Phononic heat transport in nanomechanical structures: steady-state and pumping

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by  Marcone I. Sena-Junior

Phononic heat transport in nanomechanical structures: steady-state and pumping
Marcone I. Sena-Junior,1,2,3 Leandro R. F. Lima,1 and Caio H. Lewenkopf1
1Instituto de F´ısica, Universidade Federal Fluminense, Brazil
2Instituto de F´ısica, Universidade Federal de Alagoas, Brazil
3Escola Polit´ecnica , Universidade de Pernambuco, Brazil
(Dated: January 12, 2017)
We study the heat transport due to phonons in nanomechanical structures using a phase space
representation of non-equilibrium Green’s functions. This representation accounts for the atomic
degrees of freedom making it particularly suited for the description of small (molecular) junctions
7 systems. We show that for the steady state limit our formalism correctly recovers the heuristic
1 Landauer-likeheatconductanceforaquantumcoherentmolecularsystemcoupledtothermalreser-
0 voirs. We find general expressions for the non-stationary heat current due to an external periodic
2 drive. In both cases we discuss the quantum thermodynamic properties of the systems. We apply
our formalism to the case of a diatomic molecular junction.
n
a
PACSnumbers: 65.80.-g,05.60.Gg,44.10.+i,66.70.-f
J

0
1
I. INTRODUCTION formalism to address systems under the influence of a

time dependent drive. As an example, we derive general
]
ll Significant progress has been recently achieved on the expressions for the heat current pumped by an external
a time-dependent periodic potential for a system coupled
understandingofphononicheattransferatthemolecular
h level1,2. In addition to the investigation of fundamental to two thermal reservoirs at the same temperature.
-
s aspects of the problem1,3, several authors have realized We apply our method to analyze the steady-state
e heat transport properties of a diatomic molecule cou-
that phonons, usually regarded as an energy waste, can
m
pled to thermal reservoirs by semi-infinite linear har-
bemanipulatedandcontrolledtocarryandprocessinfor-
t. mation. Exploringanalogieswithelectronsandphotons, monic chains. Next, we study the heat current pumped
a throughthesystemduetoatime-dependentdrivingforce
theoretical proposals have been put forward aiming the
m
fabrication of devices such as thermal diodes4, thermal and discuss its thermodynamic properties.
- transistors5,6, andthermallogicgates7, someofthemal- The paper is organized as follows: In Sec. II we in-
d
readyexperimentallyverified8–10. Theseideashavegiven troduce the phase space representation of the Green’s
n
o rise to the emerging field of phononics2,11. functions on which our derivations are built. We begin
Sec. III by presenting the model Hamiltonian addressed
c Thepresenceofanexternaltime-dependentdrive,such
[ as an external force or time-varying thermal bath tem- in this study. We then use the Green’s function formal-

1 perature, gives another interesting twist to the problem, ism to derive expressions for the thermal current due to
a source-drain temperature difference and the heat cur-
v making possible to explore non-equilibrium phenomena
9 such as directed heat pumping and cooling2,12–17. rent pumped by an external periodical drive of the sys-
7 One of the fundamental tools for the theoretical tem atomic degrees of freedom. In Sec. IV, we apply
7 our results to the simple model of a diatomic molecular
study of non-equilibrium properties of quantum sys-
2 junction. Finally, we present our conclusions in Sec. V.
tems is the non-equilibrium Green’s functions (NEGF)
0
theory18,19. This approach, originally developed for
.
1 fermionic systems20,21, has been nicely adapted to de-
0 II. GREEN’S FUNCTIONS IN PHASE SPACE
scribetheheattransferinsmalljunctionssystems3,22–27.
7
Despite its success, the implementation of the NEGF
1
: to calculate phonon heat currents driven by a temper- In this the section we introduce a phase space repre-
v ature difference between source and drain still has some sentationofnon-equilibriumGreen’sfunctions. Weshow
Xi caveats, like the need to symmetrize the heat current to thatthisrepresentationisveryconvenientforacanonical
r eliminateaspuriousimaginarycomponent25–27. Therel- quantization of the displacements (cid:126)u ≡ (u1,...,un) and
a evanceofNEGFforphononicscallsforadeeperanalysis their canonical conjugated momenta p(cid:126) (p1,...,pn) in
≡
of the formalism. a 2n-dimensional phase space.
The purpose of the paper is twofold. First, we present Let us consider a quadratic Hamiltonian expressed in
arigorousmethodforthedescriptionofquantumthermal terms of space phase variables ((cid:126)u,p(cid:126)) representing a sys-
transport properties due to phonon or atomic degrees of temofcoupledoscillators. ThemodelHamiltonianreads
freedom using nonequilibrium Green’s function in phase
1 1 1
space. We show that our formal developments solve the H(t)= p(cid:126)T p(cid:126)+ (cid:126)uT Kˆ(t) (cid:126)u ζT ˇ(t) ζ, (1)
2 · 2 · · ≡ 2 ·M ·
problems of the previous works25–27 and recover the well
knownLandauer-likeformulaforthestationaryheatcur- where, for the sake of compactness, we assume that the
rent in the ballistic regime28–31. Second, we extend the masses are identical and have unit value. Kˆ(t) is the
2
force constant matrix that represents the couplings of Similarly,usingEqs.(3a)and(7),weshowthatCˇK(t,t(cid:48))
the oscillators network. The dynamic variable ζ and the and Cˇr,a(t,t(cid:48)) satisfy
matrix ˇ have the symplectic structure
M (cid:18) ∂ (cid:19)
ζ =(cid:18)(cid:126)u(cid:19) and ˇ(t)=(cid:18)Kˆ(t) ˆ0(cid:19), (2) Iˇ ∂t +Kˇ(t) ·CˇK(t,t(cid:48))=0, (8a)
p(cid:126) M ˆ0 Iˆ (cid:18) (cid:19)
∂
ˇ + ˇ(t) Cˇr,a(t,t(cid:48))=δ(t t(cid:48)) ˇ, (8b)
where Iˆis the identity matrix. I ∂t K · − Q
The equation of motion for ζ reads
where ˇ is the 2n 2n identity matrix. To obtain
d ∂ Eq. (8aI), we use the×identity Cˇ>(t,t) Cˇ<(t,t) = ˇ,
ζ = ˇ H = ˇ(t) ζ, (3a) − Q
dt Q· ∂ζ −K · that follows from the canonical commutations relations.
To make the notation compact, we write the cor-
where
relation function in a block structure as (Keldysh
(cid:18) ˆ0 Iˆ(cid:19) (cid:18) ˆ0 Iˆ(cid:19) space) (symplectic space) in its irreducible representa-
ˇ = and ˇ(t) ˇ ˇ(t)= − . ⊗
Q Iˆ ˆ0 K ≡−Q·M Kˆ(t) ˆ0 tion, namely
−
(3b) (cid:18)CˇK Cˇr(cid:19)
We define the phase space correlation functions ˘(t,t(cid:48))= (t,t(cid:48))
Cˆ(τ,τ(cid:48)) on the Keldysh contour18 as C Cˇa ˇ0
σ ˇ(t,t(cid:48))+homogeneous solution, (9)
1 (cid:18)Cˆ(uu) Cˆ(up)(cid:19) ≡ 1⊗G
Cˇ(τ,τ(cid:48)) T ζ(τ) ζ(τ(cid:48)) (τ,τ(cid:48)),
≡ ı(cid:126)(cid:104) C ⊗ (cid:105)≡ Cˆ(pu) Cˆ(pp) where σ is the first Pauli matrix. Note that ˇ(t,t(cid:48)) has
1
G
(4) also a symplectic structure and satisfies (by inspection)
whereı(cid:126)Cˆ(αβ) T α(cid:126)(τ) β(cid:126)(τ(cid:48)) . Thecorrelationfunc- the equation of motion
C
≡(cid:104) ⊗ (cid:105)
tionsCˆ(αβ)(τ,τ(cid:48))areastraightforwardphasespacegene- (cid:18) (cid:19)
∂
ralization of standard Green’s functions18,19, as we dis- ˇ + ˇ(t) ˇ(t,t(cid:48))=δ(t t(cid:48)) ˇ, (10)
I ∂t K ·G − Q
cuss below.
As standard18, the greater, lesser, time-ordered, and with a self-adjoint equation
anti-time-ordered correlations functions read
(cid:32) (cid:33)
Cˇ>(t,t(cid:48))=(ı(cid:126))−1(cid:10)ζ(t) ζ(t(cid:48))(cid:11), (5a) ˇ(t,t(cid:48)) ˇ←∂−+ ˇT(t(cid:48)) = δ(t t(cid:48)) ˇ. (11)
⊗ G · I ∂t(cid:48) K − − Q
Cˇ<(t,t(cid:48))=(cid:2)Cˇ>(t,t(cid:48))(cid:3)T, (5b)
CˇT(t,t(cid:48))=θ(t t(cid:48))Cˇ>(t,t(cid:48))+θ(t(cid:48) t)Cˇ<(t,t(cid:48)), (5c) UsingEqs.(10)and(11)weobtainthefollowingidentity
− − (cid:12)
CˇT(t,t(cid:48))=θ(t(cid:48) t)Cˇ>(t,t(cid:48))+θ(t t(cid:48))Cˇ<(t,t(cid:48)), (5d) d (cid:18)∂ ∂ (cid:19) (cid:12)
where (CˇT+CˇT−Cˇ> Cˇ<)(t,t(cid:48))=−0. dtGˇ(t,t)≡ ∂t + ∂t(cid:48) Gˇ(t,t(cid:48))(cid:12)(cid:12)(cid:12)t=t(cid:48)
Alternatively, t−he cor−relation functions can be repre- = ˇ(t) ˇ(t,t) ˇ(t,t) ˇT(t). (12)
sented by their retarded Cˇr, advanced Cˇa, and Keldysh −K ·G −G ·K
CˇK components, namely Performing the Keldysh rotation19 in Eq. (9), we ob-
tainareduciblerepresentationofthecorrelationfunction
1
Cˇr(t,t(cid:48))= (CˇT+Cˇ> Cˇ< CˇT)(t,t(cid:48)) in terms of the quantities defined in Eq. (6) as
2 − −
=θ1(t−t(cid:48)) (cid:0)Cˇ>−Cˇ<(cid:1)(t,t(cid:48)), (6a) P˘·C˘(t,t(cid:48))·P˘T =(cid:18)CCˇˇ>T CCˇˇ<T(cid:19)(t,t(cid:48))
Cˇa(t,t(cid:48))= (CˇT Cˇ>+Cˇ< CˇT)(t,t(cid:48))
2 − − σ ˇ(t,t(cid:48))+homog. solution, (13)
3
=θ(t(cid:48) t) (cid:0)Cˇ< Cˇ>(cid:1)(t,t(cid:48)), (6b) ≡ ⊗G
− − where ˘ = √1 (I +ıσ ) ˇ andσ isthesecondmatrix
CˇK(t,t(cid:48))= 1(CˇT+Cˇ>+Cˇ<+CˇT)(t,t(cid:48)) of PauPli. 2 2 2 ⊗I 2
2
=(cid:0)Cˇ>+Cˇ<(cid:1)(t,t(cid:48)). (6c) Let us now introduce the frequency representation of
the correlation functions. Assuming time translational
Using Eqs. (3a) and (5) we obtain the equations of invariance, i.e., that the matrix ˇ does not depend on
motion for Cˇ≷(t,t(cid:48)) and CˇT,T(t,t(cid:48)), namely time, one defines ˇ[ω] in terms ofKthe Fourier transform
G
(cid:18) (cid:19) (cid:90) ∞
ˇ ∂ + ˇ(t) Cˇ≷(t,t(cid:48))=0, (7a) ˇ[ω]= d(t t(cid:48))eıω(t−t(cid:48)) ˇ(t t(cid:48)), (14)
I ∂t K · G −∞ − G −
(cid:18) (cid:19)
ˇ ∂ + ˇ(t) CˇT,T(t,t(cid:48))= δ(t t(cid:48)) ˇ. (7b) for ˇ(t t(cid:48)) = ˇ(t,t(cid:48)). We study the time-dependent
I ∂t K · ± − Q probGlem−in Sec. IGIIB.
3
By inserting Eq. (14) in (10) [or in (11)], we write SubstitutingEqs.(20a)and(16)intheinverseFourier
transform Eq. (14), we write the retarded component of
ˇ[ω]=(cid:0) ıωˇ+ ˇ(cid:1)−1 ˇ (t t(cid:48)) as
G − I K ·Q G −
=−Qˇ·(cid:0)ıωIˇ+KˇT(cid:1)−1, (15) ˇr(t t(cid:48))=θ(t t(cid:48))
G − −
where  sin(cid:20)√Kˆ(t−t(cid:48))(cid:21) (cid:104)(cid:112) (cid:105) 
Gˇ[ω]≡(cid:18)Gˇˇ((upuu))[[ωω]] Gˇˇ((uppp))[[ωω]](cid:19)=(cid:18) ıGωˆ[Gωˆ][ω] Gıˆω[ωGˆ][ωKˆ](cid:19), × c−os(cid:104)(cid:112)√KˆKˆ(t t(cid:48))(cid:105) (cid:112)cKoˆs sinK(cid:104)ˆ(cid:112)(tKˆ−(tt(cid:48)) t(cid:48))(cid:105)
G G − · − − − −
(16)
+ solution of homogeneous equation, (21a)
with
Gˆ[ω]=(ω2Iˆ Kˆ)−1. (17) and ˇa(t t(cid:48))= ˇr(t(cid:48) t), where ˇr,a(0±)= ˇ.
G − −G − G Q
− Similarly, the ordered and anti-ordered components
Equation (16) has been obtained in Ref. 25 by di- read
rectly taking the Fourier transform of the displacement
ui and the canonically conjugate momentum opera- ˇT,T(t t(cid:48))=
{ttohriss}s{tpria}ig.htWfoerwnaortde pthroactedduesrpeitise pbreoibnlgemveartyic,apsipnecaelitnhge, G −1 e∓ı√Kˆ|t−t(cid:48)| 1sgn(t t(cid:48))e∓ı√Kˆ|t−t(cid:48)|
ccaonnosinsitceanltlcyomdemfinuetadtiionnthreelafrteioqnusen[ucyi(dt)o,mpja(itn).] cTahnisnportobbe- ±12sg2ın√(tK(cid:48)ˆ−t)e∓ı√Kˆ|t−t(cid:48)| ±221ı(cid:112)Kˆ−·e∓ı√Kˆ|t−t(cid:48)| 
lem can be circumvented27 by performing the Fourier + solution of homogeneous equation, (21b)
transform of the phase space correlation functions, as
described above. which satisfy ˇT(0±) ˇT(0±)= ˇ.
The Green’s function Gˆ[ω] can be represented as The KeldysGh comp−onGent of th±eQcorrelation function
is, in general, more demanding to obtain. As stan-
1(cid:90) ∞ dω¯ (cid:18) 1 1 (cid:19)
Gˆ[ω]= Jˆ(ω¯) , (18) dard, theexceptionistheequilibriumcase. Inthislimit,
2 −∞ 2π ω−ω¯ − ω+ω¯ thefluctuation-dissipationtheorem33 relatestheKeldysh
component of the correlation function of a bosonic sys-
where the spectral operator Jˆ(ω¯) is tem to its retarded and advanced components as
Jˆ(ω)=2π(cid:88) 1 δ(ω ωj) j j . (19) GˆKeq[ω]=(cid:0)Gˆr[ω]−Gˆa[ω](cid:1)(2f(ω)+1), (22)
ω − | (cid:105)(cid:104) |
j
j where f(ω) = (cid:0)eβ(cid:126)ω 1(cid:1)−1 is the Bose-Einstein distri-
−
Here we have used that Kˆ is a positive-semidefinite bution function. One can also write
matrix32, which satisfies Kˆ j = ω2 j with ω (cid:62) 0 (re-
call that (cid:104)j|j(cid:48)(cid:105)=δj,j(cid:48) and (cid:80)| j(cid:105)|j(cid:105)(cid:104)jj|=| (cid:105)Iˆ). j Gˆ>eq[ω]+σGˆ<eq[ω]=ıAˆ[ω](cid:0)2f(ω)δσ,++1(cid:1), (23)
The general expression (18) does not distinguish the where σ = 1 and
retarded, advanced, ordered, and anti-ordered compo- ±
nents of Gˆ[ω]. A proper representation of the compo- ıAˆ(ω)=Gˆr[ω] Gˆa[ω]= 1 (cid:104)Jˆ(ω) Jˆ( ω)(cid:105). (24)
nents requires a regularization around the poles ω = ω¯ − 2ı − −
±
of Eq. (18), namely
As a result, the equilibrium lesser and greater Green’s
functions are given by
∞
(cid:90) (cid:18) (cid:19)
1 dω¯ 1 1
Gˆr,a[ω]= 2 2π Jˆ(ω¯) ω ω¯ ı0+ − ω+ω¯ ı0+ Gˆ<eq[ω]=ıAˆ(ω)f(ω), (25a)
−∞ − ± ± Gˆ>[ω]=ıAˆ(ω) (f(ω)+1). (25b)
(cid:104) (cid:105)−1 eq
= (ω ı0+)2Iˆ Kˆ , (20a)
± −
∞
1 (cid:90) dω¯ (cid:18) 1 1 (cid:19) III. MODEL HAMILTONIAN
GˆT,T[ω]= Jˆ(ω¯)
2 2π ω ω¯ ı0+ − ω+ω¯ ı0+
−∞ − ± ∓ In this section, we describe the heat transport proper-
(cid:104) (cid:112) (cid:105)−1 ties of a molecular junction modeled by a central region
= ω2Iˆ ( Kˆ ı0+Iˆ)2 . (20b)
− ∓ C representingananostructurecoupledbymultipleleads
connected to reservoirs in thermal equilibrium25,26. We
The Green’s functions Gˆr,a(t,t(cid:48)) and GˆT,T(t,t(cid:48)) are ob- recallthatweonlyconsiderthermaltransportduevibra-
tained by the inverse Fourier transform of Eqs. (20) and tional degrees of freedom, which is the dominant mecha-
are consistent with Eqs. (5) and (6), as they should. nism in insulator systems.
4
This partition scheme allows one to write the general region, which we refer to as “molecule”. Accordingly, we
Hamiltonian of Eq. (1) as write Eq. (26) as
(cid:88) (cid:88)
H(t)= Hα(t) +HC(t)+HT(t), (26) H(t)= Hα(t)+HM(t), (32)
α α
where the molecule Hamiltonian reads
where
H (t) H (t)+H (t). (33)
Hα(t)=Hα0 +Uαα(t), (27a) M ≡ C T
H (t)=H0 +U (t), (27b) The energy of the extended molecule is defined as
C C CC
(cid:88)(cid:104) (cid:105) EM(t)≡(cid:104)HM(t)(cid:105), namely
H (t)= U (t)+U (t) , (27c)
T Cα αC ı(cid:126) (cid:110)
α E (t)= Tr C<(pp)(t,t)+K (t) C<(uu)(t,t)
M 2 CC CC · CC
correspond to the Hamiltonian of the α-lead, central re- (cid:88)(cid:104) (cid:105)(cid:111)
+ V (t) C<(uu)(t,t) +C<(uu)(t,t) V (t) ,
gionandtunneling,respectively. Wedefinethedecoupled Cα · αC Cα · αC
Hamiltonian H0 corresponding to the a-partition as α
a (34)
1 1
H0 p(cid:126)T p(cid:126) + (cid:126)uT K0 (cid:126)u (28a) where the components of lesser functions are explicit
a ≡ 2 a · a 2 a · aa· a given by
and the coupling Hamiltonian Uab(t) between a and b- ı(cid:126)(cid:2)C<(pp)(t,t(cid:48))(cid:3) =(cid:10)[p(cid:126) (t(cid:48))] [p(cid:126) (t)] (cid:11); (35a)
partitions as ab k,k(cid:48) b k(cid:48) a k
ı(cid:126)(cid:2)C<(uu)(t,t(cid:48))(cid:3) =(cid:10)[(cid:126)u (t(cid:48))] [(cid:126)u (t)] (cid:11); (35b)
1 ab n,n(cid:48) b n(cid:48) a n
U (t) (cid:126)uT V (t) (cid:126)u . (28b)
ab ≡ 2 a · ab · b ı(cid:126)(cid:2)C<(up)(t,t(cid:48))(cid:3) =(cid:10)[p(cid:126) (t(cid:48))] [(cid:126)u (t)] (cid:11); (35c)
ab n,k b k a n
The force constant matrix in Eq. (1) is decomposed as ı(cid:126)(cid:2)C<(pu)(t,t(cid:48))(cid:3) =(cid:10)[(cid:126)u (t(cid:48))] [p(cid:126) (t)] (cid:11). (35d)
Kˆ(t)=Kˆ0+Vˆ(t),whereKˆ0 givesthedynamicalmatrix ab k,n b n a k
of the decoupled partitions with a,b= C,α , in line with Eq. (4).
{ }
Onecandefinethethermalcurrentflowingthroughan
(cid:34) (cid:35)
(cid:77) open molecule connected to multiple reservoirs by com-
Kˆ0 = K0 K0, (29)
α ⊕ C paring its energy variation
α
(cid:28) (cid:29)
dE (t) dH (t)
M M
and Vˆ(t) corresponds to the coupling between different =
dt dt
partitions, namely (cid:28) (cid:29)
ı (cid:88) ∂HM(t)
= [H,H ] + (36)
(cid:34) (cid:35) (cid:126) (cid:104) M (cid:105) ∂t
(cid:77)
Vˆ(t)= V (t) V (t) + Vˆ (t). (30) α
αα CC mixed
⊕
α with the energy continuity equation, expressed as
These definitions allow us to write the tunneling Hamil- dEM(t) (cid:88)
= J (t)+Φ(t), (37)
tonian HT(t) as dt α
α
1
HT(t)= (cid:126)uT Vˆmixed(t) (cid:126)u, (31) where one associates Jα(t) to the thermal current from
2 · · α-reservoirintothemoleculeandΦ(t)ispowerdeveloped
where (cid:126)u [(cid:76) (cid:126)u ] (cid:126)u . Note that Vˆ =VˆT and there- by the ac sources (or drives) in the molecule. Hence, by
≡ α α ⊕ C inspection one infers that
fore V =VT for all α terminals.
αC Cα
The model Hamiltonian in Eq. (26) includes V (a= i
aa J (t)= [H,H ] (38)
α,C) terms that have not been explicitly accounted for α −(cid:126)(cid:104) α (cid:105)
by previous works25–27. Neglecting V can be problem-
aa and
atic for the adiabatic switch-on picture, upon which the
(cid:28) (cid:29)
NEGF formalism is built (see Ref. 18). The absorption ∂HM(t)
Φ(t)= . (39)
ofVaa intoKα0α modifiesthefreeGreen’sfunctionsmak- ∂t
ing their calculation troublesome. This issue becomes
Aftersomealgebra, wewritethethermalcurrentfrom
clear in the formal development below as well as in the
α-reservoir into the molecule in terms of the correlation
applications in Sec. IV.
functions as
To discuss the thermodynamic properties of the sys-
tem it is convenient to describe the molecular junction J (t)=Re(cid:104)Tr(cid:110)V (t) ı(cid:126)C<(pu)(t,t)(cid:111)(cid:105), (40)
as formed by reservoirs coupled to an extended central α Cα · αC
5
while the power developed by the external time- tation. The Fourier transform of C<(up)(t,t(cid:48)) is
Cα
dependent drives reads
(cid:90) ∞ dω
C<(up)(t,t(cid:48))= e−ıω(t−t(cid:48))C<(up)[ω], (42)
(cid:20) (cid:26) Cα 2π Cα
Φ(t)=Re Tr 1V˙ (t) ı(cid:126)C<(uu)(t,t) −∞
2 CC · CC whereC<(up)[ω]=ıωG< [ω]. SubstitutingEq.(42)into
(cid:41)(cid:35) Cα Cα
(cid:88) Eq. (38), we cast the steady-state heat current as
+ V˙ (t) ı(cid:126)C<(uu)(t,t) . (41)
α Cα · αC J(S) =(cid:90) ∞ dω (cid:126)ωTr(cid:8)V G< [ω] G< [ω] V (cid:9).
α 4π Cα· αC − Cα · αC
−∞
The explicitly symmetric tunneling Hamiltonian (43)
HT(t), Eq. (27c), leads to an expression for the heat The system Green’s function Gˆ[ω] = (cid:0)ω2Iˆ Kˆ(cid:1)−1
current Jα(t) with terms depending on both VCα and satisfies the Dyson equation −
V . This ensures that J (t) accounts for processes cor-
αC α
responding to the heat flow from the central region C Gˆ[ω]=gˆ[ω]+gˆ[ω] Vˆ Gˆ[ω]
· ·
to the α-lead as well as from α to C. This combination =gˆ[ω]+Gˆ[ω] Vˆ gˆ[ω], (44)
leadstoaheatcurrentJ (t)thatismanifestlyreal,since · ·
α
one term is the complex conjugate of the other. Our re- where Kˆ = Kˆ0 +Vˆ and gˆ[ω] = (cid:0)ω2Iˆ Kˆ0(cid:1)−1. Note
sult differs from the heat current derived by Wang and −
that the free Green’s function gˆ[ω] is block diagonal in
collaborators25–27. These authors derive the heat cur-
the partitions.
rent using the Hamiltonian without taking into account
From Eq. (44) we obtain
processes associated to V (corresponding to α = L).
LC
The obtained expression for heat current depends only G [ω]=G [ω] V g˜ [ω], (45a)
Cα CC Cα α
on the hybrid Green’s function G<CL. As a consequence, GαC[ω]=g˜α[ω] V·αC G·CC[ω], (45b)
theresultingJ (t)isingeneralacomplexfunction. Fur- · ·
α (cid:16) (cid:17)−1
thermore, the absence of VαC (or VCα) in their Hamil- GCC[ω]= g˜C[ω]−1 Σ˜[ω] , (45c)
−
tonian implies that the self-energy Σ = V g V
L CL L LC
· · G [ω]=g˜ [ω] V G [ω] V g˜ [ω]
vanishes, introducing an inconsistency within their for- αβ α αC CC Cβ β
· · · ·
malism. In the case of a stationary heat flow in a two- +δ g˜ [ω], (45d)
αβ α
terminal geometry (where J = J ), the problem of
L − R where, fornotationalconvenience, weintroduceaneffec-
acomplexheatcurrentiscircumvented25–27 byimposing
tive embedding self-energy
thatJ =(J +J∗ J J∗)/4. Thisadhocsymmetriza-
L L− R− R
tion gives the well known Caroli formula for the trans- Σ˜[ω]=(cid:88)Σ˜ [ω]=(cid:88)V g˜ [ω] V , (46)
mission. However, for time-dependent systems, where α Cα· α · αC
α α
in general J (t) = J (t), the symmetrization cannot
R L
be used and the f(cid:54)orm−alism gives a heat current with an and an effective free Green’s function
imaginary component.
g˜ [ω]−1 =g [ω]−1 V with a= α,C , (47)
a a aa
As we show in the following, we fix all these problems − { }
banydprVoLpCer.lyOtaukrinfgorimntaoliascmcoduinrtectthleytleeramdssrteolattehdetCoaVrCoLli wtoheGrreegean[’ωs]fu=n(cωti2oInai−nKtha0e)r−m1.alFeoqruail=ibrαiu,mit cwoirtrhestphoenαds-
formula for the thermal transmission in the stationary reservoir at a temperature T . Hence, using Eqs. (20a)
α
regime and to a purely real heat current in the time- and (25) we write obtain
dependent case.
g<[ω]=ıA (ω)f (ω), (48a)
In the following subsections we study separately the α α α
steady-state transport (V˙ab = 0) and the heat transport g>[ω]=ıA (ω)(cid:16)1+f (ω)(cid:17), (48b)
due to pumping by an external drive (V˙ =0) for a,b= α α α
ab
α,C . (cid:54) gr,a[ω]=(cid:104)(cid:0)ω ı0+(cid:1)2 I K0(cid:105)−1, (48c)
{ } α ± α− α
whereıA (ω) gr[ω] ga[ω]istheα-lead“free”spectral
A. Steady-state transport function αand f≡α(ωα)=−(cid:0)eβαα(cid:126)ω−1(cid:1)−1 with βα =1/kBTα.
The lesser components of G and G are obtained
αC Cα
by applying the Langreth rules18,33 to Eq. (45). By in-
Let us now calculate the steady-state thermal current serting the result in Eq. (43), we obtain
flowing from the α-lead due to a temperature difference
in the reservoirs. Here, we consider the heat current ex- J(S) =(cid:90) ∞ dω (cid:126)ωTr(cid:110)G< [ω] (cid:0)Σ˜r[ω] Σ˜a[ω](cid:1)
pression (38) for a time-independent coupling matrix Vˆ. α 4π CC · α − α
−∞
Since the Hamiltonian does not explicitly depends on (cid:0)Gr [ω] Ga [ω](cid:1) Σ˜<[ω](cid:111). (49)
time, it is convenient to work in the frequency represen- − CC − CC · α
6
The self-energies are given in terms of B. Pumping transport
g˜<[ω]=ıA˜ (ω)f (ω), (50a)
α α α
The analysis of heat currents in time-dependent sys-
g˜>[ω]=ıA˜ (ω)(cid:0)1+f (ω)(cid:1), (50b)
α α α tems is far more involved for bosonic degrees of freedom
g˜r,a[ω]=(cid:104)(cid:0)ω ı0+(cid:1)2 I K (cid:105)−1, (50c) thanfortheelectronicones. Inthelattercase,theFermi
α ± α− α energy(andthecorrespondingFermivelocity)establishes
acharacteristictimescalefortheelectronicdynamics. In
where K = K0 + V and ıA˜ (ω) = g˜r[ω] g˜a[ω].
α α αα α α − α experiments34 the external driving is slow with respect
Hence,
totheelectronicdynamics,whichallowstoapproachthe
Σ˜r[ω] Σ˜a[ω]=ıV A˜ [ω] V ıΓ˜ [ω], (51a) problem using the adiabatic approximation35–38. In the
α − α Cα· α · αC ≡− α
bosonic case there is no internal characteristic time scale
where Γ˜ [ω] is the α-contact line width function. Simi-
α and analytical progress has to resort on the assumption
larly,
that the driving force is small to employ perturbation
Σ˜<[ω]=V g˜<[ω] V = ıf (ω)Γ˜ [ω]. (51b) theory.
α Cα· α · αC − α α As an example of time-dependent transport, we study
By expressing the self-energies in terms of the line
thecaseofperiodicallydrivensystemintime. Weassume
width functions, we write the heat current as
that the coupling between regions depends on time as
(cid:90) ∞ dω (cid:110) (cid:104) Vˆ(t)=Vˆ +εvˆ(t), where ε is a dimensionless parameter.
Jα(S) = −∞ 4πı(cid:126)ωTr Γ˜α[ω]· G<CC[ω] Defining an auxiliary matrix Vˇ(t) as
−fα(ω)(cid:16)GrCC[ω]−GaCC[ω](cid:17)(cid:105)(cid:111). (52) Vˇ(t)=ε (cid:18)vˆ(ˆ0t) ˆ0ˆ0(cid:19), (56)
Applying the Langreth rules to Eq. (45c) and using
wecanwrite ˇ(t)= ˇ ˇ ˇ(t)or,equivalently, ˇ(t)=
Eq. (51), we obtain ˇ + ˇ(t). IKt followKs−frQom·VEq. (10) that the DMyson’s
G< [ω]= (cid:88)Gr [ω] ıΓ˜ [ω] Ga [ω]f (ω), (53a) eMquatiVon reads
CC − CC · α · CC α
α (cid:90)
GrCC[ω]−GaCC[ω]=−(cid:88)GrCC[ω]·ıΓ˜α[ω]·GaCC[ω], Gˇ(t,t(cid:48))=Gˇ(t−t(cid:48))+ dt¯Gˇ(t−t¯)·Vˇ(t¯)·Gˇ(t¯,t(cid:48)), (57)
α
(53b) where ˇ(t t(cid:48))denotesthesteady-stateGreen’sfunction
G −
transport, givenbyEqs.(14)to(17). Weconsiderε 1
thatareinsertedinEq.(52)tofinallyarriveatthestead- (cid:28)
and treat the problem using pertubation theory. This
state heat current
is an alternative approach to the Floquet analysis used
J(S) =(cid:88)(cid:90) ∞ dω(cid:126)ω (ω)(cid:104)f (ω) f (ω)(cid:105), (54) in Refs. 15 and 17. The Green’s function deviation from
α 2π Tαβ α − β steady-state,δˇ(t,t(cid:48)) ˇ(t,t(cid:48)) ˇ(t t(cid:48)),isconveniently
β 0 G ≡G −G −
represented by
where
Tαβ(ω)≡Tr(cid:110)Γ˜α[ω]·GrCC[ω]·Γ˜β[ω]·GaCC[ω](cid:111), (55) δGˇ(t,t(cid:48))=(cid:90)(cid:90) d(ω2πd)ω2(cid:48) e−ı(ωt−ω(cid:48)t(cid:48)) δGˇ[ω,ω(cid:48)], (58)
rigorously obtaining the Landauer heat conductance put where
forward in Ref. 29. (cid:18) 1 ıω(cid:48)(cid:19) (cid:88)
Here we stress that the proper symmetric inclusion of δˇ[ω,ω(cid:48)]= εnΛˆ [ω,ω(cid:48)], (59)
G ıω ωω(cid:48) ⊗ n
the tunneling terms VαC and VCα in our Hamiltonian, − n(cid:62)1
through Eq. (27), leads to the Caroli’s formula for the
(cid:110) (cid:111)
transmission without relying on unjustified symmetriza- and the set Λˆ [ω,ω(cid:48)] is defined by the recurrence re-
n
tion procedures25–27. We also emphasize that the self lation
energy becomes ill defined without the presence of both
∞
terms V and V in the Hamiltonian. (cid:90) dν
Cα αC Λˆ [ω,ω(cid:48)]=Gˆ[ω] vˆ[ω ν] Λˆ [ν,ω(cid:48)], (60a)
The transmission coefficient αβ(ω) is interpreted as n · 2π − · n−1
the probability of an energy (cid:126)ωTto be transmitted from −∞
thereservoirαtothereservoirβ andhasthesamestruc-
with
ture of the Meir-Wingreen formula21 that describes the
electronic conductance of fully coherent systems of non- Λˆ [ω,ω(cid:48)]=Gˆ[ω] vˆ[ω ω(cid:48)] Gˆ[ω(cid:48)], (60b)
1
· − ·
interacting electrons.
It is straightforward to verify that (ω) = (ω), where Gˆ[ω]=(cid:0)ω2Iˆ−Kˆ(cid:1)−1 has been discussed in the pre-
αβ βα
which implies that in steady-state J(ST) J(S) =T J(S). vious section and
≡ L − R (cid:90) ∞
Hence, dE /dt = 0 and, as expected, the molecule en-
M vˆ[ω]= dtvˆ(t)eıωt. (61)
ergy does not change in time.
−∞
7
We model the coupling terms as where J(P) and Φ(P) are discussed in Appendix B and
α
can be cast as
v (t)=φ (t)V , (62a)
αC α αC (cid:88)∞ (cid:88) (cid:20) (cid:16) (cid:17)
vCα(t)=φα(t)VCα, (62b) Jα(P) = a(nβ)a(nγ) cos ϕn(β)−ϕ(nγ) Aαβγ(n)
vαα(t)=φα(t)Vαα, (62c) n=1 βγ
(cid:88) (cid:16) (cid:17) (cid:21)
vCC(t)= φα(t)VC(αC), (62d) −sin ϕ(nβ)−ϕ(nγ) Bβαγ(n) , (69a)
α
(cid:88)∞ (cid:88) (cid:20) (cid:16) (cid:17)
pwuhmerpeinφgα(tti)mise-addepimenednesinocneleosfstfhuencαti-olenatdh.aFtodresacrpibereisotdhice Φ(P) = a(nβ)a(nγ) cos ϕn(β)−ϕ(nγ) Dβγ(n)
n=1 βγ
pumping, i.e., φα(t+τ) = φα(t) the pumping function (cid:16) (cid:17) (cid:21)
can be expressed by a Fourier series in harmonic form as sin ϕ(β) ϕ(γ) E (n) , (69b)
− n − n βγ
∞
φα(t)= (cid:88)2an(α)cos(Ωnt+ϕ(nα)) for Ωn =n2τπ. (63) where the quantities Aαβγ(n), Bβαγ(n), Dβγ(n), and
E (n) are explicitly given by Eq. (B14) and depend on
n=1 βγ
the molecular structure.
By construction (cid:10)φ(t)(cid:11) = 0, where (cid:104)···(cid:105) ≡ 1(cid:82)τdt(...)
τ τ τ 0 Note that in Eq. (68) the first order contributions in
stands for the time average over a period. We assume
ε vanish. The second order terms J(P) and Φ(P) depend
that φ (t) =1. α
| α |max explicitly on the periodic profile φ (t).
Expanding the Dyson equation, Eq. (57), in a power α
series in ε, we write the energy E (t) of the extended
M
molecule as
IV. APPLICATION: MOLECULAR JUNCTION
E (t)=E(0)+εE(1)(t)+ε2E(2)(t)+ . (64)
M M M M ···
We investigate the consequences of our findings using
The explicitly expression for E(n)(t) are rather lengthy the molecular junction model presented in Sec. III. We
M
andaregiveninAppendixB.Foraperiodicpumpingwe consider a one-dimensional system where a central re-
show that E(0) does not depend on time and E(n)(t) = gionwithN atomsisattachedtotwosemi-infinitelinear
M M
E(n)(t+τ) for n=1,2,... (see Appendix B). chains acting as leads, as depicted in Fig. 1.
M For the sake of clarity, we consider the simplest non
We express the variation of the extended molecule en-
trivial case of a diatomic molecule, namely, N =2. The
ergy between t and t+∆t in the form of a first law of
thermodynamics, namely, ∆E(∆t) (cid:80) Q(∆t)+W(∆t). forceconstantbetweentheatomsintheleadsanditsfirst
M ≡ α α neighbors is k. The force constant between the atoms
Note that −Q(α∆t) corresponds the heat transferred from in the central region is kC while the left (right) atom
the molecule to the α-reservoir, while W(∆t) is the en- connects to the left (right) lead though a coupling k
L
ergytransferredtothemoleculethatdoesnotcomefrom (k ).
R
reservoirs, namely,
(cid:90) t+∆t (cid:90) t+∆t k k kL kC kR k k
Q(∆t) = dt¯J (t¯) and W(∆t) = dt¯Φ(t¯),
α α ··· ···
t t
(65) 2L 1L A B 1R 2R
where J (t) and Φ(t) are, respectively, the thermal cur-
α FIG. 1. (Color online) Sketch of the model system. Balls
rent flowing from α-reservoir into the molecule and the
represent the chain sites while springs represent the coupling
power developed by the ac sources. potential. The central region, formed by 2 atoms A and B
For a periodic process after a cycle of period ∆t = τ, coupled by a spring with force constant k , is connected to
C
we finding that ∆E(τ) =0, so that leftandrightsemi-infiniteleadsthroughcouplingskLandkR,
M
respectively. The leads have a constant coupling k.
(cid:88)
Q(τ)+W(τ) =0, (66)
α
In this model the inter-partition and central coupling
α
reduced matrices are
where we define (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
V = k , V = −k 0 , V = 0
LL L LC L LR
Q(ατ) =τ (cid:104)Jα(t)(cid:105)τ and W(τ) =τ (cid:104)Φ(t)(cid:105)τ. (67) V =(cid:18)−kL(cid:19), V =(cid:18)kL 0 (cid:19), V =(cid:18) 0 (cid:19),
CL 0 CC 0 k CR −k
R R
J (t) and Φ(t) can be evaluatedby using apertur-
(cid:104) α (cid:105)τ (cid:104) (cid:105)τ V =(cid:0)0(cid:1), V =(cid:0)0 −k (cid:1), V =(cid:0)k (cid:1), (70a)
bative expansion RL RC R RR R
and
(cid:104)JΦα((tt))(cid:105)τ ==εJ2α(SΦ)(+P)ε2+Jα(P()ε+3)O, (ε3), ((6688ba)) KC0C =(cid:18)−kkC −kkC(cid:19). (70b)
(cid:104) (cid:105)τ O C C
8
Here the matrices V(L) and V(R) introduced in (62d) 2 . 0 2 . 0
CC CC
read T ra n s m is s io n
1 . 5 D O S 1 . 5
(cid:18) (cid:19) (cid:18) (cid:19)
V(L) = kL 0 , V(R) = 0 0 , (70c) 1 . 0 1 . 0 S
CC 0 0 CC 0 kR O
D
0 . 5 0 . 5
and satisfy V =V(L)+V(R).
CC CC CC
The retarded and advanced components of the modi- 0 . 0 0 . 0
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
fied Green’s functions are w
1 ω2 2k ıω√4k ω2
g˜αr,a[ω]=12 ω2−(k2−kαk∓α)(cid:112)ω2ω+2(kωα2−2 4,k) |ω|(cid:54)√4k FofIGth.e2f.re(qCuoelonrcyonωlinine)uDniOtsSoafn√dktrfoarnskmLis=sioknCT=aksRf=unkct.ions
 − α− − , ω >√4k,
2 (k k )ω2+k2 | |
− α α (71) to the thermal conductance defined by
for α = L,R. Note that the property g˜r[ ω] = g˜a[ω] is
satisfied according to the Eq. (20a). Tαhe−derivatαion of σ(T)= 2kB2T (cid:90)xcdx x2 (cid:18)ωcx(cid:19) T→0 π2kB2 T
Eqs. (71) is presented in App. B. h sinh2xT xc −−−→ 3h
0
(74)
where ω √4k and x−1 2k T/((cid:126)ω ). The low tem-
A. Steady-state c ≡ c ≡ B c
perature limit of σ(T) is
Equations (70) and (71), allow us to calculate the re- π2k2
σ = B T (75)
tarded and advanced self-energies Σ˜r,a [ω] defined in 0 3h
L(R)
Eq. (46), the level-width functions Γ˜L(R)[ω] given by as theoretically precdicted28,29 and experimentally
Eqs. (51a) and (A13), and the central region Green’s observed39. From Eq. (55), it possible to verify that
functions Gr,a[ω]. The local density of states (LDOS) (0) = 1. Thus we see that at low temperatures the
CC
at the site j =A,B in the central region reads Tconductance σ(T) T and vanishes for T 0, as re-
∼ →
quired by the third law of thermodynamics.
2ω (cid:104) (cid:105)
DOS (ω)= Im Gr [ω] . (72)
j − π CC jj 1 . 0
k , k , k
L C R
The factor 2ω is present to convert the value coming di- 0 . 8 0 .1 0 ,0 .1 2 ,0 .1 0
rectly from the imaginary part of Gr [ω] into the DOS 0 .0 3 ,0 .1 2 ,0 .0 3
(cid:82) CC 0 . 6 0 .0 3 ,0 .2 0 ,0 .0 3
perunitofω,ensuringthat DOS(ω)dωequalsthenum- 0 .1 0 ,0 .1 2 ,0 .0 3
ber of propagating channels in the system. 0 . 4
For the equal force constant case we can calculate the
0 . 2
LDOS and the transmission analytically, namely
0 . 0
2 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9
DOS (ω)= Θ(4k ω2), j (73a) w
j
π√4k ω2 − ∀
−
(ω)=Θ(4k ω2). (73b)
FIG. 3. (Color online) Transmission T as a function of the
T − √
frequencyω inunitsof k intheweakcouplingregime. The
Figure 2 shows the DOS at one of the sites in the cen-
values of k , k and k are indicated in the picture in units
L R C
tral region for kL =kR =kC =k. Our formalism recov- of k. The vertical dashed lines are the frequencies given by
ers the standard DOS for a linear chain. The singularity Eq. (76).
at ω = √4k agrees with the frequency in which the dis-
persion relation of a linear chain ω = √4ksin(kxa/2) Let us now study situations where the force constants
becomes flat, i.e., at the edge of the first Brillouin zone. aredifferent. Intheweakcouplinglimit,k ,k k,k ,
R L C
Herekxisthelongitudinalmomentumandaisthelattice thecentralregionisnearlydisconnectedfromthe(cid:28)outside
parameter. Also, the transmission coefficient (ω) cor- world having only one resonant level at ω = √2k .
T C C
responds to a perfect transmission inside the frequency Thus, the conductance is only expected to be significant
band of the leads ω <√4k and it is zero otherwise. at the vicinity of ω . Instead, Fig. 3 shows one peak
| | C
In the limit of small temperatures and small tempera- at ω ω and two additional strong peaks, one at zero
C
turedifferences,namely,TL/R =T±∆T/2with∆T (cid:28)T frequ≈encyandanotherintermediatepeakat0<ω <ωC.
for T 0, the thermal current for steady-state can be The first peak at ω = 0 corresponds to the acoustic
→
written as J(S) = σ(T) ∆T, where σ(T) corresponds mode that has an infinite long wavelength so that the
L,R ±
9
short ranged “defects” introduced by k ,k ,k = k do thesteady-statecurrentfromtheα-reservoirisJ(S) =0.
L R C α
(cid:54)
not affect the transport across the system. This picture Hence, J(P) gives the leading contribution to the heat
α
is reenforced by noticing that the zero frequency peak is flow.
robust against changes in the value of kL, kC and kR in We consider the case of pumping functions with a
the weak coupling regime, see Fig. 4. phase difference ϕ, namely, φ (t)=φ (t ϕ/Ω), which
L R
−
implies that ϕ(L) ϕ(R) = nϕ and a(L) = a(R) a
n n n n n
1 . 0 k , k , k for n (cid:62) 1. Accor−ding to Eq. (69), we can expres≡s the
L C R
0 . 8 0 .0 2 ,0 .4 0 ,0 .0 5 α-thermal pumped current as
0 .0 3 ,0 .4 0 ,0 .0 5
0 .0 5 ,0 .4 0 ,0 .0 5
0 . 6 0 .0 9 ,0 .4 0 ,0 .0 5 (cid:88)∞ (cid:20)
0 .1 5 ,0 .4 0 ,0 .0 5 J(P)(Ω)= a2 α (nΩ)+cos(nϕ) α (nΩ)
0 . 4 α n Ahomo Ahete
n=1
(cid:21)
0 . 2
sin(nϕ) α(nΩ) , (77)
− B
0 . 0
0 . 0 0 . 1 0 . 2 0 . 3 0 . 4
w
where α (nΩ) Aα (n) + Aα (n), α (nΩ)
Ahomo ≡ LL RR Ahete ≡
Aα (n)+Aα (n), (α)(nΩ) Bα (n) Bα (n)forα=
FIG. 4. (Color online)√Transmission T as a function of the L,LRR. For thReLsymmBetric cou≡plinLgRcase−, i.eR.,LkL = kR =
(cid:54)
finreqthueenficgyuωrei)nwunitihtskof =k0fo.4rdainffdereknt=val0u.e0s5o.fAkRll(cinodnisctaatnetds kC, we can show that ALhomo/hete(nΩ)=ARhomo/hete(nΩ)
C L and L(nΩ)= R(nΩ).
are in units of k. The maximum transmission occurs when B −B
Similarly, the pumped power reads
k =k . The vertical dashed lines are the frequencies given
L R
by Eq. (76).
∞ (cid:20)
(cid:88)
Φ(P)(Ω)= a2 (nΩ)+cos(nϕ) (nΩ)
By coupling the diatomic molecule to leads, the reso- n Dhomo Dhete
n=1
nance level at ω is shifted and acquires broadening, as (cid:21)
C
described by the self-energy Σ˜r[ω]. Hence the peak near sin(nϕ) (nΩ) , (78)
− E
ω is very sensitive to variations in k and k . These
C L R
features, are illustrated in Fig. 3, by inspecting a set of
where (nΩ) D (n) + D (n), (nΩ)
transmission curves where we keep k constant and in- Dhomo ≡ LL RR Dhete ≡
C D (n)+D (n) and (nΩ) E (n) E (n) de-
crease k =k . LR RL E ≡ LR − RL
L R fined for 0 < nΩ < 4√k and zero otherwise. For fur-
Ontheotherhand, theremainingpeakat0<ω <ω
C ther details see Appendix B. For a symmetric setup (i.e.
depends only on the values of k and k . In the weak
L R k =k =k ), we can show that (nΩ) 0. Note that
couplingregime,asemi-classicalpictureexplainsthisad- L R (cid:54) C E ≡
Φ(P) satisfies the condition Φ(P) >0, which corresponds
ditional transmission peak. The natural interfaces fre-
to performing work on the system.
quencies ω √k , with α = L,R, are much smaller
α α
∝ The pumping function is determined by the choice of
thenω . Thelargeseparationinfrequenciessuggestthat
C
theparameterset a andphasedifferenceϕ. Westudy
the resonance close to ωC is dominated by the isolated four examples of p{umn}ping functions: single-mode repre-
molecule mode, while the other corresponds to an oscil-
sented by a = δ /2; square oscillation represented by
lation of a frozen central region. The Green’s functions n n,1
a = 2/(nπ) for n odd (and zero otherwise); triangle
of such a system gives resonances at the frequencies n
ω =(cid:118)(cid:117)(cid:117)(cid:116)k +(cid:18)kL+kR(cid:19) (cid:115)(cid:18)kL−kR(cid:19)2+k2. oonts.hcielrlawtiisoen);bsyawatnoo=th ons24cπi2ll(a−ti1o)nn−2b1y afonr=n−o1d/d(π(nan)dforzearlol
1,2 C 2 ± 2 C The thermal current absorbed by the α-reservoir,
(76) Jα(P) >0,asafunctionofpumpingfrequencyΩfordif-
−
ferent pumping profiles and phase difference is shown in
For k = k , ω = √2k +k and ω = √k that are Fig. 5. We find a suppression of thermal current accord-
L R 1 C L 2 L
plotted in Fig. 3 as vertical dotted lines matching the ingtothetypeofpumpinginthefollowingorder: square,
peaks positions. For k = k , the symmetry is broken single-mode, triangle and sawtooth. The pumping peak
L R
and the maximum transm(cid:54) ission at all the peaks, except occurs in the frequency window 2√k < Ω < 3√k and a
for the one with zero frequency, is no longer perfect. sub-peak within 0 < Ω < √k, accompanied by a weak
suppressioninthedomain√k <Ω<2√kandthestrong
suppression for Ω (cid:38) 4√k. As the phase difference is in-
B. Pumping creased, the suppression in √k <Ω<2√k is intensified.
Note that the unperturbed proposed setup is symmetric
For simplicity, let us analyze a pumping process be- (kL =kR =kC). The phase shift ϕ causes the difference
(cid:54)
tween reservoirs at the same temperature. In this case, between J(P) and J(P), see Fig. 5.
L R
10
1 .0 1 .0 1 .0
square ( a ) j = p / 4 square ( b ) j = p / 2 square ( c ) j = 3 p / 4
single-m ode single-m ode single-m ode
0 .8 triangle 0 .8 triangle 0 .8 triangle
saw tooth saw tooth saw tooth
0 .6 0 .6 0 .6
) ) )
(P (P (P
0L.4 0L.4 0L.4
J J J
- - -
0 .2 0 .2 0 .2
0 .0 0 .0 0 .0
0 1 W2 3 4 0 1 W2 3 4 0 1 W2 3 4
1 .0 1 .0 1 .0
square ( d ) j = p / 4 square ( e ) j = p / 2 square ( f ) j = 3p / 4
single-m ode single-m ode single-m ode
0 .8 triangle 0 .8 triangle 0 .8 triangle
saw tooth saw tooth saw tooth
0 .6 0 .6 0 .6
) ) )
(P (P (P
0R.4 0R.4 0R.4
J J J
- - -
0 .2 0 .2 0 .2
0 .0 0 .0 0 .0
0 1 W2 3 4 0 1 W2 3 4 0 1 W2 3 4
0 .3 0 .3 0 .3
j p jj pp j 3p
0 .2 ( g ) = / 4 0 .2 (( hh )) == // 22 0 .2 ( i) = / 4
)0 .1 )0 .1 )0 .1
(P (P (P
J0 .0 J0 .0 J0 .0
(cid:1) (cid:1) (cid:1)
-0 .1 -0 .1 -0 .1
-0 .2 -0 .2 -0 .2
square square square
-0 .3 single-m ode -0 .3 single-m ode -0 .3 single-m ode
triangle triangle triangle
-0 .4 saw tooth -0 .4 saw tooth -0 .4 saw tooth
0 1 W2 3 4 0 1 W2 3 4 0 1 W2 3 4
2 .0 2 .0 2 .0
square ( j) j = p / 4 square ( k ) j = p / 2 square ( l) j = 3 p / 4
single-m ode single-m ode single-m ode
triangle triangle triangle
1 .5 saw tooth 1 .5 saw tooth 1 .5 saw tooth
(P)1 .0 (P)1 .0 (P)1 .0
(cid:1) (cid:1) (cid:1)
0 .5 0 .5 0 .5
0 .0 0 .0 0 .0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
W W W
FIG.5. (Coloronline)Thermalcurrentabsorbedbytheleft/right-reservoir−J(P) (inunitsof(cid:126)k/ε),effectivethermalcurrent
L/R
transmitted from left- to right-reservoir ∆J(P) ≡ J(P)−J(P) (in units of (cid:126)k/ε) and power injected Φ(P) (in units of (cid:126)k/ε)
R L √
into the system as a function of the pumping frequency Ω (in units of k) using k = k = 0.5k and k = 0.25k for equal
√ L R C
temperatures with k T =k T =1 (in units of (cid:126) k) and for distinct pumping functions.
B L B R
The effective heat flux between the two reservoirs reservoir to left-reservoir and ∆J(P) < 0 to reverse di-
∆J(P)(Ω) J(P)(Ω) J(P)(Ω) for the symmetric cou- rection. Note that for ϕ = 0, 2π, 4π, 6π,..., we ob-
≡ R − L ± ± ±
pling case (i.e. k =k =k ) reads tain ∆J(P) = 0, as expected. For the single-mode case
L R C
(cid:54)
we verify that the maximum value of ∆J(P) occurs for
(cid:88)∞ ϕ = π/2, 3π/2, 5π/2,.... ∆J(P)| versus| the pum-
∆J(P)(Ω)= 2a2 sin(nϕ) L(nΩ), (79) ± ± ±
n B ping frequency Ω is represented in the Figs. 5(g)-(i).
n=1
The time-dependent drive breaks the system symme-
where∆J(P) >0correspondstotheheatfluxfromright- try. Hence, one can engineer configurations of Ω and