Nonequilibrium electron cooling by NIS tunnel junctions

1 MB·English

by  A. S. Vasenko

JLowTempPhysmanuscriptNo.
(willbeinsertedbytheeditor)
9 A.S.Vasenko F.W.J.Hekking
·
0
0 Nonequilibrium Electron Cooling by NIS
2
Tunnel Junctions

n
a
J

4
1

]
n Received:date/Accepted:date
o
c
-
r Abstract We discuss the theoretical framework to describe quasiparticle elec-
p
tric and heat currents in NIS tunnel junctions in the dirty limit.The approach is
u
based on quasiclassical Keldysh-Usadel equations. We apply this theory to dif-
s
. fusive NISS tunnel junctions. Here N and Sare respectivelya normal and a su-
t ′
a perconductingreservoirs,IisaninsulatorlayerandS isanonequilibriumsuper-
′
m
conductinglead.Wecalculatethequasiparticleelectricandheatcurrentsinsuch
- structures and consider the effect of inelastic relaxation in the S′ lead. We find
d
thatintheabsenceofstrongrelaxationtheelectriccurrentandthecoolingpower
n for voltages eV < D are suppressed. The value of this suppression scales with
o
thediffusivetransparencyparameter.Weascribethissuppressiontotheeffectof
c
[ backtunnelingofnonequilibriumquasiparticlesintothenormalmetal.

2 Keywords Electroncooling NIStunneljunctions
v ·
1 PACS 74.50.+r 74.45.+c 74.40.+k 74.25.Fy
· · ·
4
7
5
1 Introduction
.
0
1
It is well known that inNIS (Normal metal- Insulator - Superconductor) tunnel
8
junctionstheflowofelectriccurrentcarriedbyquasiparticlesisaccompaniedby
0
aheattransferfromthenormalmetalintothesuperconductor[1,2].Thishappens
:
v duetothepresenceofthesuperconductingenergygapD ,whichinducesselective
i
X tunneling of high-energy quasiparticles out of the normal metal. In the tunnel-
ing event only quasiparticles with energy E >D (compared to the Fermi level)
r
a cantunneloutofthenormalmetal.Theygeneratethesingleparticlecurrentand
A.S.Vasenko F.W.J.Hekking
LPMMC,Univ·ersite´JosephFourierandCNRS,25AvenuedesMartyrs,
BP166,38042Grenoble,France
E-mail:[email protected]
E-mail:[email protected]
2
associatedheatcurrent.AtlowerenergiesE<D chargetransferoccursviamech-
anism of Andreev reflection[3, 4]. Andreev current I generatesa Joule heating
A
I V thatisdepositedinthenormalmetalelectrodebutthiseffectdominatesover
A
single particle cooling only at very low temperatures [5]. For the temperatures
consideredinthispaperAndreevJouleheatingisnegligible.
Ithasbeenshownthatforvoltage-biasedNIStunneljunctionsheatcurrentout
ofthenormalmetal(alsoreferredas“coolingpower”)ispositivewheneV .D ,
i.e., it cools the normal metal [6]. For eV &D the current through the junction
increasesstrongly,resultinginJouleheatingIV andmakingtheheatcurrentneg-
ative.ThecoolingpowerismaximalneareV D .
≈
This effect enables the refrigeration of electrons in the normal metal. A mi-
crorefrigerator,basedonanNIStunneljunction,hasbeenfirstfabricatedbyNa-
humetal.[1].TheyhaveusedasingleNIStunneljunctioninordertocoolasmall
normal metalstrip. LaterLeivo et al. [6] have noticedthat the cooling power of
an NIS junction is an even function of an appliedvoltage, and have fabricateda
refrigeratorwithtwoNIStunneljunctionsarrangedinasymmetricconfiguration
(SINIS), which gives a reduction of the electronic temperature from 300 mK to
about 100 mK. This significant temperature reduction gives a perspective to use
NISjunctionsforon-chipcoolingofnano-sizedsystemslikehigh-sensitivityde-
tectorsandquantumdevices[7].
To enhance the performance of the NIS refrigerator it is important to under-
standtheroleofpossiblefactorsthatmayfacilitateordecreasethecoolingeffect.
Oneofsuchfactorsistheinelasticrelaxationofinjectedquasiparticlesinthesu-
perconducting lead. In non-reservoir geometries the quasiparticles injected into
thesuperconductingleadgenerateanonequilibriumdistribution.Inadiffusivesu-
perconductorbackscatteringonimpuritiesandsubsequentbacktunnelingintonor-
mal metalmay considerably reduce the net heat current out of the normal metal
electrode.
The purpose of this work is to investigatethe importance of the nonequilib-
rium quasiparticle distribution and consider the effect of inelastic relaxation in
the superconductor. Possible mechanisms of inelastic relaxation could be usual
processes,suchas electron-electronor electron-phononinteractions,but alsothe
presenceofsocalled“quasiparticletraps”,i.e.additionalnormalmetalelectrodes
connectedtothesuperconductor,whichremoveexcitednonequilibriumquasipar-
ticles from the superconducting lead [8, 9]. In this paper we do not specify the
mechanisms of inelastic relaxation and consider the relaxation time approxima-
tion approach. Effect of the nonequilibrium quasiparticle injection in NIS junc-
tionswas alsodiscussedinRef.[10],where theauthors proposed aphenomeno-
logicalmodelofquasiparticlediffusioninthesuperconductinglead.
Thepaperisorganizedasfollows.Inthenextsection,weformulatethetheo-
reticalmodelandbasicequations.Insections3and4wesolvethekineticequa-
tions andapply solutions for thecalculationof theelectricandheatcurrents,re-
spectively.InSec.5wecalculatequasiparticledistributionfunctionsinthesuper-
conductinglead.FinallywesummarizetheresultsinSec.6.
3
N S’ S
0 L x
Fig.1 Geometryoftheconsideredsystem.
2 Modelandbasicequations
ThemodeloftheN-I-S-SjunctionunderconsiderationisdepictedinFig.1and
′
consistsofavoltage-biasednormalmetalreservoir(N),aninsulatorlayer(I),asu-
perconductinglayer(S)ofthicknessLandasuperconductingreservoir(S)along
′
the x direction. The S and S leads are made from the identicalsuperconducting
′
material.We assume the S-S interface to be fully transparent. We will consider
′
the diffusive limit, in which the elastic scattering length ℓ is much smaller than
thecoherencelengthx = D/2D ,whereD isthe diffusioncoefficient(we as-
0
sumeh¯ =k =1).ThelengthLoftheS leadisassumedtobemuchlargerthan
B ′
x . The problem of currentpflow through diffusive N-I-S-S structures with short
0 ′
S superconductorleadwassolvedinRef.[11].
′
Undertheconditionsdescribedabove,thecalculationoftheelectricandheat
currentsrequiressolutionoftheone-dimensionalKeldysh-Usadelequations[13]
(seealsoreview[14])forthe4 4matrixKeldysh-GreenfunctionGˇ(x,E)inthe
×
S lead,
′
sˇ E+Dˇ, Gˇ =iD¶ Jˇ, Jˇ=Gˇ¶ Gˇ, Gˇ2=1ˇ, (1)
z
gˆR GˆK
G(cid:2)ˇ = (cid:3) , GˆK =gˆRfˆ fˆgˆA. (2)
0 gˆA −
(cid:18) (cid:19)
Here
s 0 Dˆ 0
sˇz= 0z s z , Dˇ = 0 Dˆ ,
(cid:18) (cid:19) (cid:18) (cid:19)
gˆR,A are the 2 2 Nambu matrix retarded and advanced Green’s functions, fˆ=
f +s f isth×ematrixdistributionfunction(weuse“check”for4 4and“hat”
+ z
for2 2−matrices),s y,zarethePaulimatricesintheNambuspace,Dˆ×=eis zc is yD ,
D and×c are the modulus and the phase of the pair potential, and ¶ ¶ /¶ x. In
≡
Eqs. (1) we neglectthe inelasticcollisionterm which will be takeninto account
later.Since we are interestedinsmallvoltages eV .D (when the cooling of the
normalmetaloccurs)wecanneglecttheeffectofthesuppressionofthesupercon-
ductinggapduetotheheatinginthesuperconductor.
The boundary conditions for the function Gˇ and the matrix current Jˇat the
leftnormal (x= 0) and the right superconducting (x=+0)sides of the tunnel
−
barrieraregivenbytherelation[15]
(s /g )Jˇ =Jˇ =(W/x ) Gˇ ,Gˇ , (3)
N N 0 +0 0 0 +0
− −
(cid:2) (cid:3)
4
where s and g are the normal conductivities of the N and S leads per unit
N N ′
length,respectively.InEq.(3),thetransparencyparameterW isdefinedas
W =R(x )/2R=(3x /4ℓ)G G , (4)
0 0
≫
whereR(x )=x /g isthenormalresistanceoftheS leadperlengthx andRis
0 0 N ′ 0
thejunctionresistance.IthasbeenshowninRefs.[16,17]thatthisquantityrather
thanthebarriertransparencyG plays the role ofa transparencyparameterinthe
theory of diffusive tunnel junctions (seealsothe discussioninRef. [18]). In this
paper, we will considerthe limitW 1, which corresponds tothe conventional
≪
tunneling concept. At the right transparent S-S interface all functions and their
′
firstderivativesaretobecontinuous.Weneglectpossiblesmallresistanceatthis
interface [12] since it is much smaller than the resistance of the S lead in the
′
normalstate.
The electric and heat currents are related to the Keldysh component of the
matrixcurrentJˇrespectivelyas[13,19,20,21,22],
g ¥
I= N Trs JˆKdE, (5)
z
−4e 0
Zg ¥
P=IV+ N ETrJˆKdE, (6)
4e2
0
Z
andthustheycanbeexpressedthroughtheboundaryvalueJˇ inEq.(3),
+0
1 ¥
I= Trs Gˇ ,Gˇ KdE, (7)
z 0 +0
−8eR 0 −
Z
P=IV+ 1 ¥ E(cid:2)Tr Gˇ ,G(cid:3)ˇ KdE. (8)
8e2R 0 −0 +0
Z
(cid:2) (cid:3)
For further consideration it is convenient to write the Green function in the
followingstandardway,
gˆ=s u+is v. (9)
z y
HereweneglectthephaseoftheanomalousGreenfunctionsinceitgivescorrec-
tions to the next order in diffusive barrier transparency parameterW. The func-
tions u and v determine the spectral characteristics of the system. In particular,
the quantity N(E)= uR uA /2 is the density of states (DOS) normalized to
−
its value N in the normal state. In what follows, we will express the advanced
F
Greenfunctionsthrou(cid:0)ghthere(cid:1)tardedones,(u,v)A= (u,v)R ,usingthegeneral
∗
relationgˆA= s gˆR†s ,andomitthesuperscriptR,c−onsideringretardedGreen’s
z z
−
functionsonly.
Inthispaperwewillcalculatethesingleparticlecurrent,thereforeweconsider
quasiparticle energy E >D from now on. We neglect the proximity effect since
proximity corrections to the spectral functions are of the order ofW. Therefore,
weusetheBCSdensityofstatesinbothS andSlayers.Intheleftvoltage-biased
′
normalmetalreservoir(N)wehave,
1 E+eV E eV
gˆN =s z, f N = tanh tanh − , (10)
± 2(cid:20) (cid:18) 2TN (cid:19)± (cid:18) 2TN (cid:19)(cid:21)
5
whereT isthetemperatureofthenormalmetalreservoir.Weassumethevoltage
N
V to be directly applied to the tunnel barrier and neglect a small electric field
( eVW)penetratingtheS superconductinglead.
′
∼
In the right superconducting reservoir (S) Green’s and distribution functions
aregivenbytherelations
(E,D )
gˆ =s u +is v , (u ,v )= , (11)
S z S y S S S (E+i0)2 D 2
−
E
f f =tanh , f p=0, (12)
+S x>L≡ eq (cid:18)2TS(cid:19) −S x>L
(cid:12) (cid:12)
whereT isthet(cid:12)emperatureoftheSreservoir. (cid:12)
S
In the S layer, Green’s function is given by Eq. (11) and distribution func-
′
tions f (x,E)shouldbefoundfromthekineticequations,whichfollowfromthe
S
Keldys±hcomponentofEqs.(1)andforE>0haveasimpleform¶ 2f =0within
S
our approximations. These equations have no bound solutions: both±distribution
functions f (x,E) grow linearlywith x far from the junction. Such a growth is
S
limitedinp±racticebyinelasticcollisions,whichprovide thespatialrelaxationof
f (x,E) to the equilibrium values at x l x , where l =√Dt are the
S 0
in±elasticscatteringlengths and t are the∼in±ela≫sticscattering±times. To±simplify
theproblem,insteadofincluding±complicatedinelasticcollisionintegrals,weadd
collisiontermsintherelaxationtimeapproximationtothekineticequations,
l2¶ 2f =(f f )N(E), (13)
+ +S +S− eq
l2¶ 2f = f /N(E), (14)
S S
− − −
whereN(E)=Re(u )istheBCSDOS.
S
WeshouldsupplementEqs.(13)-(14)withproperboundaryconditionsonboth
leftandrightinterfaces.Onthetunnelbarrier(x=0)theyfollowfromtheKeldysh
componentofEq.(3),
N(E)
¶ f = (f f ), (15)
x +S x=0 g R +S0− +N
N
(cid:12) 1
¶ f (cid:12) = (f f ), (16)
x −S x=0 gNRN(E) −S0− −N
(cid:12)
where f (E)aretheboun(cid:12)dary valuesof f (x,E)atx=0.Ontheright trans-
S0 S
parentin±terfacedistributionfunctionsbecom±eequilibriumfunctionsoftherightS
reservoir,
f = f , f =0. (17)
+S x=L eq −S x=L
(cid:12) (cid:12)
(cid:12) (cid:12)
3 Singleparticlecurrent
The equation for the electric current follows from Eqs. (7), (16) and for single
particlecurrentreads,
1 ¥
I= N(E)(f f )dE. (18)
N S0
eR D − − −
Z
6

e/
R
I
1

0
0 1 eV/ 2
Fig.2 (Coloronline)IVcharacteristicoftheNISSjunction.T =T =0.3T ,W =0.01.Solid
′ N S C
blacklineiscalculatedbyuseofEq.(22).OtherlinesarecalculatedbyuseofEq.(21),L=
50x0:l =10x0 (redline);l =20x0 (greenline);l =50x0 (blueline);l =100x0 (dashed
blackli−ne). − − −
Toobtain f we shouldsolvetheboundary problem (14),(16)and(17).Doing
S0
thisweget−thefollowingresult,
a
f−S0= f−Na + N(E)co−th(b / N(E)), (19)
− −
a =2Wl /x =R(l )/R, b =L/l , (20)
0 p p
− − − − −
where R(l ) =l /g is the normal resistance of the superconducting lead per
N
lengthl .−FromE−q.(19)weseethatthenonequilibriumcorrectiontothecurrent
isofthe−orderofa W,whichjustifiesourassumptionaboutneglectingterms
oftheorderofW in−t≫hekineticequation.
SubstitutingEq.(19)intoEq.(18)wefinallyobtain,
1 ¥ coth(b / N(E))
I= eRZD N(E)f−Na / N(E)+−copth(b / N(E))dE. (21)
− −
For L= 0 or l =0 the boundary vpalue f in Eq. (1p8) is equal to zero and
S0
Eq.(21)reduce−stothewell-knownequation−forsingleparticlecurrentbetweenN
andSreservoirs,connectedthroughaninsulatinglayer,
1 +¥
I= N(E eV)[n (E eV) n (E)]dE, (22)
F F
eR ¥ − − −
Z−
wheren (E)=[1+exp(E/T)] 1isaFermifunction.
F −
IntheabsenceofinelasticrelaxationintheS lead(b 1)wecanapproxi-
′
mateEq.(21)by −≪
1 ¥ N2(E)
I= f dE, (23)
eRZD −NN(E)+a L
7
wherea =2WL/x =R(L)/RandR(L)=L/g istheresistanceoftheS lead
L 0 N ′
inthenormalstate.AtzerotemperatureateV D thecurrentgivenbyEq.(23)
canbecalculatedtofirstorderinthesmallpara≫meterD /eV 1asfollows,
≪
1 eV N2(E)dE D eV/D x2dx V
I= = +I , (24)
eRZD N(E)+a L eRZ1 x√x2−1+a L(x2−1) ≈ Rtot exc
whereR =R+R(L)isthenetnormalresistanceofthejunctionandtheS lead
tot ′
andI isanexcesscurrentgivenbytherelation,
exc
D a 2a 2 1 a
I = L L arctan − L . (25)
exc eR"1−a L2 − 1−a L2 3/2 r1+a L!#
ThustheIVcharacteristicEq.(23(cid:0))exhibi(cid:1)tsavoltage-independentexcesscurrent
at large voltage, which is the manifestationof the nonequilibrium in the S lead.
′
Hereweshouldmentionthatthetotalexcesscurrentmeasuredintheexperimentis
knowntoconsistofthetwocontributions:theonecomingfromthesingleparticle
currentatlargevoltagejustcalculatedabove,andtheothercomingfromthetwo
particle current (Andreev current). In this paper we do not calculate the latter
contributionsinceitisoftheorderofW a .
L
≪
InFig.2weplottheIVcharacteristicoftheNISSjunctionfordifferentvalues
′
ofl parameter.I(V)givenbyEqs.(21),(22)isanoddfunctionofvoltageandwe
plot−itonlyforpositivevoltages.WefixT =T =0.3T ,whereT isthecritical
N S C C
temperatureofthesuperconductor.ForAluminumT =T 360mK.Weseethat
N S
≈
withthe growth ofthecharge imbalancerelaxationlengthl theelectriccurrent
decreases.Whenl >LthelengthoftheS leadLplaysther−oleofacharacteristic
′
relaxationlengtha−ndthecurrentisalmostindependentofl .
−
4 Coolingpower
TheequationforthecoolingpowerfollowsfromEqs.(8),(15)andreads
1 ¥
P= IV EN(E)(f f )dE, (26)
− −e2R D +N− +S0
Z
where I is givenby Eq. (21). To obtain f we should solve boundary problem
+S0
(13),(15)and(17).Doingthiswegetthefollowingresult,
f a N(E)+ f coth(b N(E))
+N + eq +
f = , (27)
+S0 a N(E)+coth(b N(E))
+p + p
a =2Wl /x =R(l )/R W, b =L/l , (28)
+ + p0 + p + +
≫
where R(l ) =l /g is the normal resistance of the superconducting lead per
+ + N
lengthl .
+
SubstitutingEq.(27)intoEq.(26)wefinallyobtain,
1 ¥ coth(b N(E))dE
+
P= IV EN(E)(f f ) . (29)
− −e2R D +N− eq a N(E)+coth(b N(E))
Z + p +
p p
8

2
2 /
e
R
P
0.0

-0.1
0.0 0.5 1.0 eV/
Fig. 3 (Color online) P(V) dependence of the NISS junction. T = T = 0.3 T ,W = 0.01.
′ N S C
SolidblacklineiscalculatedbyuseofEq.(30).OtherlinesarecalculatedbyuseofEq.(29),
L=50x0:l =l+=10x0 (redline);l =l+=20x0 (greenline);l =l+=50x0 (blueline);
l =l+=1−00x0(dashedblackline). − −
−
ForL=0orl =0theboundaryvalue f inEq.(26)isequalto f andEq.(29)
+S0 eq
reduces to the±well-known equation for the heat current between N and S reser-
voirs,connectedthroughaninsulatinglayer[6],
1 +¥
P= N(E)(E eV)[n (E eV) n (E)]dE. (30)
e2R ¥ − F − − F
Z−
IntheabsenceofinelasticrelaxationintheS lead(b 1)wecanapproxi-
′
mateEq.(29)by ±≪
1 ¥ EN(E)dE
P= IV (f f ) , (31)
− −e2RZD +N− eq 1+a LN(E)
where the current I is given by Eq. (23). This equation corresponds to the case
whenthelengthoftheS leadLissmallerthantheinelasticrelaxationlengthand
′
allrelaxationoccursonlyintheSreservoir.
InFigs.3,4,5weplottheP(V)dependenceoftheNISSjunctionfordifferent
′
valuesofl parameters.P(V)givenbyEqs.(29),(30)isanevenfunctionofvolt-
ageandwe±plotitonlyforpositivevoltages.FromFig.3itcanbeseenthatwith
thegrowthoftherelaxationlengthsl thecoolingpowerdecreases.Alsothemax-
imalcoolingpowershiftstotheregio±nofsmallervoltagesthanintheequilibrium
case.Weascribethissuppressiontotheeffectofbackscatteringonimpuritiesand
tunnelingofnonequilibriumquasiparticlesbacktothenormalmetalreservoir.We
canseethatforlargevaluesofl thecoolingpowerisnegativeforallvoltages.
InFig.4weplotP(V)fordi±fferentratiol /l .Forafixedl lengthwevary
+ +
the charge imbalancerelaxationlengthl . It can−be seen that the cooling power
increaseswiththegrowthofl .Thishap−pensbecauseofthedecreaseoftheterm
IV inEq.(29)duetothesupp−ressionoftheelectriccurrent(seeSec.3).
9

2
2 /
e
R
P
0.0

-0.1
0.0 0.5 1.0 eV/
Fig.4 (Coloronline)P(V)dependenceoftheNISSjunction.T =T =0.3T ,W =0.01.Solid
′ N S C
blacklineiscalculatedbyuseofEq.(30).OtherlinesarecalculatedbyuseofEq.(29),L=
50x0,l+=10x0:l =10x0(redline);l =20x0(greenline);l =50x0(blueline);l =100x0
(dashedblackline−). − − −

2
2 /
e
R
P
0.0

-0.1
0.0 0.5 1.0 eV/
Fig.5 (Color online)P(V)dependence oftheNISSjunction.T =T =0.3T ,W = 0.001.
′ N S C
SolidblacklineiscalculatedbyuseofEq.(30).OtherlinesarecalculatedbyuseofEq.(29),
L=50x0:l =l+=10x0 (redline);l =l+=20x0 (greenline);l =l+=50x0 (blueline);
l =l+=1−00x0(dashedblackline). − −
−
Finally we want to stress here the role of the transparency parameterW. In
Fig. 5 we plot the P(V) dependence for a different value of tunneling parameter
W =10 3,whichisoneorderofmagnitudesmallerthanW inFigs.3,4.Herewe
−
againseethedecreaseofthecoolingpowerwiththegrowthofl ,buttheeffectis
smallerthaninFig.3.Thisisobvioussincetheamplitudesofth±enonequilibrium
10
fS- fS+
2 01 0
1
1 1
01,0 030 01,0 030
x/ x/
1,2
1,5 E/ E/
Fig. 6 (Color online) Distribution functions f S(E,x) (left plot) and d f+S(E,x)= feq f+S
(rightplot).HereTN =TS=0.3TC,eV =0.8D ,−L=50x0,l =l+=10x0,W =0.01. −
−
distribution functions f scale with the W parameter. For very strong tunnel
S0
barriersthenonequilibri±umeffectisthereforenegligible.
Inordertoestimatethecharacteristicparametersofthejunctionsforthetrans-
parencyparametersW =10 2 andW =10 3 usedinournumericalcalculations,
− −
we will assume the junction area to be 200 200 nm and the thickness of the
×
leads as wellas the meanfree path tobe 50 nm. For Al leads,this resultsin the
sheetresistanceR 0.3W andR(x ) 0.45W atx 300nm.Then,accord-
(cid:3) 0 0
≈ ≈ ≈
ingtoEq. (4), thetunnelingprobabilityandthe junctionresistanceapproach the
valuesG 2 10 3,R 22.5W forW =10 2,andG 2 10 4,R 225 W
− − −
forW =1≈0 3×,respective≈ly. ≈ × ≈
−
5 NonequilibriumquasiparticledistributioninS
′
SolvingEqs.(14),(16)and(17)wecanobtainthefunction f (x,E)intheS lead.
S ′
Itreads −
f a sinh (b x/l )/ N(E)
f−S= a + N(E)c−oNth−b / N(E) shinh−−b /−N(pE) i. (32)
− − −
p (cid:16) p (cid:17) (cid:16) p (cid:17)
Thisequationdescribestherelaxationoftheimbalancebetweentheelectronsand
holesintheS lead(branchmixing).FromEq.(32)weseetheexponentialdecay
′
oftheimbalancefunction f atthedistancex l N(E)fromthetunnelbarrier.
AtE D thedecaylength−diverges. ∼ −
→ p
SimilarlyfromEqs.(13),(15)and(17)weobtainthefunction f (x,E)inthe
+S
S lead,
′
(f f )a N(E) sinh (b + x/l+) N(E)
eq +N + −
f = f − . (33)
+S eq−a + N(E)+coth pb + N(E) hsinh b + N(pE) i
p (cid:16) p (cid:17) (cid:16) p (cid:17)