Quantum-dot Carnot engine at maximum power

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by  Massimiliano Esposito

Quantum-dot Carnot engine at maximum power
Massimiliano Esposito
Center for Nonlinear Phenomena and Complex Systems,
Universit´e Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium.
Ryoichi Kawai
Department of Physics, University of Alabama at Birmingham,
1300 University Blvd. Birmingham, AL 35294-1170, USA.
0
1
0 Katja Lindenberg
2 Department of Chemistry and Biochemistry and BioCircuits Institute,
University of California, San Diego, La Jolla, CA 92093-0340, USA.
n
a
J Christian Van den Broeck
3 Hasselt University, B-3590 Diepenbeek, Belgium.
1 (Dated: January 13, 2010)

Weevaluatetheefficiencyatmaximumpowerofaquantum-dotCarnotheatengine. Theuniversal
]
h valueofthecoefficientsatthelinearandquadraticorderinthetemperaturegradientarereproduced.
c Curzon-Ahlborn efficiency is recovered in thelimit of weak dissipation.
e
m
I. INTRODUCTION pled systems in the presence of an additional left-right
-
t symmetry in the system [3]. Furthermore, the univer-
a
t sality of the coefficients is a direct consequence of the
s
Thepurposeofaheatengineistotransformanamount time-reversibility of the underlying physical laws. The
.
at of heat Qh, extracted from a hot reservoir at temper- coefficient1/2derivesfromthesymmetryoftheOnsager
m ature T , into an amount of work W. The efficiency matrix. Thecoefficient1/8canbeseenastheimplication
h
η = W/Q for doing so is at most equal to Carnot ef- of Onsager symmetry at the level of nonlinear response.
- h
d ficiency: η ηc, with ηc = 1 Tc/Th. Here Tc is the The above universality predictions have been con-
n temperature≤ofasecond,coldre−servoirTc Th,inwhich firmed in a number of steady-state model systems in-
o ≤
c the remainingenergyQh−W is deposited. The equality volvingclassicalparticles[4],fermions[5],andbosons[6].
[ isreachedforreversibleoperation,implyingthatthecor- UniversalityhasalsobeenobservedinvariantsoftheCA
responding power output is zero. Curzon and Ahlborn model based on Carnot cycles performed in finite time,
1 wereamongstthe firstto study the questionofefficiency even though finding the optimal driving protocol max-
v
at maximum power [1]. By considering a simple modi- imizing work extraction can be notoriously difficult [7–
2
fication of the Carnot engine and after applying the so- 13]. The connection with the steady-state analysis has
9
1 calledendo-reversibleapproximation(neglectingdissipa- been clarified by identifying the Onsager coefficient for
2 tion in the auxiliary system), they found the following a finite-time Carnot cycle in the linear regime [11]. Fur-
. beautifulexpressionfortheefficiencyatmaximumpower: thermore, agreementwith the universalquadratic coeffi-
1
0 ηca = 1− Tc/Th = 1−√1−ηc = ηc/2+ηc2/8+···. cient has also been observed in a Carnot cycle based on
0 While this formula appears to describe rather well the a (classical) Brownian particle in an harmonic trap [8].
p
1 efficiency of actual thermal plants, and is close to the Inthispaperweprovideafullanalysisofathermalen-
v: efficiency at maximum power for several model systems, gineundergoingaCarnotcycle,withtheauxiliarysystem
i it is neither an exact nor a universal result, and it is consisting of a single-level quantum dot that is switched
X neither an upper nor a lower bound. It has therefore betweenahotandacoldreservoir. We showthatthe ef-
r come as a surprise that a number of universal predic- ficiency at maximum power is again consistent with the
a
tions can be made about the expansion of the efficiency abovediscusseduniversality. Furthermore,CAefficiency
at maximum power in terms of ηc. In the regime of lin- at maximum power is obtained exactly at all orders in
ear response, i.e., at first order in ηc, it is found that ηc in the limit of low dissipation, which is distinct but
the efficiency at maximum power is at most half of the similar to the case considered in the original CA paper.
Carnotefficiency[2]. Inotherwords,theCAefficiencyis
anupperboundattheleveloflinearresponse. Theproof
wasgivenforsystemsoperatingundersteady-statecondi-
II. MODEL
tions. The upper bound is reached for so-called strongly
coupledsystems, i.e., systems in whichthe heat flux and
the work-producing flux are proportional to each other. Our heat engine model consists of a single-level quan-
More recently, it has been shown that the quadratic co- tum dot interacting with a metallic lead through a tun-
efficient, equal to 1/8, is also universal for strongly cou- neling junction. The quantum dot is assumed to have a
2
singleenergylevelεneartheFermileveloftheleadwhile
other levelsin the dotdo not contribute to the processes
described below. The state of the system is specified by
the occupation probability p(t) of having an electron in
the dot. The lead plays the role of a thermal bath at
temperature T and chemical potential µ. Electrons are
assumed to thermalize instantaneously upon tunnelling
into the lead.
When the energy level ε is modulated by an external
agent according to a given protocol, a certain amount of
energy, positive or negative, flows into the system in the
form of work and/or heat. In the case of an occupied
FIG. 1: A Carnot cycle of the model heat engine consisting
level, an amount of work equal to (εf µf) (εi µi) ofasingle-levelquantumdot interactingwith alead through
− − −
is delivered to the system, where the subscripts f and i a tunnleingjunction.
refer to final and initial values. When the electrons at
energy level ε tunnel in (out), an amount of heat equal
to Q=ε µ (Q= ε+µ) is extracted from the bath. the thermodynamic state of the system, in our case the
The ba−sic problem− that we address is the finite-time occupation probability p of the quantum level, returns
performance of this engine as it runs throughthe follow- to the same initial value after every cycle. Since there is
ingfourstandardstagesofaCarnotcycle(alsoseefigure nochangeinoccupationprobabilityduringtheadiabatic
1): stages II and IV, the change in occupation probability
from, say p to p , during process I, must necessarily
0 1
I Isothermal process be compensated by a change back from p to p during
1 0
The quantum dot is in contact with a cold lead process III.
at temperature Tc and chemical potential µc. The The time evolution of the occupation probability p(t)
energy level is raised from ε0 to ε1 according to a forthestateofthequantumdotincontactwithaleadat
certain protocol during a time interval of duration temperature β−1 (k = 1) obeys the following quantum
b
τc. Both work and heat are exchanged during this master equation:
process.
p˙(t)= ω (t)p(t)+ω (t)[1 p(t)], (1)
a b
− −
II Adiabatic process
where the ω and ω are transition rates. In the wide-
The quantum dot is disconnected from the cold a b
band approximation, these rates are given by
lead, and the quantum level is shifted from ε to a
1
new level ε2. Since the quantum dot is thermody- C
namicallyisolated,the populationofthe leveldoes ωa = e−β[ε(t)−µ(t)]+1 (2a)
not change during this process. Hence, there is no
C
heat exchange. However, the change of the energy ω = , (2b)
b e+β[ε(t)−µ(t)]+1
level releases a corresponding amount of work. We
assume that the operation time of this step is very whereCisarateconstant. Notingthatraisingtheenergy
short, in particular negligibly small compared to level is equivalent to lowering the chemical potential, we
that of the isothermal processes. introduceaneffective energylevelǫ ε µ. The master
≡ −
equation (1) can now be rewritten as
III Isothermal process
Thedotisconnectedtothehotleadattemperature C
p˙(t)= Cp(t)+ . (3)
Th and chemical potential µh. The energy level is − eβǫ(t)+1
lowered from ε to ε based on another protocol
2 3 The effective level varies along the Carnot cycle as ǫ =
during a time interval of length τ . Just as in step 0
h ε µ ǫ =ε µ ǫ =ε µ ǫ =ε µ .
I, both heat and work are exchanged. 0− c → 1 1− c → 2 2− h → 3 3− h
Notethatthechangeinthechemicalpotentialisincluded
IV Adiabatic process in the jump of the effective level during processes II and
Thesystemisagaindisconnectedfromtheleadand IV.
the level is restored from ε to the initial level ε , Wenextturntothethermodynamicdescriptionofthe
3 0
atthe costofa correspondingamountofwork. Af- model. We use the convention that heat entering the
terwards, the dot is reconnected to the cold lead. systemis(likework)positive. Theinternalenergyofthe
Again, we assume that the operation time of this system at time t is
process is negligibly small.
E(t)=U(t) µN(t)=ǫ(t)p(t), (4)
−
The above procedure defines one cycle of the thermal
where
engine, requiring a total time τ + τ . The protocols
c h
in steps I and III must be designed in such a way that U(t)=ε(t)p(t), N(t)=p(t). (5)
3
The rate of change of the internal energy, E˙, is the sum Here K is the constant of integration. Solving the
of two parts, namely, a work flux W˙ and a heat flux Q˙, quadraticequationforp˙,weobtaintwofirstorderODEs,
W˙ ǫ˙p=ε˙p µ˙p (6a)
≡ − K(1 2p) K2+4Kp(1 p)
Q˙ ǫp˙ =εp˙ µp˙. (6b) p˙ = − ∓2C(1+K) − . (12)
≡ − p
Note that the particle exchange contributes to the heat
The upper sign ( ) should be used for upwardprocesses
flux [last term in Eq. (6b)]. When the energy level is −
in which the quantum level is raised and the lower sign
below the chemical potential, the direction of heat flow
(+) for downward processes. It is worth mentioning a
is opposite to the direction of tunnelling.
useful symmetry between electrons and holes. We are
The net total work and net total heat during the pro-
using the state of an electron, ǫ(t) and p(t), to describe
cess of duration τ are obtained as functionals of the oc-
the state of system. Instead, we can also use the state
cupation probability,
of holes, ǫ(t) and 1 p(t). If p(t) is a solution for an
τ upwardpr−ocess,then1− p(t)isasolutionforadownward
W[p()] = ǫ˙(t)p(t)dt (7a) −
· process with ǫ(t). Hence we do not need to calculate
Z0 −
τ the downward process separately, as it follows from this
Q[p()] = ǫ(t)p˙(t)dt. (7b) symmetry.
·
Z0 Before turning to the solution of the differential equa-
Forcyclicprocesses,wehaveE(0)=E(τ)andhenceW+ tion (12), we examine the physical meaning of the con-
Q = 0. For mathematical simplicity, we evaluate power stant K. Eliminating p˙ in Eq. (11) by using the master
using net heat instead of net work: equation (3), the resulting quadratic equation for p(t)
W Q leads to the relation
P= − = . (8)
τ τ 1
p(t)= 1+eβǫ(t)/2√K . (13)
eβǫ(t)+1
h i
III. OPTIMIZATION: GENERAL CASE ThisequationindicatesthatwhenK =0,p(t)istheequi-
librium distribution associated with the instantaneous
Ourgoalistomaximize thepoweroutputandtoeval- value of the energy, implying that K = 0 corresponds
uatethecorrespondingefficiency. Powerisacomplicated to the quasi-static limit (τ ). As K increases, p(t)
→ ∞
functionalofthetime-dependentprotocolsinstageIand deviates from the equilibrium distribution. We conclude
III, and an exact analytical analysis looks difficult at thatK measureshowfarthestateofthesystemdeviates
firstsight. Theoptimizationcanhoweverbe doneintwo fromthequasistaticlimit. Wewillusethisinsightbelow
steps. First, we fix parameter values, τ , τ , p and p toobtainaperturbativesolutionforsmalldissipationby
c h 0 1
and maximize the power with respect to the functional assuming that K is small.
space of ǫ(t). Since the total operation time is fixed, we Next we proceed to solve Eq. (12). Separation of the
just need to maximize the heat. Next, we further maxi- variablesp and t leads to the following explicit result for
mize the power with respect to the remaining degrees of the upward processes:
freedom τ , τ , p and p . The problem of maximizing
c h 0 1
heat or minimizing work for a single-level quantum dot Ct=F[p(t);K] F[p(0);K], (14)
−
moving between given initial and final energy states has
where
alreadybeenanalysedindetailin[14]. Wereproducethe
crucial steps of this analysis for self-consistency.
1 1 1 2p
Tofindtheprotocolthatmaximizestheheat,wedonot F(p;K)= lnp+ arctan −
searchdirectly forthe optimalscheduleǫ(t), butidentify −2 √K " K+4p(1 p)#
−
the optimal occupation probability p(t). This is done by
1 2p+K+ K2p+4Kp(1 p)
expressing ǫ(t) in terms of p(t) and p˙(t), and rewriting + ln − .
the heat, Eq. (7b), as a functional of p(t) and p˙(t): 2 "2(1 p)+Kp+ K2+4Kp(1 p)#
− −
(15)
τ p
βQ[p()]= L(p,p˙)dt, (9)
· Z0 Forthedownwardprocesses,weneedtouseF(1−p;K).
where The value of K is determined by the boundary condi-
tions:
1
L ln 1 p˙(t). (10)
≡ p(t)+p˙(t) − Cτ =F[p(τ);K] F[p(0);K]. (16)
(cid:20) (cid:21) −
The extremumisfound viathe standardEuler-Lagrange
Note that K depends solely onthe operationtime τ, the
method, leading, after integration, to
probabilitiesp andp ,andthetunnelingrateC butnot
0 1
∂L p˙2 ontemperature. Unfortunately,thefunction(15)isquite
L p˙ = =K. (11)
− ∂p˙ (Cp+p˙)[C(1 p) p˙] complicatedsowecannotobtainananalyticalexpression
− −
4
for K. In general we need to solve for it numerically. which leads to the efficiency of the engine
However, an exact perturbative solution is possible, cf.
the next section. Tc∆Sc
η =1+ . (21)
Having thus obtained the optimal p(t) with K deter- T ∆S
h h
mined by (14), we insert this expression in Eq. (7b) to
obtain the correspondingmaximum heat for the optimal In the quasi-static limit, Kc 0 and Kh 0, one has
upward processes, S(p;0) = S(p) = S(1 p), he→nce ∆Sc = →∆Sh, so that
− −
Eq. (21) reduces to Carnot efficiency.
τ p(τ)
βQ= ǫ(t)p˙dt= ǫ(p)dp The above results provide the required optimization
with respect to the schedules. It remains to perform the
Z0 Zp(0)
optimization with respect to the remaining degrees of
p(τ) 2p(1 p)+K+ K2+4Kp(1 p)
freedom τ , τ , p and p . In general, this can only be
= dpln − − c h 0 1
Zp(0) " 2pp2 # done numerically since Eq. (14) only provides an im-
=S[p(τ);K] S[p(0);K]=∆S, plicit equation for the time-dependence of the optimal
− schedule. We are, however,mainly interested in the ver-
(17)
ification of universal features of efficiency at maximum
where power. We therefore proceed with a perturbative analy-
sis for which analytic solutions can be obtained.
2(1 p)p+K K2+4Kp(1 p)
S(p;K)=pln − − −
2p2
" p #
1 2p IV. WEAK DISSIPATION LIMIT
√Karcsin −
− √K+1
(cid:20) (cid:21)
The deviation from Carnot efficiency can be investi-
2(1 p) K K2+4Kp(1 p)
ln − − − − . gated using the theory of linear irreversible thermody-
− " p2 # namics where Th Tc is assumed to be smaller than the
−
(18) temperatures Th and Tc of the reservoirs. However, for
finite time thermodynamics, a different kind of expan-
Forthedownwardprocesses,S(p;K)isreplacedbyS(1 sion, directly related to the irreversibility caused by fi-
−
p;K). nite operation time, is more natural. As mentioned in
Out of equilibrium, S is different from the system en- the previous section, K is a direct measure of the devi-
tropy S(p) = plnp (1 p)ln(1 p). Indeed, ∆S is ation from the quasi-static limit. Hence, it is natural to
− − − −
the entropy flow and is related to the system entropy expand thermodynamic quantities in K. Since this is an
change ∆S = S(p(τ)) S(p(0)) via the always-positive expansion about the reversible case of zero dissipation,
entropy production ∆i−S = ∆S ∆S 0. It is only in we will refer to this as the limit of weak dissipation.
− ≥
the quasi-static limit, where K 0 and thus ∆iS = 0, We expand Eq. (15) in a series in √K. The leading
that S(p;K) reduces to S(p). →
term is
We are now ready to apply the above results to our
heat engine. To make the connection with the left/right arcsin(1 2p)
F(p;K)= − . (22)
symmetry required for the universality of the coefficient √K
in the quadraticterm, cf. the discussionin the introduc-
tion, it will be of interest to consider an asymmetry in With this approximation, we are able to solve Eq. (19)
the rate constant: we will use the subscripts Cc and Ch for K to obtain
fortherateconstantC whenincontactwiththecoldand
hotreservoir,respectively. RecallingthatprocessesIand φ1 φ0
K = | − |, (α=c,h) (23)
α
IIIareupwardanddownwardprocesses,respectively,the C τ
α α
boundary condition (16) leads to p
where
C τ =F(p ;K ) F(p ;K ) (19a)
c c 1 c 0 c
−
Chτh =F(1−p0;Kh)−F(1−p1;Kh), (19b) φi =arcsin(1−2pi), (i=0,1). (24)
which determine the integration constants Kc and Kh, The present expansion is thus valid under the following
respectively. condition of weak dissipation:
Substituting K and K into Eq. (17), we obtain the
c h
amount of heat that enters the system during the pro- C τ φ φ , (α=c,h). (25)
α α 1 0
cesses I and III: ≫| − |
Notethatitcaneasilybesatisfiedinourmodelsincethe
Q =T [S(p ;K ) S(p ;K )]=T ∆S (20a)
c c 1 c − 0 c c c right hand side is bounded by π.
Q =T [S(1 p ;K ) S(1 p ;K )=T ∆S ,
h h 0 h 1 h h h OncewefindthevalueofK,theremainingcalculation
− − −
(20b)
is straightforward. Equation (14) leads to the optimal
5
protocols: dissipation. That is, for the process III the following
inequality must be satisfied:
1 t
p (t) = 1 sin φ φ φ (26a)
pc(t) = 21(cid:20)1+−sin(cid:18)τtc|φ1− φ0|−+φ0(cid:19)(cid:21). (26b) |φC1h−τhφ∗0| = 2|φ1∆−Sφ0|Th(1+ThT−cCThc/ThCc) ≪1.
h 1 0 1 (32)
2(cid:20) (cid:18)τh| − | (cid:19)(cid:21) This can be achieved in two wayps. The first one cor-
Expanding in a Taylor series with respect to √K, Eq. responds to the usual condition for linear irreversible
thermodynamics, (T T )/T 1. The alternative
(18) is approximated by the two lowest order terms as h − c h ≪
is ∆S/(φ φ ) 1. In this limit, our result remains
1 0
| − | ≪
S(p)=S(p) arcsin(1 2p)√K. (27) valid even for large temperature differences.
− − Withtheoptimizedoperationtimes(31),theresulting
power is written as a function of p and p :
Inserting the value of K, we obtain the maximum heat 0 1
∗ (φ1 φ0)2 P = (Th−Tc)2 D(p ,p ), (33)
Q = T ∆S T − (28a) 0 1
c − c − c Ccτc 4( Th/Ch+ Tc/Cc)2
Q∗ = T ∆S T (φ1−φ0)2 (28b) where p p
h h − h C τ
h h
∆S2
where ∆S = S(p0) − S(p1) is the reversible entropy D(p0,p1)= (φ1 φ0)2 . (34)
change. The second term on the right hand side is the −
irreversible heat, which has to be small under the con- The power reaches its maximum when D(p ,p ) takes a
0 1
dition (25) of weak dissipation. In the quasi-static limit maximum value, D = 0.439 at p = p = 0.0832 or
max 0 1
(τ ), the second term vanishes and the efficiency p =p =0.9168. Atthese conditions,the optimaloper-
→ ∞ 0 1
(21) reaches the Carnot efficiency, as expected. ationtime (31)andthemaximumheat(28)bothvanish.
When the operation time is too short, the irreversible However, the power remains finite. This final optimiza-
heat becomes dominant and the net heat becomes nega- tion thus leads to a singular and unrealistic situation.
tive. Equation (28) indicates that positive power can be We note, however, that since Eq. (34) does not depend
obtained only if on the system parameters,the efficiency does in fact not
depend on this final optimization step. Therefore, we
(T T )∆S T T
h− c > c + h . (29) proceed to evaluate efficiency without further reference
(φ1 φ0)2 Ccτc Chτh to optimal occupation probabilities.
−
Using the maximum heat (28) and the optimal time
This inequality is consistent with the condition of the
(31), we finally obtain the following remarkable result
asymptoticexpansion(25)andcanthusbesatisfiedeven
for the efficiency at the maximum power:
for a large temperature difference.
Sofar,wehavemaximizedthepoweronlyforthefixed
η (1+ C T /C T )
∗ c h c c h
operation times τ and τ and the boundary values p η = ,
c h 0
2(1+ C T /C T ) η
and p of the occupation probabilities. Now we further ph c c h c
1 −
maximize the power η η2 η3
= c + p c + c +o(η4) (35)
2 4(1+√r) 8(1+√r) c
Q +Q
c h
P = (30)
τc+τh with r =Ch/Cc. When r=1, the efficiency (35) exactly
coincides with the Curzon-Ahlborn efficiency η =1
CA
with respect to the operation times. It is easy to find −
T /T . Note also that the efficiency is bounded below
c h
that the power is a maximum when
by η /2 for C /C and bounded above by η /(2
c h c c
p →∞ −
η ) for C /C 0. These limits can be realized without
τ∗ = 2(φ1−φ0)2Tc(1+ ThCc/TcCh) (31a) vciolatinghthecc→ondition of weak dissipation.
c C ∆S(T T )
c hp− c
2(φ φ )2T (1+ T C /T C )
∗ 1 0 h c h h c
τh = − C ∆S(T T ) . (31b) V. DISCUSSION
h hp− c
This optimization reflects the usual competition with Wehavecalculatedtheefficiencyηatmaximumpower
the denominatorofthe powerpreferringfasteroperation of a Carnot cycle with a single level quantum dot as
whereasthenumeratorsuggestsaslowerscheduletostay the operational device. Our calculation is in agreement
closer to Carnot efficiency. withknownuniversalityproperties. Inparticular,theef-
For the asymptotic expansion to be valid, the optimal ficiencyatmaximumpowerisequaltohalfoftheCarnot
operation times must satisfy the condition (25) of weak efficiency in the regime of linear response, η =η /2+....
c
6
In the case of a left/right symmetry, corresponding to tion is more general. It remains to be explored whether
equal exchange rate coefficients C =C of the dot with this observation implies a wider range of validity of CA
h c
theheatreservoirs,thecoefficientofthequadratictermis efficiency. Inparticular,itcouldexplainwhyobservedef-
alsogivenbyitsuniversalvalue1/8,η =η /2+η2/8+.... ficiencies at maximum power are not very different from
c c
However,we need to stress that this result was obtained CA efficiency in a wide range of systems under opera-
not by an expansion in η but in the limit of weak dissi- tional conditions far from linear response.
c
pation. In fact, this calculation adds a new perspective
concerning the occurrence of Curzon-Ahlborn efficiency
itself. Indeed,inthepresenceofleft/rightsymmetry,the
efficiency is actually exactly equal to the CA efficiency
Acknowledgments
η =1 √1 η ,inthelimitofweak dissipation. This
CA c
− −
limit is reminiscent of the original derivation of CA effi-
ciency,andisinthepresentmodelformallysimilartothe M.E.issupportedbytheBelgianFederalGovernment
assumption of a linear conduction law between reservoir (IAP project “NOSY”). This research is supported in
and quantum dot. Howeverthe concept of weak dissipa- part by the NSF under grant PHY-0855471.
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