Low-Temperature Linear Thermal Rectifiers Based on Coriolis forces

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by  Suwun Suwunnarat

Low-Temperature Linear Thermal Rectifiers Based on Coriolis forces
Suwun Suwunnarat1, Huanan Li1, Ragnar Fleischmann2 and Tsampikos Kottos1,2
1Department of Physics, Wesleyan University, Middletown, Connecticut 06459, USA
2Max Planck Institute for Dynamics and Self-organization (MPIDS), 37077 Go¨ttingen, Germany
We demonstrate that a three-terminal harmonic symmetric chain in the presence of a Coriolis
force, produced by a rotating platform which is used to place the chain, can produce thermal
rectification. The direction of heat flow is reconfigurable and controlled by the angular velocity
Ω of the rotating platform. A simple three terminal triangular lattice is used to demonstrate the
6
proposed principle.
1
0
2 PACSnumbers: 44.10.+i,05.60.-k,66.70.-f

n
a I. INTRODUCTION in Ref. [12].
J
In this paper we focus on the development of a new

2
concept for the creation of unidirectional thermal valves
1 In the last few years considerable research effort has
that control and direct heat currents on the nanoscale
been invested in developing appropriately engineered level. Specifically we propose to create a ballistic sym-
]
l structures that display novel transport properties not metricthermalrectifierwhichreliesonCoriolisforces. In
l
a found in nature. In the thermal transport framework, contrasttopreviousstudiesourproposaldoesnotrelyon
h this activity has recently started to gain a lot of atten-
any structural asymmetry. Rather, an asymmetry is in-
- tion. Apart from the purely academic reasons, there is
s duced in a very controllable way by the Coriolis force
e a growing consensus on its practical implications in the allowing a high flexibility in the rectification properties
m efforts of the society to manage its energy resources ef-
without a change of the structural setup. The system
. ficiently. Thus research programs that aim to propose consists of a symmetric harmonic lattice placed on a ro-
t
a new methods, designs or materials that allow to harness tating platform with angular velocity Ω. The direction
m and mold the heat flow at the nanoscale level in ways
of heat current is controlled on the sign of Ω, thus allow-
that can affect society’s energy consumption needs are
- ingforareconfigurableheatcurrentmanagement. Using
d at the forefront of the research agenda. Some of the tar-
the non-equilibrium Green’s function formalism we de-
n
gets that are within our current nanotechnology capa-
rive the conditions of optimal operation of the structure.
o
bilities include the generation of nanoscale heat-voltage
c Subsequently we have verified these predictions via de-
[ converters, thermal transistors and rectifiers, nanoscale tailednumericalcalculationsusingasimplevariantofthe
radiation detectors, heat pumps and even thermal logic proposed structure.
1
gates [1–6].
The paper is organized as follows. In section II we
v
1 In spite of these efforts the understanding of thermal presentthetheoreticalmodel. Themathematicalformal-
6 transportandmoreimportantlythemanipulationofheat ism for the evaluation of heat current and the conditions
7 current is still in its infancy. This becomes more obvi- for non-reciprocity are discussed in section III. Finally
2 ous if one compares with the tremendous achievements a numerical confirmation of our predictions are given in
0
of the last fifty years in understanding and managing section IV. Our conclusions are given in the last section
.
1 electron transport. In contrast, for example, the impor- V.
0
tant problem of thermal rectification is addressed only
6
by a handful of researchers. The majority of these pro-
1
: posals relies on the interplay of system non-linearities II. THEORETICAL MODEL
v
with structural asymmetries [7–9]. At the nanoscale
i
X limit where the phonon mean-free path is comparable We consider a harmonic lattice of N particles coupled
to the size of the devices the inherent nonlinearities are together with spring constants kC. All particles are as-
r
a irrelevant and transport is fully dominated by ballistic sumedtohavethesamemassm. Below,withoutanyloss
phonon transport (ballistic limit). However recently it of generality, we assume that all masses are m=1. The
wasshowninRef. [10–12]thatintheballisticregimeone lattice is placed on a platform in the X−Y plane. Thus
canachievethermalrectificationunderveryspecificcon- each particle can move on this platform and has D = 2
ditions: low temperature limit associated with quantum degrees of freedom. The connectivity of the lattice is
harmonicsystemsinthenonlinearresponsedomain, and uniquely defined by the N = N ∗D dimensional sym-
thepresenceofathirdreservoirwhichactsasaprobeand metric force matrix KC. We will assume the most gen-
it is coupled asymmetrically with the rest of the system eralcasewherethecouplingisnotnecessarilyconfinedto
in order to breakanyspatial symmetry. The importance be only between nearest (in space) masses. Moreover we
ofasymmetryofthesystemitselfwasfurtherhighlighted will assume that all masses are coupled harmonically to
2
a central post which is parallel to the Z axis and placed WearenowreadytowritedownthetotalHamiltonian
in the middle of the platform. of the bath-lattice system which takes the form:
The lattice rotates around the post with a constant
→− (cid:88) (cid:88)
H =H + H + H (4)
angular velocity Ω. It is useful to describe the mo- tot C α αC
tion of the system in the rotating frame. In this frame α α
the particles are characterized by their equilibrium po-
where H =uTVαCu describes the coupling between
sition vector R0 = (R0,··· ,R0 )T, by their displace- αC α C
1 N the lattice particles and the heat baths.
mentsvectorS =(Sx,Sy,Sz,··· ,Sx,Sy,Sz )T andthe
1 1 1 N N N Belowwewillbeassumingthattheinitialconditionof
associated conjugate canonical momenta vector p =
C the total system is a direct product state
(px ,py ,pz ,··· ,px ,py ,pz )T [21]. The Hamilto-
C1 C1 C1 CN CN CN
nianthatdescribesthesystemintherotatingframetakes (cid:89) e−Hα/(kBTα) e−HC/(kBTC)
the form: ρˆini(t0)= ⊗Tr(cid:0)e−Hα/(kBTα)(cid:1) ⊗ Tr(cid:0)e−HC/(kBTC)(cid:1)
α
H = 1pTp + 1STKCS−(cid:0)R0+S(cid:1)T Ap (1) (5)
C 2 C C 2 C which after sufficient long time will be relaxing to a
steady state ρˆss. Note that the temperature of the two
where A is the N ×N block-diagonal matrix baths T are kept constant and the steady-state thermal
α
current will not depend on the initial temperature of the
A=diag{A˜ };A˜ =(cid:20)A˜D=2 0(cid:21);A˜ =(cid:20) 0 Ω(cid:21) lattice model TC. The value of the temperature of the
D D=3 0 0 D=2 −Ω 0 probe in the NESS will be determined by the tempera-
(2) ture of the other baths.
with the property AT = −A. The last term in Eq. (1)
describes the Coriolis force in the rotating frame.
TherotatinglatticedescribedbyEq.(1)iscoupledwith III. FORMALISM
three equivalent co-rotating heat baths. The latter are
always attached to the same three particles of the lat- A. Heat Current
tice α = R ,R ,R . We will assume that two of these
1 2 3
baths are at high Thot and low Tcold constant tempera- We want to calculate the steady-state thermal current
tures,respectively. Thethirdbathactsasaprobeandits flowingfromthehotreservoirT towardsthecoldreser-
hot
temperature TR3 = Tp is adjusted self-consistently such voir Tcold. To this end we will employ the standard
that the net heat flux from it is zero. In general this can nonequilibrium Green’s function technique [15]. The
leadtodifferentTp valuesintheforward(TR1 =Thot and steady-state current out of the heat bath α is
T =T )andbackward(T =T andT =T )
R2 cold R1 cold R2 hot (cid:34) (cid:35)
process. dHˆ (t)
For simplicity, the heat baths are described quasi- Iα ≡ −Tr ρˆss dαt
classically, i.e. we promote the relative momenta p of
theα−bathparticlesandtheirdisplacementsuα,withαre- = (cid:88)(cid:90) ∞ dω(cid:126)ωT [ω](f −f ). (6)
spect to the rotating frame, to conjugate canonical pairs 2π γα α γ
γ(cid:54)=α 0
[13]. Nevertheless, we assume that the statistical prop-
erties of the phonons in these thermal reservoirs are not where f = f(ω,T ) = {exp((cid:126)ω/k T )/−1}−1 is the
α α B α
affectedbytheCoriolisforceandrespectquantumstatis-
Bose distribution associated with the particles at the
tics [20]. The three baths, each consisting of an infinite
bath α which have fixed temperature T and T is the
α γα
number of harmonically coupled bath particles, are dis-
transmission coefficient from the α-th bath to the γ-th
cribed by Hamiltonians
bath. The hot and cold reservoirs, with temperatures
T and T , will be attached to the particles R ,R
1 1 hot cold 1 2
H = pTp + uTKαu , α=R ,R ,R (3) while the third - probe - reservoir with varying tempera-
α 2 α α 2 α α 1 2 3
ture T will be always attached to the R -particle.
p 3
with a (semi-infinite) harmonic force matrix Kα = During the forward process we attach the hot reser-
(Kα)T contains additional −(cid:12)(cid:12)(cid:12)→−Ω(cid:12)(cid:12)(cid:12)2 ·I terms due to the vreosierrtvooitrh,etopatrhteicplearRti1clseoRth2astoTtRh1at=TTRh2ot=anTdcotldh.e cTohlde
centrifugalforce(I denotesthesemi-infiniteidentityma- associated heat current If out of bath R2 is then given
trix). from Eq. (6)
Finallywehaveassumedthateachparticleispinnedto
(cid:88) (cid:90) ∞ dω
thesubstrateviaaquadraticpotentialwiththecoupling I = (cid:126)ωT [ω](f(T )−f(T )). (7)
f 2π γ,R2 R2 γ
constant k0. This pinning potential guarantees that the γ=R1,R3 0
lattice particles have an equilibrium position and it can
be incorporated easily in all force matrices KC,Kα, see wherethebathtemperatureTf associatedwiththeprobe
p
Eqs. (1) and (3). reservoir has to be evaluated from the requirement that
3
the net current I flowing in the probe reservoir is zero which lead us to the conclusion that for Ω(cid:54)=0 the time-
p
i.e. reversal symmetry of the system is broken due to the
rotation.
I = (cid:88) (cid:90) ∞ dω(cid:126)ωT [ω](cid:0)f(Tf)−f(T )(cid:1)=0. In order to highlight the importance of the Coriolis
p γ=R1,R2 0 2π γ,p p γ force in thermal rectification we will be considering set-
(8) upswhichinvolvestructuralrotationalsymmetry. Inthis
Thebackwardprocessisassociatedwiththereversecon- case,anyasymmetricheattransportphenomenathatwe
figuration i.e. T = T and T = T . The associ- willfindareduetoCoriolisforceonly(contrastthiswith
R2 hot R1 cold
ated heat flux I out of bath R and the corresponding the studies of Refs. [7–9] where thermal rectification is
b 2
temperature of the probe Tb are derived using similar due to structural asymmetries). As an outcome of this
p
relations as the ones shown in Eqs. (7,8). rotational symmetry assumption we have that the trans-
mission coefficients T satisfy the following relations
α,γ
B. The Transmission Coefficient TR2R1 =TR3R2 =TR1R3, TR1R2 =TR2R3 =TR3R1. (14)
Theaboveexpressionscanallowustocalculatethetrans-
We proceed with the evaluation of the transmission
mission coefficient T [ω], and thus to proceed with the
coefficient T . The latter is needed for the calculation γα
γα evaluation of the forward and backward currents I ,I .
of the forward and backward currents, and therefore for f b
the evaluation of the rectification parameter R via Eq.
(15). The transmission coefficients can be expressed as
C. The Rectification Parameter
T [ω]=Tr[Gr Γ Ga Γ ]; Γ ≡i[Σr −Σa] (9)
γα CC γ CC α α α α
Below we will quantify the rectification effect by the
whereGr(a) isthetheretarded(advanced)Green’sfunc- rectification parameter R defined as
CC
tions of the lattice. Σαr(a) = (cid:0)VαC(cid:1)T gαr(a)VαC is the −∆I
associated self-energy which can be expressed in terms R= max{|I |,|I |}; ∆I =If +Ib. (15)
of the equilibrium Green’s function gr(a) of the isolated f b
α
heat bath α and the coupling matrix VαC. Moreover When ∆I >0 (R<0) the system acts as a thermal rec-
using the definition Eq. (9) we can deduce that Γα is a tifierfavoringthedirectionfromthebathplacedatR2 to
symmetric matrix i.e. ΓTα =Γα. the bath attached at R1 while the reverse is true in case
The retarded Green’s function Gr for the central that∆I <0(R>0). WewillassumeT =T ±∆T
CC hot/cold 0
junction in the frequency domain takes the form andanalyzethedependenceofRonthetemperaturedif-
ference ∆T.
Gr [ω]=(cid:104)(cid:0)ω+i0+(cid:1)2−KC −Σr[ω]−A2−2iωA(cid:105)−1 Using Eqs. (6,8,14) we can evaluate the rectification
CC
parameter R up to the second order in the temperature
(10)
whereΣr =(cid:80) Σr denotesthetotalretardedself-energy difference∆T. Thecurrentdifference∆I isgivenby[20]:
α α
due to the interaction with all the heat baths. The as-
K
sociated advanced Green function can be expressed in ∆I =(1−a2) ∆T2 (16)
F (T )+F (T )
terms of Gr as R2R3 R1R3
CC
where a,F and K are defined as follows: the parameter
Ga [ω]=(Gr [ω])†. (11)
CC CC a is defined as
Using expression Eq. (10) we can deduce that F (T )−F (T )
a= R1R3 R2R3 (17)
F (T )+F (T )
Gr(a)[ω,−Ω]=Gr(a)[ω,Ω]T (12) R1R3 R2R3
CC CC
and its absolute value is always smaller than unity i.e.
. |a| ≤ 1. Therefore the coefficient (1−a2) appearing in
Using the cyclic property of the trace, the symmetric Eq. (16)isalwaysgreaterthanzeroi.e. (1−a2)>0. The
snhaotwurethoaftfurther ΓTα = Γα and Eq. (12), we can easily pFo(sTitive)d=efin(cid:82)it∞e dfuωn(cid:126)cωti(cid:16)on∂fs(ωF,T()T(cid:17)αR3T) > ,0 aαre=defiRne,dRas.
αR3 0 2π ∂T T0 αR3 1 2
T [ω,−Ω]=T [ω,Ω](cid:54)=T [ω,Ω] (13) Finally K in Eq. (16) is defined as
γα αγ γα
4
(cid:90) ∞ dω (cid:90) ∞ dω
K = 1 2W(ω ,ω ,T )T(ω ,ω ); (18)
2π 2π 1 2 0 1 2
0 ω1
(cid:126)4ω ω (cid:18)∂f(ω ,T)∂f(ω ,T)(cid:19)
W = 1 2 1 2 {ω (1+2f(ω ,T ))−ω (1+2f(ω ,T ))}
k T2 ∂T ∂T 2 2 0 1 1 0
B 0 T0
with the weight function W and T(ω ,ω ) =
1 2
(cid:110) (cid:111)
TR2R3[ω1]TR2R3[ω2] TTRR12RR33[[ωω22]] − TTRR12RR33[[ωω11]] .
From Eq. (16) and the subsequent discussion we can
draw a number of conclusions: Since the linear term
in the ∆T expansion is zero we immediately conclude
that in the linear-response regime or in the classical
limit our system has zero rectification parameter, i.e.
R = 0,and therefore one needs to consider the quantum
regime beyond linear response. Second, we realize that
since 1−a2 > 0 and F (T ) > 0, the sign of ∆I (and
αR3
thus the sign of the rectification parameter R) is com-
pletelydeterminedbythesignofK,seeEq. (18). Below
we will be confirming these predictions for a simplified
version of our model Eq. (4).
FIG. 1: (Color online) A schematic of the thermal rectifier
IV. AN EXAMPLE OF THERMAL due to rotation when Nb = 1: on each edge of the trian-
RECTIFICATION DUE TO CORIOLIS FORCE gle masses are coupled together with equal harmonic springs
up to the next-nearest neighbor. The masses are also at-
tached to a post with similar springs and they move on a
We will demonstrate the rectification effect, induced
platform at the X −Y plane which rotates with a counter-
by the Coriolis force (Ω (cid:54)= 0), using a simple version of
clockwise(CCW)angularvelocityΩ. a)ThemassR iscou-
1
the general model, Eq. (4). The underlying symmetric pledtoahigh-temperaturebathwithtemperatureT =T
R1 hot
harmonic lattice has an equilibrium configuration asso- while the mass R is coupled to the low-temperature bath
2
ciated with an equilateral triangle. Each of the three with temperature T = T . The mass R is connected
R2 cold 3
corners of the triangle R10, R20, R30 are occupied by equal to the probe bath with temperature TR3 = Tp. b) The
massesR1, R2, R3,coupledwiththesameharmoniccou- mass R1 is coupled to a low-temperature bath with temper-
pling kC to a post which is placed at the center O of ature TR1 = Tcold while the mass R2 is coupled to the high-
temperature bath with temperature T = T . The mass
the triangle. On every edge of the triangle, say for ex- R2 hot
R isagainconnectedtotheprobebathwithadifferenttem-
ample edge R0R0, (by excluding the masses at the ver- 3
1 2 perature T =T(cid:48).
texes) there are additional N equal masses, which are R3 p
b
connected with the center O of the triangle by springs
dividing the angle ∠R0OR0 equally. An equal harmonic
1 2
coupling up to the next-nearest neighbor is considered
between the masses on each edge of the triangle. Fur-
thermore, the masses R , R , R are attached to three
1 2 3
equivalent 1D Rubin baths [19]. The 1D Rubin baths
α = R ,R ,R are made up of a semi-infinite spring
1 2 3
chainwithKα =δ (2kα−Ω2+k )−kαδ . Here
nm nm 0 n±1,m
an additional coupling k with the substrate is assumed.
0
An illustration of our model with Nb = 1 is shown in ative displacement of the particles with respect to their
Fig. 1. equilibrium positions. The local Euclidean coordinate
The symmetric force matrix KC in Eq. (1) can be ob- system for each particle is specified in the following way:
tained systematically as follows. Each spring (harmonic its origin is taken as the equilibrium position of the par-
coupling) will contribute to the matrix KC with an ad- ticle and its x-axis and y-axis respectively coincide with
ditiveharmonicterm. Thesespringcontributionscanbe theradial(r)andtangentialdirection(θ)offixedequilib-
classified in three categories. First, we consider the cou- rium position of the specific particle. Specifically, using
pling between two masses R and R of the system. We this local Euclidean coordinate system, the contribution
j k
uselocalEuclideancoordinatesystemtodescribetherel- of the spring connecting the particles R ,R to the ma-
j k
5
trix KC is
[KC] = [KC] =[KC] =[KC] =k
jrj,jrj jθj,jθj krk,krk kθk,kθk C
[KC] = [KC] =−k cos(θ0−θ0)
jrj,krk krk,jrj C j k
[KC] = [KC] =−k sin(θ0−θ0)
jrj,kθk kθk,jrj C j k
[KC] = [KC] =k sin(θ0−θ0)
jθj,krk krk,jθj C j k
[KC] = [KC] =−k cos(θ0−θ0). (19)
jθj,kθk kθk,jθj C j k
where (cid:0)r0, θ0(cid:1) and (cid:0)r0, θ0(cid:1) is the polar coordinate for
j j k k
the equilibrium positions R0 and R0 of two masses R
j k j
and R . Second, we evaluate the contribution to the
k
matrix KC due to the spring coupling between the mass
R and the center O. We have that [KC] =
j jrj,jrj
[KC] = k . Similarly the coupling between the
jθj,jθj C
mass R and the platform provides the contribution
j
[KC] = [KC] = k . Third, due to the cou-
jrj,jrj jθj,jθj 0
pling with the bath α additional terms are added to
the matrix KC. For example, for α = R they are
1
[KC]1r1,1r1 =[KC]1θ1,1θ1 =kR1.
Finally the nonzero elements of the coupling matri-
ces (see below Eq. (4) for a definition) are respec-
tively (cid:2)VR1C(cid:3) = −kR1,(cid:2)VR2C(cid:3) = −kR2 and
(cid:2)VR3C(cid:3) =1,1−r1kR3. In the follow1in,2gr,2 kR1 = kR2 =
1,3r3
kR3 =k is assumed for simplicity.
FIG.2: (Coloronline)a)PlotsofrectificationRversusangu-
The thermal rectification effect is present when the
larvelocityΩfordifferentaveragetemperaturesT . b)Aplot
(structurally) symmetric system is rotating with an an- 0
ofthetransmissioncoefficientratio TR1R3 versusfrequencyω,
gular velocity Ω around the post situated at the center TR2R3
O. To be specific, first the bath coupled to the mass R withthefixedangularvelocityΩ=0.03. Theinsetshowsthis
1
ratio from ω = 0.45 to 0.75. Other parameters are N = 4,
is set to the higher temperature T = T = T +∆T b
R1 hot 0 k =k=1 and ∆T =2.5%T .
and the bath coupled to the mass R is set to the lower C 0
2
temperature T = T = T −∆T (forward process).
R2 cold 0
The temperature T of the probe bath coupled to the
p
mass R is determined by the zero flux condition as dis-
3
cussed above. We then calculate the forward thermal
current I using Eq. (7). Second, by reversing the tem-
f
perature bias, i.e., T = T and T = T , and
R2 hot R1 cold
adjusting the temperature of the probe bath R to be
3
T(cid:48), the thermal current I for the backward process is
p b
evaluated. In the following natural units are assumed,
i.e. (cid:126)=1, m=1, k =1, k =1.
B
OurnumericalresultsfortherectificationparameterR
versus the angular velocity Ω and different average tem- FIG. 3: (Color online) A density plot of the weight function
peratures T0 are shown in Fig. 2a. These data clearly W(ω1,ω2,T0)versusω1 andω2 fortwodifferentaveragetem-
demonstrate that the rectification effect is present once peratures a) T0 = 0.1 and b)T0 = 0.4. Clearly, when the
average temperature T increases, the dominant part of the
Ω (cid:54)=0. Figure 2a confirms that for high average temper- 0
magnitude of the weight- function |W(ω ,ω ,T )| moves to
atures T , corresponding to the classical limit, the recti- 1 2 0
0 the higher frequency.
fication effect is suppressed (compare e.g. the curves for
T = 0.1 and T = 3) as we have predicted in subsec-
0 0
tion IIIC.
function |W (ω ,ω ,T )| is large will shifts towards high
From Fig. 2a we further see that the rectification pa- 1 2 0
frequencies. Bearing this fact in mind, we are now able
rameter R for a fixed angular velocity may change sign
to explain the change of sign in the rectification. For
when the average temperature T changes from 0.1 to
0 example, for Ω=0.03, we find that the transmission ra-
0.4. This can be understood from the perturbation re-
sult in Eq. (16). Figure 3 demonstrates that, when the tio TTRR12RR33 initially decreases as a function of frequency
average temperature T changes from 0.1 to 0.4, the fre- while for larger frequencies it increases, see Fig. 2 b). As
0
quency region for which the absolute value of the weight a result K (see Eq. (18)) changes sign from negative to
6
Finally, a 3D plot for the rectification R versus k
C
and the angular velocity Ω is shown in Fig. 4. In these
calculationsthespringconstantthatcouplestheparticles
R ,R ,R with the heat baths is taken to be k =1. We
1 2 3
seethatthemaximumrectificationofR =1.6×10−3
max
isobservedfork ≈0.65andΩ≈±0.02neartheorigin.
C
Thenon-monotonicbehaviourofRversusk isassociated
c
with an impedance missmatch between the two chains
associatedwiththecentraljuntionlattice(withcoupling
constant k ) and the one-dimensional chain of the bath
c
(which involves coupling constants k =1).
FIG. 4: (Color online) A 3D plot of rectification R versus
angular velocity Ω and spring constant k in the center lat- V. CONCLUSIONS
C
tice. Other parameter are N = 4, T = 0.4, k = 1 and
b 0
∆T =2.5%T . At k ≈0.65 and Ω≈±0.02 the rectification
0 c We have proposed the use of the Coriolis force in or-
gets its maximum value.
dertobreaktime-reversalsymmetryandinducethermal
rectification in a ballistic three terminal nano-junction.
Two of these terminals are attached to a hot and a cold
positive so that the rectification parameter R Eq. (16)
reservoir, respectively, while the third one is attached to
changes also sign from positive to negative.
a probe reservoir whose temperature is self-consistently
Furthermore, we see that, the rectification direction is
adjusted such that the net current towards this reser-
controlled by the sign of the angular velocity Ω, thus al-
voir is zero. The nano-junction consists of a symmetric
lowingagreaterflexibilitytoreconfigure“onthefly”the
lattice which is placed on a rotating platform. Using
direction of the heat current. This can be understood
from the equality TR1R3[ω,Ω]=1/(TR1R3[ω,−Ω]) which non-equilibrium Green’s function formalism we have cal-
TR2R3 TR2R3 culated the rectification effect up to second-order in the
is directly derived from Eq. (13) and Eq. (14). Specifi-
temperature difference between the two reservoirs. We
cally, due to this equality, in the frequency region where
conclude that the Coriolis-induced rectification effect is
|W(ω ,ω ,T )| is large (for a fixed average temperature
1 2 0 ofquantummechanicalnatureanditsdirectionandmag-
T0), the behaviour of TTRR12RR33 as a function of ω will turn nitude depend on Ω. It will be interesting to extend this
from decreasing (increasing) function to increasing (de- study beyond the ballistic limit and investigate the ef-
creasing) function when Ω → −Ω. Thus the function K fects of non-linearity and disorder. The latter phenom-
(defined in Eq. (18)) and correspondingly the rectifica- ena can be important to any realistic structure where
tion parameter R will change sign when we reverse the phonon-phononinteractionsandstructuralimprefections
angular velocity Ω. are generally also present.
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[2] A. Dhar, Adv. Phys. 57, 457 (2008). 020101(R) (2015).
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