Table Of ContentTransient behavior of heat transport in a thermal switch
Eduardo C. Cuansing and Jian-Sheng Wang
Department of Physics and Centre for Computational Science and Engineering,
National University of Singapore, Singapore 117542, Republic of Singapore
(Dated: 22 December 2009)
Westudythetime-dependenttransportofheatinananoscalethermalswitch. Theswitchconsists
ofleftandrightleadsthatareinitiallyuncoupled. Duringswitch-onthecouplingbetweentheleadsis
abruptlyturnedon. WeusethenonequilibriumGreen’sfunctionformalismandnumericallysolvethe
0
constructedDysonequationtodeterminethenonperturbativeheatcurrent. Atthetransientregime
1
we find that the current initially flows simultaneously into both of the leads and then afterwards
0
oscillates between flowing into and out of theleads. At later times the oscillations decay away and
2
the current settles into flowing from the hotter to the colder lead. We find the transient behavior
n to be influenced by the extra energy added during switch-on. Such a transient behavior also exists
a evenwhen thereisnotemperaturedifferencebetween theleads. Thecurrentat thelong-timelimit
J
approaches thesteady-state valueindependentlycalculated from theLandauerformula.
9
2 PACSnumbers: 44.10.+i,63.22.-m,66.70.-f,66.70.Lm
]
l
l The physics of generation, dissipation, and manipula- -2 -1 0 1 2 3
a
tion of heat in nanoscale systems is an important topic
h
- that has recently gathered attention. Experiments on semi-infinite left lead semi-infinite right lead
s molecular junctions1 found that the heat generated in
e
m current-carrying metal-molecule junctions is substantial coupling is switched on at t=0
andcanaffecttheintegrityofthe device. Understanding
.
t how to efficiently dissipate extraneous heat in nanoscale FIG. 1: (color online) An illustration of a quantum thermal
a
m systemsisthusimperativeintheconstructionofdevices. switch. The labels of the first 3 sites in each lead are shown.
In addition, the movement of heat may be harnessed for The coupling between sites 0 and 1 is switched on at t=0.
- informationprocessing2. Recentexperimentshaveshown
d
n theviabilityofthermaltransistors3,thermalrectifiersus-
o inginhomogeneouscarbonandboronnitridenanotubes4, andrightleadsareuncoupledandareinthermalequilib-
c andconductance-tunablethermallinksconsistingofmul- riumwithtemperaturesTLandTR,respectively. Attime
[ tiwalled carbon nanotubes5. t = 0 the coupling potential, in the form of an interpar-
ticle harmonic potential with the same spring constant
2 Most of the above-mentioned work, however, focus on
k, is switched on, i.e., the potential between the masses
v
theexaminationofsteady-statephenomena. Incontrast,
3 labeled 0 and 1 in Fig. 1 is suddenly switched on. We
any physical device must function in a time-dependent
1 thenwantto knowhow the time-dependent heatcurrent
environment. Although some work has been done in the
8 behaves in such a setup. In experiments on molecular
4 study of time-dependent electronic transport6, the in- junctions1 ascanningtunnelingmicroscopetipisusedto
. vestigationof time-dependent behavior in quantum heat
0 stretchamoleculeuntilabondinthemoleculebreaks. In
transport has not yet attracted much attention7. De-
1 the thermal switch we can think of the switch-on as the
9 veloping theoretical tools and computational methods inverseprocess,i.e.,abondisinducedthroughproximity.
0 for such problems are thus essential to the progress of
The leads follow the Hamiltonian
: the field. In this paper we study the time-dependent
v 1 1
i heat current in a junction system we call a thermal Hα = u˙αu˙α+ uαKαuα, α=L,R, (1)
X switch. We generalize the nonequilibrium Green’s func- 2 i i 2 i ij j
Xi Xij
r tion formalism8, which is developed for steady-state sit-
a uations, to the time-dependent case with a well-defined where the sums are over all sites in the lead, the trans-
initialthermalstate. To getnonperturbativeresults,the formed coordinates are given by ui = √mxi, xi is the
key steps we take are to construct and numerically solve relative displacement of the i-th atom of mass m, and
aDysonequationthatdoesnotsatisfytime-translational Kα is the spring constant matrix. The KL and KR ma-
invariance. trices are semi-infinite tridiagonal matrices with 2k+k0
along the diagonal and k along the off-diagonal. The
Fig.1showsaone-dimensionalchainhavingacoupling −
Hamiltonian for the switched coupling is
that can be switched on and off. The semi-infinite left
andrightleadsarelinearchainsofmassesm. Eachatom HLR = uLVLRuR. (2)
i ij j
interacts with its left and rightneighbors throughan in-
Xij
terparticle harmonic potential having spring constant k.
Anon-siteharmonicpotential,withspringconstantk ,is The coupling constant matrices VLR and VRL are zero
0
alsoexperiencedbyeachatom. Duringtimet<0theleft matrices during t < 0. After the switch-on, VLR has
2
one non-zero element VLR = k and VRL has the lone R L R R L L
non-zero element VRL =01 k,−where the matrix indices τ τ = τ τ τ
correspondto the la1b0els of−the massesinthe leads. Note 1 GRL 2 1 gR a gL 2
thatHLR =HRL. Thetime-dependentgoverningHamil-
tonian therefore is H(t)=HL+HR+HLR θ(t), where R R L L R L
+
θ(t) is the Heaviside step function. τ τ τ τ
The current flowing out of the left lead IL(t) = 1 gR a gL b GRL 2
dHL/dt ,i.e.,itistheexpectationvalueoftherateof
−
ch(cid:10)angein H(cid:11)L. Whenthe switchis turned onthe current FIG. 2: (color online) Diagram representation of the Dyson
is given by equation. Each line is labelled by the Green’s function it
represents. Each concentricdot representsa coupling vertex.
∂GRL,<(t ,t )
1 2
I (t)=¯hk Im , (3)
L
(cid:20) ∂t (cid:21)
2 t1=t2=t
gR(τ ,τ ) = i T uR(τ )uR(τ ) , and a dashed line
where “Im” stands for the imaginary part. The lesser 1 2 −h¯ c 1 1 1 2 0
represents the eq(cid:10)uilibrium Green’s(cid:11)functions for the left
Green’s function that appears in the formula is defined
lead, gL(τ ,τ ) = i T uL(τ )uL(τ ) . The sub-
as GRL,<(t1,t2) = −h¯i uL0 (t2)uR1 (t1) , where the sub- script 0 im1pli2es that−th¯h(cid:10)e acve0rag1e is0tak2en(cid:11)0with respect
scripts 0 and 1 corresp(cid:10)ond to the lab(cid:11)els of the masses.
to equilibrium distributions that are maintained when
Notethatthisisatwo-timecorrelationfunctionthatdoes
t<0 before the switch-on.
not satisfy time-translational invariance, i.e., its time-
Rewriting the diagram equation in Fig. 2 as contour
dependence can not be written as a difference t t .
1 − 2 integrals, we have
Similarly, the current flowing out of the right lead,
I (t) = dHR/dt , when the switch is turned on is in
R
the form−of(cid:10)Eq.(3)(cid:11)except that the Rand L superscripts GRL(τ1,τ2) = dτa gR(τ1,τa)VRL gL(τa,τ2)
Z
C
are swapped.
Todetermine the full, nonperturbative,currentweare + dτ dτ gR(τ ,τ )VRL
a b 1 a
Z Z
going to solve the associated Dyson equation. First, we C C
define the contour-orderedGreen’s function9 gL(τ ,τ )VLR GRL(τ ,τ ). (6)
a b b 2
×
i
GRL(τ ,τ )= T uR(τ )uL(τ ) , (4) Applying Langreth’s theorem to Eq. (6) and then
1 2 −¯h c 1 1 0 2 iterating9,weshouldobtainanexpressionforGRL,<. To
(cid:10) (cid:11)
calculate the current in Eq. (3) we need the time deriva-
where the uR1 anduL0 areHeisenberg operators,Tc is the tive of GRL,<, and so we differentiate it to get
contour-ordering operator, and τ and τ are complex
1 2
variables on the contour C. To determine the current at ∂GRL,<(t ,t ) ∂GRL,<(t ,t )
time t we employ a Keldysh contour C that goes from 1 2 = 1 1 2
∂t ∂t
time 0 to t and then back to 0. Since the left and right 2 2
leads are uncorrelated before the switch is turned on at k tdt GRL,r(t ,t )∂GR1L,<(ta,t2)
a 1 a
time 0, the contour does not have a complex tail after it − Z ∂t
0 2
goes back to time 0. t ∂GRL,a(t ,t )
Converting to the interaction picture the contour- − kZ dta GR1L,<(t1,ta) ∂t a 2
ordered Green’s function shown in Eq. (4) becomes 0 2
t t
+ k2 dt dt GRL,r(t ,t )
GRL(τ1,τ2)=−¯hi(cid:28)Tce−h¯i RCdτ′HLR(τ′)uR1 (τ1)uL0(τ2)(cid:29). Z0 aZ0 ∂bGRL,a(t ,1t )a
(5) × GR1L,<(ta,tb) ∂t b 2 , (7)
2
A perturbative calculation can then be done by expand-
ing the exponential as an infinite series. We can also use where
this expansion in constructing the Dyson equation. In
t
theseries,thezeroth-ordertermvanishesbecauseitdoes GRL,<(t ,t ) = k dt gR,r(t t ) gL,<(t t )
not contain the coupling potential. Furthermore, all the 1 1 2 − Z0 a(cid:8) 1− a a− 2
even-ordered terms also vanish because there will be an + gR,<(t t ) gL,a(t t ) (8)
1 a a 2
extra uR-uL pair without a connecting coupling poten- − −
(cid:9)
tial. Thus, only the odd-ordered terms survive. is the first-order term in the perturbation series in
We can use diagrams in constructing the Dyson equa- Eq. (5). The analytic expressions for the equilibrium
tion. Shown in Fig. 2 is the resulting diagram equa- surface Green’s functions gR,r, gR,<, gL,<, and gL,a are
tion when we expand the series in Eq. (5). A double- known in frequency space10. To determine their time-
line diagram represents GRL, a single line represents dependencewenumericallycalculatetheircorresponding
the equilibrium Green’s functions for the right lead, Fourier transforms.
3
The other unknowns in Eq. (7) involve the retarded their steady-state values. The dots shown at the right
and advanced versions of the full Green’s function. We edges of the plots are steady-state values calculated in-
can apply Langreth’s theorem again to Eq. (6) to deter- dependently from the Landauer formula I = I =
mine expressions for these unknowns. We get 1 ∞ dω ¯hω(f f )θ˜(ω), where f andLf a−reRthe
2 −∞ 2π L− R L R
BoRse-Einstein distributions of the left and right leads,
GRL,β(t ,t ) = k tdt GRL,β(t ,t )GRL,β(t ,t ) respectively, and θ˜(ω) is 1 within the phonon band,
1 2 − Z0 a 1 1 a a 2 k0 < ω2 < 4k+k0, and 0 otherwise10. At steady-state,
+ GRL,β(t ,t ), (9) heat should flow from the hotter to the colder lead, i.e.,
1 1 2 fromthe left to the rightlead. Thus, the signofthe cur-
rent flowing out of the left lead should be positive and
where β =r,a, and the first-order term is
for the right lead negative. However, during the tran-
t sienttime,thecurrentcanflowinunexpecteddirections.
GR1L,β(t1,t2)=−kZ dta gR,β(t1−ta) gL,β(ta−t2). Just after the switch-on, the current actually does not
0 flow from the hotter to the colder lead. In Fig. 3 we
(10)
see the current to flow simultaneously into both of the
To solve Eq. (9) we discretize the time variable into N
leads. There appears to be an energy source in-between
segments and thus transforming the integral into a sum.
This results in a linear problem of the form A~x = ~b, the leads that supply the current. Recall that there is
no coupling between the leads before the switch-on. By
where the unknown ~x is determined by performing an
turningontheswitchweactuallyaddenergy,intheform
LU decomposition on A and then using ~b in the back-
of the switched coupling potential, to the system. This
substitution. The ∂GRL,a(t ,t )/∂t term required in
a 2 2 added coupling energy supplies the current that flows
Eq. (7) can also be calculated by first differentiating
into both leads. In Fig. 3 we also compare the results
Eq. (9) and then finding the solution to the resulting
fromfirst-orderperturbationto the results fromnonper-
equation with the time discretized. By determining the
turbative calculations. Unlike the perturbative results,
time derivative of the full Green’s function in Eq. (7)
nonperturbativeresultsapproachthe steady-statevalues
thecurrentthatwecalculateisanonperturbativeresult.
at later times. Since the switched coupling has the same
We follow the same steps to independently calculate the
strength as the interparticle potential we indeed expect
current flowing out of the right lead.
correctionsto perturbativecalculationsto be significant.
Shown in Fig. 3 are plots of the time-dependent cur-
Shown in Fig. 4 are plots of the sum of the currents,
rentflowing outof the leads whenthe left andrightlead
I = I +I , as functions of the time and the average
temperatures are T = 330 K and T = 270 K, respec- S L R
L R
temperature of the leads. I fluctuates aroundzero with
tively. The average temperature is thus T = 300 K S
a fluctuation amplitude that decayswith time. Fig. 4(b)
with offsets of 10%. The currents oscillate at a fre-
± shows I to vary strongly with temperature at the tran-
quency comparable to the highest phonon frequencies S
sient regime. As time goes on I slowly loses its de-
available in the system and then gradually decay to S
400
80 80
first-order W] 0
40 all orders 40 n
[S -400 (a)
W] 0 0 I
nt [n -40 all orders -40 -800 0 10 time [2100-14 s] 30 40
e
curr -80 first-order -80 800 6
W] 600 (b) W] 3
--116200 (a) (b) --116200 |I| [nS 240000 I [nS- 03 (c)
0 10 20 30 40 0 10 20 30 40 0 -6
time [10-14 s] time [10-14 s] 0 500 1000 20 30 40
temperature [K] time [10-14 s]
FIG. 3: The time-dependent current flowing out of the (a) FIG. 4: (a) Plots of IS = IL+IR as functions of time when
leftleadand(b)rightlead. The(blueonline)datapointsare T = 10 K (orange online), T = 300 K (red online), and
resultsfromsolvingtheDysonequationwhilethe(redonline) T =1000K(blueonline). Thetemperatureoffsetsare±10%.
lineistheresultfromthefirst-orderperturbationcalculation. (b) Plots of |IS| as functions of T at time t = 0.8 [t] (brown
The average temperature between the leads is T = 300 K. online), t= 4.1 [t] (yellow online), t= 6.0 [t] (violet online),
Theinterparticlespringconstantisk=0.625eV/(˚A2u)while and t = 23.0 [t] (green online), where [t] = 10−14 s. (c) An
theon-site spring constant is k0 =0.0625 eV/(˚A2 u). enlarged view of (a) for time t=20 [t] to t=40 [t].
4
ever, since energy is added to the system during the
60
switch-on, at the transient regime we see a fluctuating
40
current with temperature-dependent amplitude flowing
20
within the system.
0
nW] -20 The method can also be generalized in a straightfor-
nt [ -40 T = T = T ward manner to deal with a time-varying coupling that
curre --8600 L R iass,onforduerxianmgptl>e,0k.(tF)o=r akmtainldhl(yfitn),crtehaesitnrgancsoiuenptlincgursruecnht
-100 T = 10 K also initially flows simultaneously into both of the leads.
T = 300 K Thistransientbehaviorpersistsbecauseswitchingonthe
-120
couplingintroducesenergy,thathastodissipateintothe
-140
0 10 20 30 40 baths, into the system.
time [10-14 s] To summarize, we have shown an exact nonperturba-
tivemethodtocalculatethetime-dependentheatcurrent
in a thermal switch using nonequilibrium Green’s func-
FIG. 5: The current flowing through the junction when the
left and right leads have the same temperature. T = 10 K tions. The Dyson equation is constructed using Keldysh
(red online) and T =300 K (blueonline). contours and the real-time Green’s functions needed to
calculate the current are determined by applying Lan-
greth’s theorem to the Dyson equation and numerically
pendence on temperature. Fig. 4(c) shows that at later solving the equation with the discretized time variable.
times fluctuations in I still appear but are significantly We set the strength of the switched coupling to be the
S
smallerthanthoseatthetransientregime. Wedoexpect same as the interparticle spring constant. Nonperturba-
that at the steady state, of which the long-time limit of tive results are thus significantly different from pertur-
ourdataapproaches,allofthe currentflowingoutofthe bative ones. We find the transient current just after the
left lead should flow into the right lead and thus result- switch-ontobeinfluencedbytheextraswitchedcoupling
ing in I =0, regardlessof the value of the temperature. energy. In particular, the initial reaction after switch-on
S
However,sinceenergyisaddedduringtheswitch-on,this is for the current to flow simultaneously into both the
extraenergyinfluences the transientbehaviorofthe sys- leftandrightleads. Thecurrentthenoscillateswitham-
tem until it eventually dissipates out to the heat baths. plitude that decayswith time. Inthe long-timelimit the
Fittingthe envelopetoanexponentialfunctionwefinda current approaches the expected steady-state values cal-
characteristic decay time of about 10 10−14 s. At any culated independently from the Landauer formula. We
×
particulartimetwecandeterminetheenergythesystem alsonote that the theory presentedhere is not restricted
absorbs or emits from HLR(t) = t(I +I )dt. only to one-dimensional chains but is applicable to any
0 L R
Suppose we set the(cid:10)tempera(cid:11)tureRs of the leads to be junction system where a thermal switch-on occurs.
the same. Shown in Fig. 5 is the time-dependent cur- We are grateful to Lifa Zhang, Jin-Wu Jiang, Meng
rent, in either lead since the leads are indistinguish- LeeLeek,andJoseGarciaforinsightfuldiscussions. This
able, for such a situation. At the steady state there workissupportedinpartbyanNUSresearchgrantnum-
should be no current flowing within the system. How- ber R-144-000-257-112.
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