Table Of ContentSpintronics-based mesoscopic heat engine
J. Atalaya and L. Y. Gorelik1
1Department of Applied Physics, Chalmers University of Technology, G¨oteborg Sweden, SE-412 96
Weconsiderananowiresuspendedbetweentwospin-polarizedleadsandsubjecttoanonuniform
magnetic field. We show that a temperature drop between the leads can significantly affect the
nanowire dynamics. In particular, it is demonstrated that under certain conditions the stationary
distribution of the mechanical subsystem has Boltzmann form with an effective temperature which
is smaller than the temperature of the ”cold” lead; this seems rather counter-intuitive. We also
find that a change of the direction of the temperature gradient results in generation of mechanical
vibrations rather than heating of the mechanical subsystem.
2
1
PACSnumbers: 73.63.–b,73.23.Hk,75.76.+j,85.85.+j
0
2
n Nanomechanical resonators are devices which are be-
(a) Magnetic STM (b)
a ing employed not only to develop new technological ap- M
J
1 plications, such as ultra-sensitive sensors [1–4], but also Z M z
to shed light on fundamental questions such as the tran- B
1 sition from the classical to the quantum mechanical de- Y d x
] scription of macroscopic objects [5]. Investigation of a u
l A
l system where a mechanical degree of freedom controls
a T T
thepropertiesofamesoscopicjunctionbetweentwobulk L R
h
- leads is an important line of research in nanomechan-
s ics[6–8]. Insuchstructures, themechanicalpartmaybe (c) Nanotube
e
m consideredasananoenginewhoseoperationiscontrolled ε ε
bythestatesoftwobulkthermodynamicreservoirs. Itis ε0 Δ
. ε
at well known that a macroscopic mechanical system may Vibration quanta ħω F
m be driven into cyclic motion if it is coupled to reservoirs
- held at different temperatures. A Stirling engine, oper- fL(ε) TL TR fR(ε)
d atingbycycliccompressionandexpansionofairorother
Left-lead Right-lead
n gas and placed between hot and cold spaces, is one ex-
o
ample.
c FIG. 1: (a) A nanotube suspended between two spin-
[ By decreasing the size of the heat engine to the polarized leads and in proximity to a magnetic STM tip
nanoscale level, quantum mesoscopic effects come into with magnetization M. The leads have opposite polariza-
2
v playanddeterminethebehaviorofboththeworkingsub- tions along the z-direction. (b) Nonuniform magnetic field,
2 stance and the mechanical subsystem. This opens new B, created by the magnetic tip. The nanotube (circles) de-
7 possibilities for the operation of a heat engine; for in- flection,u,isinthex-direction. (c)Adoublyspin-degenerate
8 stance, reduction or even suppression of the mechanical electronic level exists in the nanotube at energy (cid:15)0. The ap-
3 fluctuations—effective cooling of the mechanical subsys- plied magnetic field splits this level into two levels σz =↑,↓
2. tem. Recently, it was shown that suppression, leading to separated by an energy ∆ ∝ BzµB. The leads are held at
different temperatures T with a zero bias voltage.
1 ground state cooling, may be achieved if reservoirs, pre- L,R
1
sented by normal or superconducting metal leads, are
1 held at different electrochemical potentials [9–12]. It mentally[15])andsubjecttoanonuniformmagneticfield
:
v was also demonstrated that a temperature drop between B (cf. Fig. 1b). The field can be generated, for example,
Xi reservoirs can also generate this effect if one assumes a byamagneticSTMtipintheformofawedgewhichisin
very special ”three particles” interaction inside a junc- proximity to the nanotube (cf. Fig. 1a). If the nanotube
r
a tion[13,14]. Inthisletterweinvestigateheating,pump- is not too long (length L (cid:46) 1µm), it can be considered
ingandcoolingofamechanicalmodeinarealisticnano- asaquantumdotcoupledtotheelectrodesthroughtun-
junctionwhereonlya”twoparticle”interactionbetween neljunctions. Weassumethatinsidethenanotubethere
mechanicaldegreeoffreedomandworkingsubsystemex- is only one doubly degenerate, with respect to spin, and
ists. We show that a temperature drop between linked spatially quantized electronic level which participates in
leadscangeneratecoolingorexcitation,dependingonits electron exchange with the ferromagnetic leads.
direction, of the mechanical subsystem. If the nanotube is straight and positioned below the
To be specific, we consider a carbon nanotube sus- STM sharp end, as shown in Fig. 1b, the applied mag-
pended between La Sr MnO ferromagnets of oppo- netic field is directed towards the nanotube (z-direction)
1−x x 3
site polarizations (a structure recently realized experi- and Zeeman-splits the degenerate electronic level of the
2
nanotube. A two-level system (TLS) is thus formed in describes the TLS with energy levels (cid:15) = (cid:15) ± ∆/2,
σ 0
the latter with levels σ =↑,↓ and energies (cid:15) , respec- where ∆ is the Zeeman-splitting energy proportional to
z ↑,↓
tively. Deflection of the suspended part of the nanotube the z-component of the applied magnetic field B , and
z
in the x-direction generates interlevel transitions (spin- (cid:15) is the zero-field energy, which is measured relative to
0
flip)intheTLS.Consequently,thenonuniformmagnetic the leads Fermi energy. The last term in Eq. (2) de-
fieldinducescouplingbetweenmechanicalandelectronic scribes the spintronic coupling. It is proportional to the
√
subsystems. In what follows, we will refer to this mech- oscillator displacement, uˆ = (ˆb† + ˆb)/ 2, and to the
anism of interaction between mechanical and electronic spin-flip operator. The coupling parameter g is equal
subsystem as spintronic coupling. to Cµ x ∂ B (0) where µ is the Bohr magneton, x is
B 0 x x B 0
Weassumethattheleftandrightleadsarecompletely thezero-pointvibrationalamplitude,∂ B (0)isthefield
x x
polarizedalongthez-direction(i.e.,thedensityofstates gradient along the x-direction, and C is a numerical fac-
are ν↑(↓) = 0 and ν↓(↑) ≡ ν > 0) and the energy tor ∼A/L, where L is the nanowire length and A is the
R(L) R(L) R(L)
length of the STM wedge. This factor accounts for the
difference between levels of the TLS is ∆ ≡ (cid:15) −(cid:15) >
↑ ↓
electronic state inside the nanotube being extended over
0. If there is no spintronic coupling and the intrinsic
thewholelengthofthenanotubewhilethemagneticfield
relaxation time τ of the TLS is larger than the dwell-
is concentrated only in the region below the STM. The
timeofanelectroninthenanotube, thentheoccupation
field gradient ∂ B (0) induced by a Fe-based magnetic
number of the spin up (down) state inside nanotube is x x
(cid:0) (cid:1) tip7.5nmthickwithmagnetizationM =1.75 ·106 A/m
n (cid:39) f ((cid:15) −(cid:15) )/T ≡ f , where f
↑(↓) F ↑(↓) FL(R) L(R) L(R) F
at a distance 7.5 nm is of the order of 35 mT·nm−1. For
is the Fermi-distribution function, (cid:15) and T are
FL(R) L(R)
this value of field gradient and for a nanotube with vi-
theFermienergyandtemperatureoftheleft(right)lead,
brational frequency ω = 2π ·100 MHz, mass 1 ag and
respectively.
A/L=0.1, g is of the order of 2π·106 Hz. The term H
The spintronic coupling generates spin-flip transitions l
inEq.(3)describestheleftandrightleads. ThetermH
between the energy levels of the TLS. These transitions t
in Eq. (4) describes the tunneling of electrons from the
are inevitably accompanied by absorption or emission of
nanowire to the leads, and v.v., and t are tunneling
mechanical quanta. The cooling (pumping) process of L(R)
amplitudes [16].
the mechanical subsystem is the result of an electron
transition from the lower (upper) energy level to the up-
To analyze the performance of the system, we start
per (lower) energy level of the TLS. In order to effec-
from the Liouville-von Neumann equation for the total
tively cool (pump) the mechanical mode, it is required
densityoperator(cid:37)ˆandtheneliminatetheleadelectronic
that n (cid:28)n (n (cid:29)n ). This can be realized by apply-
↑ ↓ ↑ ↓ degrees of freedom [17]. If the temperatures of the leads
ing a bias voltage or by subjecting the leads to different are much greater than (cid:126)ω/k , the Fermi-distributions
B
temperatures. Below, we consider cooling and pumping
f ((cid:15)) are smooth functions within the energy interval
L,R
of the mechanical subsystem generated only by a tem- (cid:126)ω. As a result one gets the following Lindblad master
perature gradient between the leads.
equation(5)forthereduceddensitymatrixρ=Tr (cid:37)ˆ.
R+L
To perform a quantitative analysis of the system de-
The latter describes the mechanical degree of freedom
scribed above, we consider the Hamiltonian
and the electronic state of the TLS of the nanotube.
H =H +H +H , (1)
nw l t
i(cid:2) (cid:3)
∂ ρ=− H ,ρ +L [ρ]+L [ρ], (5)
H = (cid:126)ωˆb†ˆb+ (cid:88) (cid:15) nˆ +guˆ(dˆ†dˆ +dˆ†dˆ ), (2) t (cid:126) nw L R
nw σ σ ↑ ↓ ↓ ↑
σ=↑,↓
H = (cid:88)(cid:15) (k)aˆ† aˆ +(cid:15) (k)aˆ† aˆ , (3) where
l L k,↑,L k,↑,L R k,↓,R k,↓,R
k
Ht = (cid:88)tLaˆ†k,↑,Ldˆ↑+tRaˆ†k,↓,Rdˆ↓+h.c., (4) Lα[ρ] = Γα[(1−fα)dˆσαρdˆ†σα +fα(dˆ†σαρdˆσαρ)−
k
−(1/2−f ){nˆ ,ρ}]. (6)
α σk
whereaˆ (aˆ† )anddˆ (dˆ†)beingtheannihila-
k,σ,L(R) k,σ,L(R) σ σ
tion (creation) operators for electrons in the left (right)
Here, α = (L,R), σ =↑ (↓) and Γ = 2π|t |2ν /(cid:126)
leads and in the nanotube, respectively, and nˆ =dˆ†dˆ . L(R) α α α
σ σ σ are the tunneling rates, ν is the density of states, and
ThefirstterminEq.(2)describesthenanotubemechan- α
{Aˆ,Bˆ} denotes anticommutator. The collision integrals
ical degrees of freedom, which we restrict to the funda-
L [ρ] describe the decoherence in the electronic subsys-
mental flexural mode. This mode is described as a sim- α
tem induced by the bulk electronic reservoirs.
ple harmonic oscillator with vibrational frequency ω and
ˆb(ˆb†)beingtheannihilation(creation)operatorforanele- If the resonant condition ∆ = (cid:126)ω is fulfilled and ω (cid:29)
mentary excitation(vibron). Thesecondterm inEq.(2) g/(cid:126),Γ ,therotating-wave-approximationcanbeusedto
L
3
25 0.04
1 -Log(Pm(n)/Pm(0)) 0.015 C
Teff
3 B
P (n) 1 A
m
2
T 0
L 0 2 0 n 300
0 n 30
3
T
0 L Θ* 1 5 10 00 n 300
FIG. 2: Effective temperature Teff of the vibrational mode FIG. 3: Vibron stationary distribution P (n) when Θ<Θ∗.
m
as function of the ratio Θ = T /T . For Θ > 1: Teff < WeincludeabosonicbathwithtemperatureT andcoupling
R L b
min{T ,T }. For Θ∗ < Θ < 1: Teff > max{T ,T }. For parameter γ. The stationary distribution exhibits a peak at
L R L R
Θ<Θ∗: nostationarydistributionexitsunlessadditionaldis- n , which increases as γ gets smaller. We use n γ(cid:126)/g (cid:29)
max b
sipation mechanisms are included. Inset: Vibron stationary 1 (curve 1), n γ(cid:126)/g = 0.021 (curve 2) and n γ(cid:126)/g = 0.01
b b
distributioninlogarithmicscalewhenthemechanicalsubsys- (curve3). Inset: ThewidthofthedistributionP (n)scales
m
tem interacts only with ferromagnetic leads (curve 1), only inversely with T . T (A) = 1.5 K, T (B) = 0.3 K, T (C) =
b b b b
with a bosonic bath at temperature T and coupling param- 0.06 K and n γ(cid:126)/g=0.01 for all curves A, B and C.
b b
eter γ (curve 2) and, both the fermionic and bosonic baths
(curve 3). We use ω/2π = 100 MHz, T = 0.02 K, Θ = 10,
√L
T =(T +T )/2, (cid:15) =k T /2, Γ = 2g/(cid:126), Γ =1.62g/(cid:126) temperature is given by
b L R 0 B R L R
and γ =0.002g/(cid:126).
Teff =T (cid:126)ω(cid:104)1−Θ−1+ (cid:126)ω(cid:0)1+Θ−1(cid:1)(cid:105)−1, (9)
L (cid:15) 2(cid:15)
obtain the following rate equations 0 0
where Θ = T /T . This solution has physical mean-
P˙m(n) = g˜(cid:0)P↑i↓(n)−P↑i↓(n+1)(cid:1), (7) ing only if TeRff >L 0. Negative effective temperature
P˙ (n) = −g˜Pi (n+1)−Γ f P (n)+Γ f P (n) indicates that there is permanent energy pump into the
↑ ↑↓ R R ↑ L L 0
mechanical subsystem. In this case in order to achieve a
−Γ (1−f )P (n)+Γ (1−f )P (n),
L L ↑ R R 2 stationary regime one has to introduce additional exter-
P˙↓(n) = g˜P↑i↓(n)+ΓRfRP0(n)−ΓLfLP↓(n) nal sources of dissipation.
+ΓL(1−fL)P2(n)−ΓR(1−fR)P↓(n), From Eq. (9) one can see that for (cid:15)↓ > 0 and TR >
T , the vibrational mode is effectively cooled to a final
P˙ (n) = −Γ (1−f )P (n)−Γ (1−f )P (n) L
2 L L 2 R R 2 temperature which is smaller than the temperature of
+ΓLfLP↓(n)+ΓRfRP↑(n), the ”cold” lead (Teff < min{T ,T } = T )—cooling
L R L
2P˙i (n) = g˜n(cid:0)P (n−1)−P (n)(cid:1)−(Γ +Γ )Pi (n), regime. Fromthisequationalsofollowsthatintheinter-
↑↓ ↑ ↓ L R ↑↓ val1>Θ>Θ∗ ≡(cid:0)1−(cid:126)ω/2(cid:15) (cid:1)/(cid:0)1+(cid:126)ω/2(cid:15) (cid:1)theeffective
0 0
√
whereg˜=g 2/(cid:126). HereP (n),P (n),P (n)arethejoint temperatureisgreaterthanthetemperatureofthe”hot”
σ 0 2 lead (Teff >max{T ,T }=T )—heating regime.
probabilities to find a vibrational mode in a Fock state L R L
Note that whena temperature difference is used tore-
withnvibronicquantaandoneelectrononthenanotube
duce mechanical fluctuations of the nanotube, the mini-
with spin σ, empty nanotube and two electrons on the
mum effective temperature of the mechanical subsystem
nanotube, respectively. Therefore, the total probability
is always greater than Teff = 2T /(1 + 2(cid:15) /(cid:126)ω), ap-
to find n vibronic quanta is Pm(n) ≡ P0(n)+P↑(n)+ min L 0
proaching this value as Θ → ∞. As a result, the aver-
P (n)+P (n). The off-diagonal elements of the density
↓ 2 (cid:80)
matrix Pi (n) = Im(cid:104)0|dˆ b(n−1)ρ(ˆb†)ndˆ†|0(cid:105)/(n−1)! de- age vibron number (cid:104)n(cid:105)= nnPm(n)∼exp(−(cid:15)0/kBTL)
↑↓ ↑ ↓ is finite but exponentially small. This is a consequence
scribe the quantum entanglement between the electronic
of the finite temperature of the ”cold” lead which leads
and mechanical subsystems generated by the correlation
to infrequent pumping processes and does not allow to
betweenspin-flipofelectronandachangeinthenumber
achieve absolute ground state cooling.
of vibronic quanta.
From Eq. (7) with T and T such that Teff < 0
L R
Equation (7) always has a stationary solution where
(Θ < Θ∗), it follows that d(cid:104)n(cid:105)/dt > 0 and the aver-
Pst(n) has a Boltzmann form
m age number of vibrons increases with time. However,
in a real physical situation the mechanical subsystem
Pst(n)=Z−1exp(cid:0)−(cid:126)ωn/k Teff(cid:1), (8)
m B is also coupled to the phononic thermal baths of the
leads. Toaccountforthedissipationduetothiscoupling
where Z = (cid:2)1−exp(−(cid:126)ω/k Teff)(cid:3)−1 and the effective we add to the left side of Eq. (5) a Lindblad operator
B
4
γL [19]. Here, the temperature of the phononic bath is magnetic field may result in suppression or generation
γ
(cid:0) (cid:1)
T ∼ T +T /2 and γ =ω/Q is the coupling parame- of mechanical vibrations when the leads are held at dif-
b L R
tercharacterizedbytheresonatorqualityfactorQ,which ferent temperatures. In particular, it was demonstrated
can be very large Q(cid:39)105 at low temperatures [20]. We that under certain conditions the stationary distribution
properlymodifythesystemofequations(7),andsolveit of the mechanical subsystem has Boltzmann form with
numerically. The resulting shape of Pst.(n) is depicted effective temperature which is smaller than the temper-
m
in Fig. 3. From this figure one can see that at small n ature of the ”cold” lead. This counter-intuitive result
theprobabilitiesPst.(n)increasewithn. Thisisbecause is a consequence of the Fermionic nature of the ther-
m
for small n < Γ/g and γ < g, the pumping rate (∝ gn) malbathscoupledtomechanicalsubsystem. Noticethat
is larger than the dissipation rate (∝ γn). Then, the coupling to bosonic baths results in an effective temper-
pumping rate eventually saturates to Γ (cid:39) minΓ at ature of the vibrational mode equal to the mean value
L(R)
n=n ∼Γ/g while the the dissipation rate governed by of the baths temperature. Also changing direction of the
s
interaction with phonon reservoirs continue to increase temperaturegradientresultsingenerationofmechanical
linearly with n and finally overcomes the pumping rate. vibrations rather than heating of the mechanical subsys-
Asaresult,thedistributionfunctionreachesamaximum tem. Finally, for partial spin polarization in the leads,
at n = n ∼ Γ/γ and then exponentially decays for ground state cooling of the mechanical vibration can be
max
larger n(cid:29)n . achievedatrealisticphysicalparametersiftheleadshave
max
Finally, we discuss conditions for ground state cooling (cid:38)50% spin polarization.
of the mechanical subsystem; i.e., a stationary regime
This work was supported in part by the Swedish VR
with a final vibron number (cid:104)n(cid:105) (cid:46) 1. Thus far, we have
and SSF and by the EC project QNEMS (FP7-ICT-
considered the situation of completely polarized leads.
233952). We also acknowledge Dr. Andreas Isacsson for
However, in order to analyze the conditions for ground
helpful discussions.
statecooling,itisnecessarytoestimatetheeffectofpar-
tial spin-polarization in the leads. To quantify the de-
gree of polarizability of the right lead, we introduce a
parameter η = 1−ν↓(↑) /ν↑(↓) and find that, for a
L(R) L(R) L(R)
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5
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