermostatics, 2nd ed. (John Wiley & Sons, New York,
ηr √y+1 1 1985).
ε= C − , (20) [3] H. B. G. Casimir, Rev.Mod. Phys.17, 343 (1945).
x √y+1+1
[4] C.Goupil,W.Seifert,K.Zabrocki,E.Mu¨ller, andG.J.
Snyder,Entropy 13, 1481 (2011).
where ηCr = TR/(TL − TR) ≈ 1/TFqL is the efficiency [5] G. D. Mahan and J. O. Sofo,
of an ideal refrigerator. Like for ηmax and η(Pmax), the Proc. Natl. Acad. Sci. USA 93, 7436 (1996).
maximumε∗ ofεisattainedfor4y =h(x). Theresulting [6] T. E. Humphrey, R. Newbury, R. P. Taylor, and
bound is plotted in Fig. 3. Again we observe that the H. Linke,Phys. Rev.Lett. 89, 116801 (2002).
[7] T. E. Humphrey and H. Linke,
efficiency deteriorates rapidly as x deviates from 1.
Phys. Rev.Lett. 94, 096601 (2005).
In conclusion, we have investigated the thermoelec-
[8] C. J. Vineis, A. Shakouri, A. Majumdar, and M. G.
tric transport properties of the most general version of Kanatzidis, Adv.Mat. 22, 3970 (2010).
the paradigmatic three terminal model in the presence [9] L. E. Bell, Science 321, 1457 (2008).
of broken time reversal symmetry. We have derived a [10] G.J.SnyderandS.Toberer,NatureMater.7,105(2008).
5
[11] B. M. S. Dresselhaus, G. Chen, M. Y. Tang, R. Yang, [17] M. Bu¨ttiker, IBM J. Res. Develop.32, 317 (1988).
H.Lee, and D. Wang, Adv.Mat. 19, 1043 (2007). [18] M. Bu¨ttiker, Phys. Rev.B 33, 3020 (1986).
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Phys.Rev.Lett. 106, 230602 (2011). [20] U. Sivan and Y. Imry,Phys.Rev.B 33, 551 (1986).
[13] M. Horvat, G. Benenti, and G. Casati, (2012), [21] See SupplementalMaterial at [URL] for more details.
arXiv:1207.6014v1. [22] F. L. Curzon and B. Ahlborn,
[14] O. Entin-Wohlman and A. Aharony, Am. J. Phys. 43, 22 (1975).
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Supplemental Material for
Strong bounds on Onsager coefficients and efficiency
for three terminal thermoelectric transport in a magnetic field
Here, we prove that the matrix the matrix elements K′ (K′) (l,m = 1,...,4) can
lm ≡ lm
be written in the form
K′ (L′+L′t)+i√3(L′ L′t) (1)
≡ −
3
1 as introduced in Eq. (11) of the main text is positive K′ = e2T ∞ dE F (E)v†(E)V(E,B)v (E) (3)
20 semidefinite on C4. To this end, we first notice that by lm h Z−∞ l m
virtue of the multi-terminal Landauer formula [cf. Eq.

n (9) and Eq. (10) of the main text]
with
a
J
3 L′ = e2T ∞ dE F (E) 1 E−eµ 2 1 E−µ 0 0
] AB h Z−∞ E−eµ E−eµ ! v1 ≡ 0 , v2 ≡ 0e , v3 ≡ 1 , v4 ≡ E−µ
h (cid:16) (cid:17) 2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) e (cid:19)(4)
c δAB SAB(E,B) , (2)
× −| | and the Hermitian matrix
e
(cid:16) (cid:17)
m
-
at 2(1 S 2) S 2+ S 2+i√3(S 2 S 2)
st V= SLP 2+ SPL 2 −i√| 3L(LS|LP 2 SPL 2)| LP| | PL2|(1 SPP| 2L)P| −| PL| . (5)
. (cid:18)| | | | − | | −| | −| | (cid:19)
t
a
m
- From (4) onwards we notationally suppress the explicit of the determinant yields
d
dependence of the v , V and the elements of the scatter-
n l DetV
o ing matrix on energy E and magnetic field B. For any =(1 SLL 2)(1 SPP 2)
[c z≡(z1,z2,z3,z4)t ∈C4 we have 4 −| SL|P 4 −|SPL 4| + SLP 2 SPL 2. (8)
−| | −| | | | | |

1 Now, we distinguish two cases. First, we assume
v 4 S 2 S 2 and rewrite (8) as
LP PL
2 z†K′z= zl∗zmKl′m |Det|V≥| |
49 l,Xm=1 4 =(|SLP|2+|SLR|2)(|SLP|2+|SRP|2)
0 = e2T ∞ dE F(E) 4 z∗v† V 4 z v (6) SLP 4 SPL 4+ SLP 2 SPL 2
01. e2hT Z−∞∞ Xl=1 l l! mX=1 m m! =|SL−P||2|SR|2P−|2+| |S2L|R|2||SLP2||2+| |SL|R|2|SRP|2
3 = dE F(E)y†Vy + SPL (SLP SPL ) 0.
1 h Z−∞ | | | | −| | ≥ (9)
:
v
Here, we used the sum rule
i
X with 2 2
S = S =1, (10)
AB BA
r | | | |
a A=XL,R,P A=XL,R,P
4 whichisadirectconsequenceoftheunitarityofthescat-
y≡ zmvm. (7) tering matrix S. On the other hand, if |SLP|2 < |SPL|2,
m=1 we can use (10) to write
X
DetV
2 2 2 2
=(S + S )(S + S )
PL RL PL PR
The last line of (6) reveals that if V is positive semidefi- 4 | | | | | | | |
nite, the same holds for K′. SLP 4 SPL 4+ SLP 2 SPL 2
−| | −| | | | | |
2 2 2 2 2 2
InordertoshowthatVisindeedpositivesemidefinite, = SPL SPR + SRL SPL + SRL SPR
| | | | | | | | | | | |
we firstobservethat both diagonalentries of V areobvi- + S 2(S 2 S 2) 0.
LP PL LP
| | | | −| | ≥
ously non-negative. It thereforesuffices to showthat the (11)
determinant ofV is non-negative. The directcalculation
In conclusion, we have shown that DetV 0 and there-
foretheproofthatK′ispositivesemidefini≥teiscomplete.