Particle-exchange heat engine working between bosonic and

PHYSICAL REVIEW E 83, 062104 (2011)

Particle-exchange heat engine working between bosonic and fermionic reservoirs Peng Li1,* and Beike Jia2 1

Department of Physics, Zhejiang Ocean University, Zhoushan 316000, China 2 Division of Mathematics and Natural Sciences, Gordon College, Barnesville, Georgia 30204, USA (Received 20 February 2011; revised manuscript received 3 May 2011; published 27 June 2011) We propose a mesoscopic particle-exchange heat engine realized via coupled quantum dots. The engine is able to extract work from a bosonic reservoir and can be used as a refrigerator. The electric current is derived from a master-equation method and is readdressed under a thermodynamic context. The quantum-mechanical result is found to be consistent with thermodynamic analysis. DOI: 10.1103/PhysRevE.83.062104

PACS number(s): 05.70.−a, 03.65.Yz, 73.23.−b, 82.60.Qr

At zero temperature, when a quantum dot (QD) is attached to two contacts, an electric current is induced if the QD-state energies sit within the transport window [1]. At finite temperature, the electrons at each contact constitute a thermal reservoir under local equilibrium. Thus, the FermiDirac statistics of the electron reservoirs plays an important role in electron transport [2]. In the simplest case, consider sequential tunneling via a single QD state. The direction of the current depends solely on the difference between the two reservoirs’ Fermi-Dirac distribution functions at the energy of the QD state. The interplay between the bias voltage, the reservoir temperatures, and the QD-state energy thus exhibits an interesting thermodynamic division of the electron transport [3]: (i) a heat engine that absorbs heat from the hot reservoir and outputs electric work; (ii) a refrigerator that dissipates electric work and absorbs heat from the cold reservoir; (iii) a work-to-heat convertor that transforms work to heat and deposits it to one or both of the reservoirs. The above thermodynamics in QD electron transport provides a playground for many proposed nanoscale heat engines [4]; for example, the cooling of a mesoscopic metallic island [5] or a two-dimensional (2D) electron gas [6]. A common characteristic shared by these studies is that only the temperatures of electrons are exploited in the transport (i.e., the heat engines only work between fermionic reservoirs). The outer environment (e.g., lattice phonons) of the contacts and the QDs serve solely as a “super bath” for the electrons. In this sense the environment only establishes and maintains an equilibrium temperature of the electron reservoirs and plays no significant role in the electron transport. Here we propose a particle-exchange heat engine which can extract work directly from an external bosonic reservoir. Such a heat engine is realized via a series of three coupled QDs, where two outer QDs are coupled to two fermionic reservoirs and the central QD is coupled with a bosonic reservoir. Using a quantum master-equation method, we derive an analytical result for the steady-state tunneling current and describe its thermodynamic content. The refrigerator mode of this heat engine can be used to cool a bosonic reservoir. Thus, it may find an application in the ground-state cooling of a micromechanical system [7].

*

[email protected]

1539-3755/2011/83(6)/062104(4)

We note that heat engines in quantum-optical systems have been extensively discussed (see, e.g., the quantum amplifier studied in [8–10]). Both the quantum amplifier and our model can be viewed as a multistate quantum heat engine operating in a continuous mode with an efficiency bounded by the Carnot limit. However, our model differs from the quantum amplifier not only in its application area, but also in three major aspects: (i) It works in a particle-exchange manner in which carrier particles are transported from one reservoir to another, whereas the quantum amplifier only exchanges energies with the reservoirs. (ii) It outputs power via a dc current instead of by electromagnetic waves. (iii) It uses three thermal reservoirs, although the two fermionic ones are assumed to have equal temperatures in our analysis. Moreover, we emphasize that this engine is not a conventional thermionic or thermoelectric device [11,12]. In those devices the electron current is driven essentially by a temperature gradient; therefore, the electron reservoirs must be kept at different temperatures to generate power or refrigerate. The terms thermionic and thermoelectric merely differentiate between a ballistic and a diffusive current. In contrast, our engine operates with the two electron reservoirs at the same temperature and exploits heat from a bosonic reservoir. The underlying physics is different and can be best termed as thermal-boson-assisted electron current at a single temperature. The model consists of three coupled QDs named a, b, and c, as sketched in Fig. 1. The QDs a and c are attached to two contacts named source Rl and drain Rr , respectively, and are both coupled to QD b via coherent tunneling. QD b, besides being coupled with a and c, is thermally in contact with the bosonic reservoir Rb . |a and |c are the only 1-electron states in a and c, respectively, with energies Ea and Ec . |b and |b are 1-electron states of b with energies Eb and Eb , and an electron can flip between |b and |b by interacting with the bosonic reservoir Rb . The tunneling strengths for |a ↔ |b and |b ↔ |c are denoted 1 and 2 , respectively (here the off-resonant tunnelings for |a ↔ |b and |c ↔ |b are omitted). We assume that the whole coupled QD system operates in the sequential tunneling regime (i.e., only one electron is simultaneously allowed in a, b, and c). The basic idea under this model is to transport electrons from source to drain (and vice versa) via the route |a ↔ |b ↔ |b ↔ |c, where |b ↔ |b is designed to exchange energy between the transport electron and the bosonic reservoir.

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©2011 American Physical Society

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Source

|a >

μ

Drain

|b'>

Ω1

μ

Ω2 |c >

|b >

a

b

c

FIG. 1. (Color online) The system consists of three coupled QDs a, b, and c. Two contacts (source and drain) are attached as fermionic reservoirs Rl and Rr . QD b is coupled with bosonic reservoir Rb .

In the formalism of master equations [13,14], a division between the “system” and the “environment” is required [15,16]. We consider the QDs to be the “system,” while the fermionic and the bosonic reservoirs are the “environment.” The Hamiltonian of the whole is given by H = HS + HB + HFl + HFr + HI ,

where Lb ≡ |bb |, La ≡ |ha|, and Lc ≡ |hc|. is the zero-temperature decay rate for |b → |b, while γa (γc ) is the zero-temperature tunneling rate for an electron from a (c) to Rl (Rr ). The symbol N¯ is the mean number of bosons for the mode in resonance with |b ↔ |b in Rb , and abides by the Bose-Einstein distribution as N¯ =

(1)

where the right-hand-side terms stand for the Hamiltonian of the QDs, the bosonic reservoir, the left fermionic reservoir (source), the right fermionic reservoir (drain), and the systemenvironment interaction, respectively. The Hamiltonian of the QDs is defined by HS = HS0 + HS1 ,

state, and structureless reservoirs—we derive a Lindblad-type master equation [17] for the system density matrix ρ as below (¯h = 1): ρ˙ = −i HS1 ,ρ † † + N¯ Lb ρLb − 12 Lb Lb ,ρ † † + 1 + N¯ Lb ρLb − 12 Lb Lb ,ρ + γa n¯ a L†a ρLa − 12 La L†a ,ρ + γa (1 − n¯ a ) La ρL†a − 12 L†a La ,ρ + γc n¯ c L†c ρLc − 12 Lc L†c ,ρ (6) + γc (1 − n¯ c ) Lc ρL†c − 12 L†c Lc ,ρ ,

(2)

where HS0 = n=a,b,b ,c,h En |nn|, with |h being the state for no electrons in the QDs, and HS1 = 1 (|ab | + |b a|) + 2 (|cb| + |bc|). The Hamiltonian of the bosonic reservoir is given by HB = εν Bν† Bν , (3)

where μl and μr are the chemical potentials and Tl and Tr are the temperatures of Rl and Rr , respectively. According to Eq. (6), the elements of ρ evolve as ρ˙aa = i1 (ρab − ρb a ) − a→h ρaa + h→a ρhh , ρ˙bb = i2 (ρbc − ρcb ) − b→b ρbb + b →b ρb b , ρ˙b b = i1 (ρb a − ρab ) − b →b ρb b + b→b ρbb , ρ˙cc = i2 (ρcb − ρbc ) − c→h ρcc + h→c ρhh , ρ˙hh = −(h→a + h→c )ρhh + a→h ρaa + c→h ρcc ,

where Bν† (Bν ) is the boson creation (annihilation) operator for each mode ν. Likewise, the Hamiltonians of the fermionic reservoirs are given by † HFl,r = k F k F k , (4)

ρ˙bc = i(Ec − Eb )ρbc + i2 (ρbb − ρcc ) − 12 (b→b + c→h )ρbc , ρ˙ab = i(Eb − Ea )ρab + i1 (ρaa − ρb b ) − 12 (b →b + a→h )ρab ,

k∈Rl,r †

HI =

HIb

+

HIl

+

HIr ,

(9)

where b→b ≡ N¯ , b →b ≡ (1 + N¯ ), γh→a ≡ γa n¯ a , γa→h ≡ γa (1 − n¯ a ), γh→c ≡ γc n¯ c , γc→h ≡ γc (1 − n¯ c ).

(5)

with each term defined by HIb = ν∈Rb ων (Bν† |bb | † + Bν |b b|), HIl = k∈Rl ωkl (Fk |ha| + Fk |ah|), and † HIr = k∈Rr ωkr (Fk |hc| + Fk |ch|), where ω denotes the coupling strength of the specific interaction. Now the QDs can be viewed as an open quantum system subject to interaction with the environment. Under the Born-Markov approximation—a weak system-environment interaction, short reservoir correlation times compared to the system relaxation time, a separable system-environment initial

(7)

where T is the temperature of Rb . Likewise, n¯ a and n¯ c are the mean numbers of the fermions with energies Ea and Ec in Rl and Rr , respectively, and abide by the Fermi-Dirac distributions as 1 1 , n¯ c = (E −μ )/(k T ) , (8) n¯ a = (E −μ )/(k T ) a l B l c r B r + 1 e +1 e

ν∈Rb

where Fk (Fk ) is the electron creation (annihilation) operator for each momentum k in Rl or Rr . The interaction between the QDs and the reservoirs is decomposed as

1 , e(Eb −Eb )/(kB T ) − 1

(10)

In Eqs. (9) we have omitted the equations for off-diagonal elements (e.g., ρhc ) other than ρbc and ρab (besides their hermite conjugates, which abide by the conjugate equations) because those elements only decay and do not affect the steady-state solution discussed hereafter. Now the system can be treated as an artificial atom consisting of five states |a, |b , |b, |c, and |h, with coherent and incoherent inner transitions as depicted in Fig. 2.

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|a> Ω1 |b'>

|b >

h a

Γb

b'

Γb'

|h >

b c h

Ω2

a h

h c

|c > FIG. 2. (Color online) The five-state-atom model of the QDs.

By setting ρ˙ = 0, Eqs. (9) reduce to a set of linear equations which can be solved analytically with n=a,b,b ,c,h ρnn = 1, and the result is the steady-state solution for Eqs. (9). Let us denote this solution as ρ; ¯ then the steady-state tunneling current (in number of electrons per unit time) is given by I¯ ≡ γh→a ρ¯hh − γa→h ρ¯aa = γc→h ρ¯cc − γh→c ρ¯hh .

(11)

For electron transport from source to drain we have I¯ > 0, and for the reverse direction I¯ < 0. The general solution of I¯ is too complicated to unveil the physical picture. Therefore, we adopt the simplifications below: γ ≡ γa = γc , ≡ 1 = 2 , T ≡ T l = Tr , 0 ≡ Ea − Eb = Eb − Ec ,

≡ Eb − Eb > 0,

(12)

δ ≡ Ea − μl = −(Ec − μr ), and obtain the concise result 42 (xv 2 − u2 y) I¯ = , A+B

(13)

where

where Y ≡ 2δ/ and X ≡ T /T . Note the sign of Y is determined solely by δ because > 0. Next we discuss the thermodynamics conveyed by this conclusion. To begin with, first let us draw an overall picture on the heat flow between the QD system and the reservoirs. An electron traveling along |b → |b absorbs heat from Rb , and deposits the same amount of heat back to Rb for the reverse direction |b → |b. Meanwhile, at the contacts each electron-tunneling event is accompanied by a reservoir relaxation process that affects the heat flow. For example, with an ideal “filtering” [8,12], an electron-tunneling event for |a → Rl transfers energy Ea from a to Rl , accompanied by an electron with energy μl being removed from Rl to the external circuit. The overall effect is a net heat δ delivered to Rl . Conversely, an electron tunneling event for Rl → |a indicates that an amount of heat δ is taken away from Rl . Heat flow for tunneling events between |c and Rr takes place in the same manner. Secondly, to clarify the electric work done by the current, we specify two realms of bias voltages: (i) μl > μr for positive bias and (ii) μl < μr for negative bias. Under positive bias, I¯ > 0 reflects an amount of work W = μl − μr = − 2δ being dissipated by each electron, and I¯ < 0 indicates that the same amount of work being output by each electron. Similarly, under negative bias each electron dissipates or outputs an

= μr − μl = 2δ − (W

= −W ), for amount of work W ¯ ¯ I < 0 or I > 0, respectively. Then, the QD system can operate in three distinct thermodynamic modes: a heat engine, a work-to-heat convertor, or a refrigerator. These modes are illustrated by the regions on the X-Y plane shown in Fig. 3, where (I) and (III) correspond to heat engines, (II) and (IV) correspond to refrigerators, and the others correspond to work-to-heat convertors. Details in these regions are discussed below. Positive bias (Y < 1). To realize a heat engine that outputs work requires I¯ < 0; thus Y > X because of inequality (15). Furthermore, from X < Y < 1 we obtain T < T (i.e., Rb is hotter than Rl and Rr ). Then region (I) corresponds to a heat engine that absorbs heat from Rb , deposits heat 2δ to Rl and Rr , and outputs work W = − 2δ with an efficiency η = 1 − 2δ/ . Comparatively, the efficiency of an

¯ u = γ (1 − n), ¯ v = γ n, δ/(kB T ) n¯ = [e + 1]−1 , x = 1 + N¯ ,

Y=2δ /Δ work-to-heat convertor

¯ y = N,

/(kB T ) ¯ N = [e − 1]−1 ,

( IV )

Y=X

heat engine ( III)

1

A = 42 [2(u + v)(u + v + x + y) + xv + yu], B = (x 2 + y 2 )uv + u2 x(2x + u) + v 2 y(2y + v)

heat engine ( I)

+ (x + y)[u2 (u + v + y) + v 2 (x + u + v)]. (14) Clearly, the sign of I¯ is dictated by the term xv 2 − u2 y, which shares the same sign as e /(kB T ) − e2δ/(kB T ) . Then we arrive at the conclusion that I¯ < 0 I¯ > 0

refrigerator

if Y > X, if Y < X,

(15) 062104-3

work-to-heat convertor

( II)

0

refrigerator

1

X=T ' /T

work-to-heat convertor

FIG. 3. (Color online) Thermodynamic modes of the QDs.

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ideal Carnot engine working between the same temperatures is ηc = 1 − T /T . Then the second law of thermodynamics, η < ηc , says nothing but Y > X, as required for I < 0. In contrast, if I > 0 (i.e., Y < X), each electron will dissipate work W , absorb heat 2δ from Rl and Rr , and deposit heat

to Rb . If Y < 0, (the the region below the X axis), it simply means the work W completely transforms to heat. But if 0 < Y < 1, a refrigerator that cools Rl and Rr is possible if X < 1 [i.e., in region (II)]. Indeed, inequality Y < X just means the cooling efficiency η = 2δ/W is less than that of a Carnot refrigerator, ηc = 1/ηc − 1. For the region at 0 < Y < 1 and X > 1, although heat is still absorbed from Rl and Rr , it is not a refrigerator but a work-to-heat convertor because T > T . Negative bias (Y > 1). Now I¯ > 0 is needed to realize a heat engine, thus Y < X and we are in region (III). From X > Y > 1 we obtain T > T . Then this heat engine outputs work

= 2δ − , absorbs heat 2δ from Rl and Rr , and deposits W η = 1 − /2δ. Compared heat to Rb , with an efficiency

with the efficiency of a corresponding Carnot engine

ηc = 1 − T /T , again the second law of thermodynamics

η
0. In contrast, in region (IV) I¯ < 0 and T < T , a refrigerator that cools the

, reservoir Rb is realized. This refrigerator dissipates work W absorbs heat from the colder reservoir Rb , and deposits heat 2δ to the hotter reservoirs Rl and Rr with a cooling efficiency

. Given the efficiency of the Carnot refrigerator

η = /W

ηc − 1, it is simple to verify that the condition for

ηc = 1/

ηc . Finally, the region I¯ < 0 (i.e., Y > X) is equivalent to

η <

at Y > 1 and X < 1 corresponds to a plain work-to-heat convertor because T < T . In summary, we propose a coupled-QD system which can operate in various thermodynamic modes. The regions (I) and (IV) discussed above are of particular interest because they allow us to extract work from a hot bosonic reservoir, or cool it down below the electron temperature. We also show that the second law of thermodynamics manifests itself via the border Y = X, which coincides with the quantum-mechanical result. Finally, we note the assumption that at most one electron occupies the entire three-dot system is used primarily to simplify the theoretical treatment. Experimentally, it would be more desirable to permit a multielectron configuration but require that each individual dot be occupied by at most one electron. This will bring in a more complicated multistate atom model, which can be treated by essentially the same method. Detailed investigation of this configuration is under way and will be published elsewhere.

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This work is supported by NSFC Grant No. 11047155 and the Research Fund at Zhejiang Ocean University. B.J. is grateful to Professor Philip Brooks at Rice University for his constant support and encouragement.

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