Interface-mediated pairing in field effect devices

0.28 MB·

by  V. Koerting

Interface-mediated pairing in field effect devices
V. K¨orting1, Qingshan Yuan1,2,3, P. J. Hirschfeld1,4, T. Kopp1, and J. Mannhart1
1Center for Electronic Correlations and Magnetism, EP6, Univ. Augsburg, Augsburg Germany
2Texas Center for Superconductivity and Advanced Materials, Univ. of Houston, Houston, TX 77204 USA
3Pohl Institute of Solid State Physics, Tongji University, Shanghai 200092, P.R. China
4Department of Physics, University of Florida, PO Box 118440, Gainesville FL 32611 USA
(Dated: February 2, 2008)
7
0 We consider the pairing induced in a strictly 2D electron gas (2DEG) by a proximate insulating
0 filmwithpolarizablelocalizedexcitations. Withinamodelofinteracting2Delectronsandlocalized
2 two-level systems, we calculate the critical temperature Tc as a function of applied voltage and for
different materials properties. Assuming that a sufficient carrier density can be induced in a field-
n
a gateddevice,wearguethatsuperconductivitymaybeobservableinsuchsystems. Tc isfoundtobe
J a nonmonotonic function of both electric field and theexcitation energy of the two-level systems.

5
PACSnumbers: 74.25.Fy,74.25.Jb,74.40.+k,74.81.-g
1

]
l Shortly after the publication of the BCS theory of su- tal objection to even higher charge densities, since com-
e
perconductivity, W. A. Little proposed a possible pair- plexoxidedielectricsandferroelectricoxidescanachieve
-
r ingmechanismforelectronsinlongorganicmoleculesin- polarizations in the range of ∼ 0.5 [8]. In fact, very re-
t
s volving localized electronic excitations in the molecules’ centlythe firstobservationoffield-inducedsuperconduc-
.
at side chains [1]. The suggested advantages of such an ar- tivity wasreportedfor a device with a Nd1.2Ba1.8Cu3O7
rangementincludedthelargecharacteristicenergyofthe epitaxial film grown on SrTiO substrates [12].
m 3
couplings, which might lead to high temperature super-
Therefore one may legitimately ask the question what
-
conductivity (HTSC), and the separation of the excita-
d critical temperature can be achieved in a system where
n tionsthemselvesfromthescreeningeffectsoftheelectron the pairing comes entirely from the excitations in the
o gas. Such a pair mechanism has never been realized, proximate insulating layer, and how T is likely to vary
c
c presumably due to large fluctuation effects in quasi-1D
with field in this case. Alternatively, one can assume
[
systems. Later, Ginzburg [2] and Allender, Bray and the existence of a 2D superconducting layer with pre-

3 Bardeen [3] proposed a similar excitonic 2D mechanism existing pair interaction and bare critical temperature
v in superconductor-semiconductorsandwiches. The enor- T0, and ask by how much the presence of the insulat-
4 mous body of early work on this problem has been re- c
ing layer enhances T . While in a strictly 2D system the
2 viewed in [4]. One problemwith these schemes is clearly c
T ’s referredto cannotcorrespondto true long-rangeor-
6 c
that the localized nature of the excitations implies that
2 der, the creation of a field-tuned superconducting state
at most one or two atomic layers of the intercalating in-
0 with algebraic order at finite temperatures would be of
4 sulator can contribute to the pairing, meaning that the considerable interest and applicability in small devices.
0 enhancementofpairinginthe relativelythicknearbysu-
We beginby consideringaninsulatingamorphousfilm
/ perconductor is negligible.
t L1inproximitytoacorrelatedinsulatorL2(drain-source
a
Here we propose that a similar scheme might work for
m (DS) channel of a field effect device), similar to Ref. 12,
an insulating layer in proximity with a superconducting
with a small density of localized charge carriers. Ap-
- layerofnear-atomicthickness. Suchsystemscaninprin-
d plying an electric field as shown in Fig. 1 sweeps charge
n ciplebepreparedinseveralways,butthemostpromising carriers to the interface with the film L1 where they ac-
o is perhaps the field effect method pioneered in the early cumulate[13]. Theformationofthe2Dbandinducedby
c 90’s with the intent of increasing the carrier density and anelectric field has been studied by Poilblancet al. [14].
v: hence Tc in HTSC cuprate devices [5, 6, 7, 8, 9]. This Weassumeforthemomentthatintheabsenceofthefilm
i workhasshownincontrovertiblythatTcandothersuper- L1therearenopairinginteractionsbetweentheelectrons
X
conducting properties in metallic samples can be influ-
in the material L2. Qualitatively, we expect the follow-
r encedbyanappliedgatevoltage. Whiletheexperiments
a ing picture to apply: virtual excitations in the dielectric
have most often been interpreted in terms of electronic
L1 induce Cooper pairing in the adjacent layer L2. The
structure variations near the interface, they are not un-
criticaltemperature mustincreaseinitially with the field
derstood in detail.
since carriers are being injected into the system. With
It is interesting to ask if new physics can result from increasing electric field the larger level splitting of the
field doping of insulators. In the most recent reports on two level system leads to a suppression of the polariza-
field-doping of SrTiO , Pallecchi et al. [10] and Ueno et tion fluctuations and the pair potential decreases. T is
3 c
al. [11] were able to achieve an areal carrier density of thereforeexpected to reacha maximumata characteris-
∼ 0.01–0.05per unit cell. It is difficult to achieve higher tic field strength;it is ourobjectiveto estimate the scale
densities due to electrical breakdown in the insulating of possible T ’s through this process, as well as the field
c
layerat highfields. But there seems to be no fundamen- strength required to attain it.
2
with
H = −t c† c (3)
t i,σ j,σ
<ij>,σ
X
1
H = ∆ (p†p −s†s ) (4)
2l 2 sp i i i i
i
X
H = E (p†s +s†p ) (5)
ext sp i i i i
i
X
H = V c† c (p†s +s†p ) (6)
int sp i,σ i,σ i i i i
i,σ
FIG. 1: Our model of a field effect transistor is represented X
by two layers (L1 and L2): H = −µ c† c (7)
µ i,σ i,σ
L1: dielectric layerwith local dipoles of dipolemoment edsp, i,σ
presented by two-level systems; X
L2: metallic DS-channelwith a 2D electron gas. He−e = U c†i↑ci↑c†i↓ci↓. (8)
i
X
Here H describes a band ofnoninteracting 2D electrons
t
We now propose a crude but concrete framework on a square lattice, H2l the energies of the localized two
within which one can calculate these effects. Roughly level system, Hext the coupling of the electric field to
speaking,thedielectriccanhavetwoeffects onTc. First, these orbitals, Hint the Coloumb interaction between
itcanreducetheCoulombpseudopotentialofelectronsin electrons in the metallic layer and the two level sys-
the field-doped layer due to the large dielectric constant tem,andHµ thechemicalpotential. Thedirectelectron-
of the amorphous insulator L1. We are more concerned electron interaction term He−e is taken to be local and
here with the secondeffect, namely anadditional contri- repulsive. Very similar models have been used recently
butiontothe residualpairinginteractionattheinterface to discuss dielectric properties of bulk cuprates [15], the
due to virtual polarization of the dielectric itself. competition between charge density wave and supercon-
ductivity in twodimensionalelectronic systems[16],and
superconductor-ferroelectric multilayers [17]. The sys-
tem is not exactly soluble, but will be treated under the
assumption that the polarization through the charges in
I. MODEL L2isnotlargeenoughtodrivethe systemintothe ferro-
electric state. We first diagonalize H +H , and then
2l ext
express the corresponding quasiparticle operators in a
The Hamiltonian we consider describes a single layer
pseudospin representation where Sz measures the occu-
L2 of electrons c†i hopping on a square lattice with lat- pation of the 2-level system and Si± induces transitions
tice constant a, nearest neighbor hopping t, and a set of i
between the two eigenlevels. The occupation constraint
localized two-level systems in an adjacent layer L1 with
on the two-level system is preserved by the usual spin
ground state s† and excited state p† separated in en-
i i algebra. The (exact) final form of the Hamiltonian is
ergy by ∆ , and their respective dipole moment is ed
sp sp
(Fig. 1). Note these states simply designate ground and
excited states of a localized two level system, and need 1
H +H = −2 E2 +( ∆ )2 Sz (9)
not correspond to actual atomic s– and p– orbitals. To 2l ext sp 2 sp i
r i
ensure that the two-levelsystem is occupied by only one X
electron,theconstraints†isi+p†ipi =1mustbeenforced. Hint = −Vz c†i,σci,σSiz (10)
The electric field i,σ
X
+V c† c (S++S−) , (11)
x i,σ i,σ i i
E ≡Esp/(edsp) (1) Xi,σ
where
is assumedto populate the metallic layerand simultane- E
sp
ouslypolarizethetwolevelsystemsbydrivingtransitions Vz = 2Vsp
E2 +(1∆ )2
betweenthe s andpstates,whicharecoupledtothe free sp 2 sp
electrons in the layer L2 by a contact interaction Vsp. q 1∆
V = V 2 sp .
The Hamiltonian contains, according to our assump- x sp
E2 +(1∆ )2
tions, the following elementary processes: sp 2 sp
q
The system of interacting spins-1/2 and fermions may
H =H +H +H +H +H +H (2) now be treated within linear spin wave theory, an ap-
tot t 2l ext int µ e−e
3
proximation which is justified a posteriori by the ob- by rewriting the fourth term of Eq. (12)
servation that the occupation of the higher-energy 2-
level state is very small, due to the sufficiently large V c† c b†b
z i,σ i,σ i i
2-level splitting [18]. Introducing the usual Holstein- i,σ
X
Primakoffbosons[19],wemaketheapproximatereplace-
=−NV nn +V n b†b +V n c† c
ments S+ →b , S− →b†, Sz → 1 −b†b to find z b z i i z b i,σ i,σ
j j j j j 2 j j i i,σ
X X
+V c† c −n/2 b†b −n (14)
z i,σ i,σ i i b
Xi,σ h ih i
1
H = −t c†i,σcj,σ−µr c†i,σci,σ−E2l (2 −b†ibi) where nb = hb†ibii is the number of bosons (the rela-
<Xij>,σ Xi,σ Xi tive number of inverted 2-level systems per site), n =
+Vz c†i,σci,σb†ibi+Vx c†i,σci,σ(bi+b†i)+He−e (1/N) i,σhc†i,σci,σi is the chargecarrierdensity in layer
i,σ i,σ L2,andN isthenumberofsitesinL2. Thesecondterm
X X P
on the right hand side accounts for a density-dependent
(12)
renormalizationof the two-level splitting:
E⋆ =E +nV . (15)
We have introduced the energy splitting of the two-level 2l 2l z
system
In a first step we apply a Lang-Firsov (LF) transfor-
mation of the Hamiltonian H in order to identify the
pairing interaction, and in a second step we control the
1
E =2 E2 +( ∆ )2 (13) decoupling of the interaction terms through Feynman’s
2l sp 2 sp
r variational principle which also fixes the parameters of
the LF transformation.
TheLFtransformationoftheHamiltonianH˜ =U†HU
and the chemical potential has been renormalized µ =
r
is achieved by the following unitary operator U:
µ+V /2.
z
We now declare that the Hamiltonian H is as good a U =exp[−S1(θ)] exp[−S2(γ)] (16)
starting point as (3)–(8) provided hb†b i≪1. We there-
j j with
fore proceed to apply a Feynman variational procedure
to the exact H, Eq. (12), after a unitary transforma- 1
S (θ) = − θ (b†−b ) (17)
tion,andattempttoextractthepairingmechanism. The 1 2V i i
x
dominant term for this mechanism is the term with Vx Xi
which, as in the phonon-induced superconductivity, pro- S (γ) = Vx γ c† c (b†−b ) (18)
duces Cooper pairing in second order perturbation the- 2 E⋆ i,σ i,σ i i
2l i,σ
ory [1]. However, for the field strengths which we will X
consider, the term with Vz is also not negligible and it The parameters θ and γ will be fixed through the vari-
willalterpairinginonesignificantrespect: sinceit mod- ation of the free energy. The standard form of the
ifies the 2-level splitting when a charge carrier occupies LF transformation for the Holstein model takes θ = 0
the site coupled to the considered 2-level system, it will and γ = 1. With a “zero-phonon” approximation the
either increase the pairing interaction (for reduced split- (phononic) Holstein model accounts for exact results in
ting, i. e., negativeVz) or decreasepairing(for enhanced the antiadiabatic limit E2l/t ≫ 1. The variation of
splitting, i. e., positive Vz) — an observation which may the parameters θ and γ has been devised in order to
be derived straightforwardly from a mean field decou- reproduce the adiabatic limit of the Holstein model as
pling. Forthe physicalsystemthatweconsider,Esp and well[21,22,23,24];thestaticcaseisrealizedwithγ →0
Vsp always have the same sign, and consequently Vz is and with a finite θ which signifies a displacement field.
positive [20]. All of the above considerations can also For the Holstein model, a second canonical transforma-
be derived in terms of states where charges in L1 are lo- tion of Bogoliubov type U = exp −α (b†b†+b b )
calized in states of definite position with respect to the B i i i i i
interface, as depicted schematically in Fig. 1. In this is sometimes applied to allow for thhe aPnharmonicity oif
the lattice fluctuations. For our model, we have to sup-
representation, the V interaction may be shown to cor-
z
pressstateswithtwobosons(correspondingtoourinitial
respond to the repulsion of the electron in layer L2 and
constraint for the 2-level system), and subsequent vari-
the field-induced dipole in L1.
ation of the free energy shows that in fact α = 0. Also
Inordertoincorporatethiseffectwewillincludebelow the “static displacement” should be suppressed since we
the thermal average V hc† c ib†b = V n b†b do not intend to consider a finite static polarization. We
z i,σ i,σ i,σ i i z i i i
explicitly in the energy of the two-level system before verified from the minimization that indeed θ ≃0 for the
P P
handlingthebosonicdegreesoffreedom. Thisisachieved considered range of small to intermediate V /4t. For
sp
4
this reason, and to simplify the notation, we fix θ and α 6 1
to zero.
5
With the LF transformation U = exp[−S (γ)] we ob-
2
0.75
tain
4
4t
H˜ =E0+HLF+HVx +HVz +HVxVz (19) V / eff3 0.5γ
∆ / 4t = 3.75
where E0 =−12NE2l−NVznnb, and 2 ∆sspp / 4t = 2.5 0.25
∆ / 4t = 1.25
HLF = −t c†i,σcj,σ eEV2⋆xlγ(b†i−bi)e−EV2⋆xlγ(b†j−bj) 1 sp
<ij>,σ 0 0
X 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
−Veff c†i,↑ci,↑c†i,↓ci,↓ Esp / 4t Esp / 4t
i
X
−µ c† c + E⋆ b†b (20) FIG.2: Leftpanel: EffectivepairinteractionVeff/4tvs.nor-
LF i,σ i,σ 2l i i malized electricfield energy Esp/4t forvariousexcitation en-
Xi,σ Xi ergies ∆sp of the local two-level systems in layer L1. The
dielectricconstantǫ=100controlstheincreaseofchargeden-
with
sity in themetallic layer L2 with electric field (cf. Eqs. (38)–
(39)),wherebywechoseabandwidth4t=400meV,adipole
HVx = (1−γ)Vx c†i,σci,σ(b†i +bi) (21) length dsp =2 ˚A (for charge transfer excitations), and a lat-
i,σ tice constant a = 4 ˚A. The coupling of the dipoles to the
X
H = V c† c −n/2 b†b −n (22) conduction electrons is fixed at Vsp/4t = 1.89 which corre-
Vz z i,σ i,σ i i b sponds to a spatial distance r/a = 1.5 (cf. Eq. (40)). Right
Xi,σ(cid:0) (cid:1)(cid:0) (cid:1) panel: variational parameter γ vs. normalized electric field
H =−γ VxVz c† c −n/2 c† c (b†+b ) Esp/4t, where γ is found from the minimization of the free
VxVz E⋆ i,σ i,σ i,σ′ i,σ′ i i energy (rhs of Eq. (26)).
2l i,σ,σ′
X(cid:0) (cid:1)
(23)
We begin by setting U = 0 and investigate the mag-
where the effective chemical potential is given by nitude of the attractive interaction obtainable by polar-
izing the two level systems. In Fig. 2 we illustrate how
µ =µ + V /2 + γ(2−γ)Vx2 Veff depends on applied electric field. The main point is
LF z E⋆ physicallyobvious: iftheelectricfieldissufficientlylarge,
2l
V 2 it polarizes the electric dipoles in L1 and suppresses the
x
−(1−n)V γ −V n (24) Little-type mechanism. We also note that if the bare ex-
z E⋆ z b
(cid:18) 2l(cid:19) citation energy ∆sp of the dipoles in the insulating layer
is small, the intrinsic low-field pairing strength can be
The induced electronic interaction V is now
eff quite large, up to several electron volts. However ∆
sp
V2 V should not be set to considerably smaller values than in
Veff =2Ex⋆ γ (2−γ)−γ(3−n)Ez⋆ −U, (25) Fig. 2 in order to guaranteethe condition that the occu-
2l (cid:20) 2l(cid:21) pation of the excited state is negligible.
In the second step we introduce an exactly solvable
where we note that the Hubbard term H has been
e−e
test Hamiltonian H and determine the fields in H
unaffected by the above transformations of the electron- test test
througha Bogoliubovinequality for the free energyF of
twolevelsystemcouplingterms,duetothefactthatitis
the model system [25]:
simply a product of local densities. The local repulsion
thereforemerelydiminishestheeffectiveattractionbyU.
F ≤F +hH˜ −H i (26)
Ifγ =1,thefirsttermwithinbracketsinEq.(25)coin- test test test
cideswiththewell-knownresultfromtheexactmapping where h i signifies the thermodynamic average with
test
oftheHolsteinmodeltotheattractiveHubbardmodelin the test Hamiltonian. We already anticipate the result
thehigh-frequencylimit(E2l →∞). Thesecondtermin of this variational scheme [26] and write
the brackets ∝ V is always repulsive and suppresses T
z c
for increasing field. The electric field dependence enters H = E − t c† c − µ c† c
test 0,test eff i,σ j,σ LF i,σ i,σ
through Vx, Vz, E2⋆l and the implicit dependence of γ on <Xij>,σ Xi
E .
sp
We have now achieved the desired result of expressing + ∆ c† c† +h.c. − ∆ hc† c† i
i,↑ i,↓ i,↑ i,↓
theHamiltonianintermsofaneffectiveBCS-likeinterac- (cid:18) i (cid:19) i
X X
tionbetweenelectrons,atthecostofintroducingbosonic +E⋆ b†b (27)
2l i i
phase factors into the hopping matrix elements.
i
X
5
1 Since we have chosenE so as to guaranteethe re-
0,test
1e-06 lationhH˜−H i =0,thevariationreducestofinding
test test
0.8 the optimal value of γ from the minimization of F :
test
1e-08
0.6
t / teff0.4 1e-10nb γ = n(n+2) + n(n+n1()n(2+−2n))(Vz/E2⋆l) + δ(γ) (34)
∆sp / 4t = 3.75 1e-12 where
∆ / 4t = 2.5
0.2 sp
∆ / 4t = 1.25
sp 1e-14 ε¯ βE⋆
0 δ(γ) = 2 coth 2l (35)
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 E⋆ 2
E / 4t E / 4t 2l
sp sp (cid:0) (cid:1)
and
FIG. 3: Left panel: Effective hopping teff/t vs. normalized
electric field energy Esp/4t at the transition temperature Tc; εk
all paramaters are identical to those of Fig. 2. Right panel: ε¯= (36)
1+eβξk
bosonic occupation number nb (the relative number of in- Xk
vertedtwo-levelsystemspersite)vs. normalizedelectricfield
Forthe rangeofparametersconsideredinourevaluation
energy Esp/4t at thetransistion temperature Tc.
the function δ(γ) can be neglected with respect to the
other terms in the denominator of Eq. (34). This allows
with us to calculate γ algebraically from Eq. (34). The valid-
N ity of this approximation has been proved by iterating
E0,test = E0− n2Veff (28) the implicit Eq. (34). Fig. 2 (right panel) displays the
4
decrease of γ with increasing E for various values of
V 2 βE⋆ sp
t = t exp − x γ coth 2l (29) ∆sp at fixed Vsp/4t=1.89.
eff " (cid:18)E2⋆l (cid:19) (cid:18) 2 (cid:19)# FinallytheassumptionbelowEq.(11)thatthebosonic
occupation number n = hb†b i is very small is in fact
We have assumed s-wave pairing for simplicity. The as- b i i
justifiedaposteriorifortemperaturesuptothetransition
sociated gap equation then reads
temperatureT . Intheconsideredparameterrangeofthe
c
∆ Ek two-level splitting ∆ and of the applied gate field E ,
∆=V tanh (30) sp sp
eff k 2Ek 2T nbisalwayssmallerthan10−5. Typicalfielddependences
X of n are displayed in the right panel of Fig. 3.
b
and the Bogoliubov quasiparticle dispersion
6
Ek = ξk2 +∆2 (31) ∆ / 4t = 1.25
ξk = εqk−µLF 5 ∆sp / 4t = 2.5
sp
εk = −2teff(coskx+cosky) 4 ∆sp / 4t = 3.75
To understand qualitatively the physics of the corre-
n
ltahteio2n-sleivnedluscyesdtembysiinnteLr1a,ctwioendoisfptlhaye minetthaellilcefltaypearnwelitohf ∂µ / ∂ 3
Fig. 3 the magnitude of the renormalized hopping t . 2
eff
The band narrowing can be rather significant for small
excitationenergies∆ butfortheparametersofgreatest 1
sp
interest will turn out to be only a factor of 1–5.
0
Thefieldsstilldependontheparameterγ;correspond-
0 0.2 0.4 0.6 0.8 1
inglytherhsofrelationEq.(26)hastobeminimizedwith
n (band filling)
respect to γ. For this purpose we need the explicit form
of F :
test FIG.4: Thederivativeofthechemicalpotentialwithrespect
1 toparticledensity,∂µ/∂n,versusparticledensitynatthere-
Ftest =E0,test+ β N ln 1−e−βE2⋆l +FBCS (32) spectiveTc,calculatedinSec.II.Allparamatersareidentical
(cid:16) (cid:17) to those of Fig. 2. For a positive derivative, the normal and
with superconducting states are globally stable in the vicinity of
2 thetransition.
F = − ln 1+e−βEk
BCS
β
Xk (cid:0) (cid:1) Theeffectivemodeldoesnotimplementthelong-range
− Ek−ξk+∆hc−k,↓ck,↑i (33) partoftheCoulombinteraction. Amodelwithpurelylo-
cal interactions, however, has a tendency towards phase
k
X(cid:16) (cid:17)
6
separation. In order to investigate if the transition from We note that dynamical screening of the interaction
the normal to the superconducting state is a transition V should be included in order to get more precise es-
sp
between global minima of the free energy, we evaluated timates. Although this will reduce somewhat the high
the derivative of the chemical potential with respect to values of T in our evaluation, it will not alter the qual-
c
carrier density. In this work, we focus on the transi- itative behavior of the T -dependence on the gate field
c
tionintothesuperconductingstateandnotontheevolu- (see below).
tion of the superconducting state at lower temperatures. Finally, the excitation energy of the dipoles has to be
Thenitissufficienttodiscussthestabilityinthevicinity identified. Agenericdielectricisnotcomposedofasingle
ofT . Fig.4illustratesourconclusionthatthederivative speciesofwell-defined2-levelsystems. Dipoleexcitations
c
ofthe chemicalpotentialwith respectto particle density at various energies are always present. We now address
is always positive for ∆ /4t larger than approximately excitations in three different energy ranges: for 10 eV
sp
1.4. Correspondingly, the transition into the supercon- (low atomic excitations with dipole length d ≃ 1 ˚A),
sp
ductingstateisnotpreemptedbyacompetingtransition inthe1eVrange(chargetransferexcitationswithd ≃
sp
intoaphaseseparatedstate. Atsomewhatlowervaluesof 2˚A),andinthemeVrange(ionicdisplacementinatomic
∆sp/4t,thephaseseparationwillprobablybesuppressed clusters with dsp ≃0.1 ˚A).
by the non-local part of the Coulomb interaction. As
discussed in the following section, the relevant values of
∆ /4t for a sufficiently strong interface-mediated pair- A. Ionic displacement
sp
ing appear in the regime where ∂µ/∂n is positive.
In dielectrics a displacement of ions, well localized at
atomic positions, usually accounts for the high dielec-
II. RESULTS
tric constant. Such displacements in atomic clusters
typically correspond to a dipole length of the order of
In order to present the numerical results in the phys- d ≃ 0.1 ˚A and a small excitation energy. In this case,
sp
ically relevant range of control parameters, we have to the excitation energy is related to ǫ and d through
sp
relate microscopic quantities to laboratory parameters, ∆ = 8πe2d2 /(a3(ǫ−1)). This relation follows di-
sp sp
such as the dielectric constant ǫ of the insulating layer, rectly from the polarization density hPi of the dielectric
the dipole length dsp of the two-level systems, and the (with a cubic unit cell of volume a3) in linear response:
distance r from the two-level systems to the sites of the hPi = ed hs†p +p†s i/a3 = (2e2d2 /a3∆ )E where
sp i i i i sp sp
conducting layer. Furthermorewehavetoestablishare- hPi=(ǫ−1)/4πE holds. Ionic displacements may have
lation between the electric field energy Esp (see Eq. (1)) energiesofthe orderof10meV (forǫ ofthe orderof20).
and the number of charge carriers in L2. In our calcula- However, the induced pair interaction is far too small
tionweassume thatthe chargecarrierdensity is alinear (see Fig. 5) to find sizable transition temperatures. It is
function of the electric field with a field independent ca- therepulsiveterminV /4tofEq.(25)whichveryeffec-
eff
pacitance C of the dielectric L1: Q = CV, where V is tively impedes a transition to the superconducting state
the voltage drop across the dielectric and Q is the total, for such small values of ∆ .
sp
accumulated charge at the interface in L2. Then, the
charge per square unit cell is
B. Charge transfer excitations
Ed a2
n=C (37)
e A
In order to realize a sizable T of the order of 100 K,
c
excitations at intermediate energy and sufficiently large
where a is the lattice constant,Ais the areaof L2andd
dipole length have to be available for polarization.
its thickness (cf. Fig. 1). With C = ǫ ǫA/d and Eq. (1)
o
Charge transfer excitations in the dielectrics are found
we establish
at these energies — e.g. for the transition metal (TM)
Esp oxides at energies of the order of 1 eV, the charge trans-
n = c (38)
4t fer gap. Although these excitations of, for example, a
TM-oxygen plaquette or octahedron become delocalized
with
through hybridization, we assume that most states near
a2 4t the interface have been localized by disorder and local
c = ǫoǫ e2 d (39) strain [27]. Furthermore localized bound states below
sp the charge transfer gap are conceivable. It is not the
focus of our present investigation to identify these local
Theinteractionenergybetweenthedipolesnexttothe
excitations for a specific material. We rather point out
interface and the electric field of a charge carrier on the
that such processes are possible and then evaluate T as
nearest site is c
a function of their respective energy.
1 e2d We intend to focus on this latter type of localized
sp
V = . (40)
sp 4πǫ r2 charge transfer excitations. For the evaluation we now
0
7
6 0.0001 0.2
ε V / 4t
eff n = 0.5
5 0 n = 0.1
0 20 40 60 80 100 n = 0.01
0.15
4
-0.0001
/ 4tVeff23 0.0001 T / 4tc 0.1
5e-05
1 E / 4t
sp V / 4t
eff 0.05
0 0
0 0.0005 0.001 0 0.1 0.2
-1
Esp / 4t 0
0 5 10 15
∆ / 4t
sp
FIG. 5: Ionic displacement in atomic clusters: induced pair-
ing interaction Veff/4t versus electric field energy Esp/4t for
ǫ = 10, dsp = 0.1 ˚A and r/a = 1 (which corresponds to FIG. 6: Charge transfer excitations: transition temperature
Vsp/4t =0.2125). Left and lower right panels show different Tc/4t versus excitation energy ∆sp/4t. The three curves
scales. Upper right panel: Pairing interaction versus dielec- present Tc for small and intermediate band filling. The di-
tric constant ǫfor charge density n=0.5. electric constant is ǫ = 100, which implies c = 1.87 for the
considered set of parameters (see Eq. (41)). The coupling of
the dipoles to the conduction electrons is fixed to Vsp/4t =
1.89 which corresponds to a spatial distance r/a = 1.5 (cf.
fix the following parameters: the bandwidth, the dipole
Eq.(40)).
length, and the lattice constant:
4t = 400meV band filling
dsp = 2˚A 0.20 0.5 1
a = 4˚A (41)
Here, we chose a bandwidth and lattice constant typi- 0.15
cal of the high-T cuprates and many other oxides, and
c
we take a dipole length which corresponds to the inter-
aotfommoirceotxhyagnen0-.T2MeVd,istthaencper.imFaorryexpcoiltaartiizoantieonneprgrioecse∆ssseps T / 4tc 0.1 ∆sp / 4t = 3.75
are electronic in nature. ∆ / 4t = 2.5
sp
In Fig. 6 the transition temperature to the supercon- ∆ / 4t = 1.25
0.05 sp
ducting state is displayed as a function of the excita-
tion energy. In the range of small, increasing values of
∆ we observe a strong enhancement of T whereas the
sp c
transitiontemperaturedecreaseswithexcitationenergies 0
0 0.1 0.2 0.3 0.4 0.5
∆sp/4t above ≈ 2.5. The latter observation is expected E / 4t
sp
sincethepairinginteractionV isinverselyproportional
eff
totheexcitationenergyforlarge∆ . Forsmall∆ ,the
sp sp
repulsive term (second term in V , cf. Eq. (25)) domi-
eff FIG. 7: Charge transfer excitations: transition tempera-
nates andsuppressesthe transitionto the superconduct- ture Tc/4t versus electric field energy Esp/4t and band fill-
ing state for a finite value of ∆sp. Note that the finite ing n, lower axis and upper axis, respectively. The three
longitudinal pseudospin-charge Vz is decisive for the de- curves present different dielectrics which are characterized
cayofT atsmallexcitationenergies,outside the regime by different excitation energies only, other parameters are
c
where the Holstein model is appropriate. fixed for comparison. The dielectric constant ǫ is 100 and
For small increasing electric field, the transition tem- Vsp/4t=1.89 (r/a=1.5).
perature is raised due to the accumulation of charge in
themetalliclayer(cf.Fig.7). Theelectricfieldstrengthis
directlyrelatedtothebandfillingorinducedarealcharge T athigherfields. Thelowerscaleissetbytherepulsive
c
density. Strong electric fields with sizable band filling terminV andisalsoresponsiblefortheobserveddecay
eff
lowerthe transitiontemperatureastherepulsivetermin of T in the two further curves (with ∆ /4t = 2.5 and
c sp
V is enhanced and, moreover, the effective level split- 3.75). Inthisregime,T isbeingsuppressedprimarilyby
eff c
ting is enlarged. In fact, as seen from the ∆ /4t=1.25 theincreasingrepulsionV betweenthepolarizeddipoles
sp z
curveinFig.7therearetwoscalesforthesuppressionof andthe2Delectrons,whichscaleswiththe appliedfield.
8
0.2 band filling
0 0.5 1
ε = 100 0.4
ε = 50 r/a = 1
ε = 10 r/a = 1.25
0.15
r/a = 1.5
r/a = 1.75
0.3
4t
T / c 0.1 4t
/ c0.2
T
0.05
0.1
0
0 5 10 15
∆ / 4t
sp 00
0 0.1 0.2 0.3 0.4 0.5
E / 4t
sp
FIG. 8: Charge transfer excitations: transition temperature
Tc/4t versus excitation energy ∆sp/4t. The three curves
present different dielectrics (ǫ = 10,50,100). The band fill- FIG.9: Chargetransferexcitations(dsp =2˚A,∆sp/4t=2.5,
ing is fixed at n = 0.1, and the interaction Vsp/4t = 1.89 ǫ = 100): transition temperature Tc/4t versus electric field
corresponds to r/a=1.5. energyEsp/4t,parameterizedbyr/a. ThecorrespondingVsp
is in the intermediate coupling range: Vsp/4t = 4.25 (corre-
spondstor/a=1),Vsp/4t=2.72(r/a=1.25),Vsp/4t=1.89
The larger scale, which is responsible for the slow decay (r/a=1.5), Vsp/4t=1.39 (r/a=1.75).
at even higher fields, is set by ∆ /4t, and corresponds
sp
to the eventual saturation of the dipole moment of the
2-level systems. occurs for n = 0.2 at about 2×108 V/m. Thus the re-
The nonmonotonic dependence on filling or electric quired fields for the maximum Tc are only about 5 times
field is reflected in the varying height of the three dif- higher than those already realized in this system. Given
ferent curves in Fig. 6. However more striking is the that breakdown occurs due to “pinhole”-type defects in
small variation of the position of the maxima in Fig. 6 the films,itseemsto usthatthe manufactureofsamples
with filling (from 0.01 to 0.5). A value of ∆ ≃ 1 eV with the requiredbreakdownfields is challengingbut far
sp
seems to be optimal for a bandwidth of 4t = 400 meV. from impossible.
Thisoptimalexcitationenergyinunitsofthebandwidth We conclude this subsection with a brief discussion
is approximately of the consequences that a nonzero local interaction U
within the charge-carrier layer L2 has on the reduction
∆osppt/4t ≃ 2.5 (42) of Tc in a weak-coupling evaluation. As shown in Eq.
(25), sucha repulsiveHubbard-type interactiondoesnot
∆opt is weakly dependent on the parameters V , d modify the field dependence of the effective interaction
sp sp sp
and ǫ which mostly influence the maximum value of Tc. Veff, but just adds a constant −U. For weak interac-
Forfixedchargedensity,dielectricswithlargerǫdisplaya tion U/4t <∼ 1, the effective attraction Veff is reduced
highertransitiontemperature(forgiven∆ ),seeFig.8, but not fully suppressed for small to intermediate fields
sp
astheelectricfieldnecessarytocreatethechargedensity (cf. Fig. 2) and corresponding filling. This observation
is smaller, accounting for a larger Veff. is reflected in the field and filling dependence of Tc (see
An increasing dipole charge-carrier interaction Vsp is Fig. 10): Tc is reduced for small to intermediate fields
not only responsible for an enhancement of T but, for and suppressed for strong field-induced doping. As a
c
sufficientlylargeVsp,alsofora“retarded”initialincrease consequence of the field-dependent reduction of Tc, the
of T with electric field (see Fig. 9). Again, this obser- maxima are shifted towards lower values of filling, from
c
vation may be traced back to the field and interaction about n = 0.2 at U = 0 to n = 0.1 at U/4t = 1. The
dependence of the repulsive term in Veff. requiredfieldsforthemaximumTc arereducedsimilarly.
For direct comparison with experimental devices, we
note that the gate voltage is related to the electric field
energy plotted in the figures through E /4t = ed V/d C. Atomic excitations
sp sp
where d is the thickness of the dielectric gate. As a con-
creteexampleforadielectricgatelayer,wetakeSrTiO . Athighenergies,weencounteratomicexcitations,such
3
Breakdownfieldsof4×107V/marereportedforthisma- as the 2p to 3s transition of O2− states in the metal ox-
terial with a low-T dielectric function of order 100 [28]. ides. Thesestatesaretreatedinexactlythe samewayin
ThemaximumoftheT versuselectricfieldenergycurves the currenttheory,but arecharacterizedby smalldipole
c
9
band filling band filling
0 0.5 1 0 0.5 1
0.2
U/4t = 0 0.01
U/4t = 0.2
U/4t = 0.4
0.15
U/4t = 0.6
0.001
U/4t = 0.8
4t U/4t = 1.0 T / 4tc rr//aa == 11.25
T / c 0.1 0.0001 r/a = 1.5
1e-05
0.05
1e-06
0
0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25
E / 4t E / 4t
sp sp
FIG.10: Charge transfer excitations: transition temperature FIG. 11: Atomic excitations (dsp = 1 ˚A, ∆sp = 10 eV, ǫ =
Tc/4t versus electric field energy Esp/4t and band filling n, 100): transitiontemperatureTc/4tversuselectricfieldenergy
lower axis and upper axis, respectively. A local electronic Esp/4t, parameterized by r/a: Vsp/4t = 2.125 (corresponds
interactionU (withinlayerL2)isincludedinaweakcoupling to r/a = 1), Vsp/4t = 1.36 (r/a = 1.25), Vsp/4t = 0.94
evaluation. Thedielectricconstantǫis100andVsp/4t=1.89 (r/a=1.5).
(r/a=1.5).
insulator,wecalculatedthesuperconductingcriticaltem-
lengths d ∼ 1˚A and large 2-level splittings ∆ ∼10
sp sp perature, which displays as a function of applied field a
eV. Polarizing such dipoles is difficult and T′s are cor-
c steepinitialriseandsubsequentdecay. Theriseiscaused
respondingly small. Comparison of Fig. 11 for parame-
by the increasing density of charge carriers swept to the
ters consistent with atomic polarizations with Fig. 9 for
interface by the electric field and the decay is due to the
charge transfer excitations makes the distinction of the
interaction of the field-induced dipoles with the charge
two scenarios evident. In the atomic case, T is so small
c carriers. The optimal values of the field or gate voltage,
thatanobservableeffectmaybefoundonlyforthesmall-
as well as sample dimensions and dielectric properties,
est value of r/a. Moreover, the excitation energy is so
were discussed in some detail within the framework of
highthatadecreaseofT isnotseen,evenforthelargest
c this model.
electric fields. Of course, when band filling is above half
The goal of these model calculations was to establish
filling, the transition temperature will decrease with in-
theplausibilityofsuperconductivityinfield-effectdevices
creasing field. However these field strengths are beyond
enhanced or induced by the presence of a polarizable in-
electrical breakdown.
terface, and to investigate the magnitude of the criti-
cal temperature and likely dependence on external field
and materials properties. We recognize, however, that
III. CONCLUSIONS
several potentially important aspects of the physics are
not present in the model used. These include long-range
We have proposed that the field-induced 2DEG in a Coulomb interactions, dynamics of the dielectric screen-
field effect device may become superconducting entirely ing,andcorrelationsinthe2DEG.Weanticipatethatthe
duetopairinteractionswithaproximateinsulatinglayer effectsoflong-rangeCoulombinteractionsarelessimpor-
with high polarizability. In the general case, properties tantinthepresentcasethanin3Dmetallicsuperconduc-
of a DS-channel embedded in an oxide field effect tran- torsduetothelargerdielectricconstantsintheinsulating
sistor will then be controlled not only by the electronic material,butthismustclearlybejustifiedbyalegitimate
andstructuralpropertiesoftheDS-channel,asisusually calculationofthetrueCoulombpseudopotentialentering
assumed, but can furthermore be strongly influenced by the MacMillan formula. Screening of the bare electron-
the gate insulator or by other adjacent dielectric layers. dipole interaction V must also be included; we expect,
sp
The choice of the gate dielectric layer may then influ- however, that the form of the dependence of T on the
c
ence the device behavior of a field effect transistor far electric field will not change significantly (see Fig. 9).
beyond controlling the maximum gate polarization and The inclusion of a repulsive local interaction U does not
gate current. qualitativelychangethis formeither. Howeveritreduces
For a concrete model of electrons in a 2D layer in- T (byafactorof2forU/4t=0.8)anditshifts theopti-
c
teracting with 2-level systems at the interface with the mal value of the field-induced doping to lower levels (cf.
10
Fig. 10). teraction besides the interface-mediated pairing. In this
Ifthe effective on-siteCoulombrepulsioninthe drain- case we expect a qualitatively similar behavior with a
source layer is much larger than we have considered, s- nonmonotonic T .
c
wave superconductivity will be completely supressed. It According to our calculations, devices with thickness
is still tempting, however, to regard the starting Hamil- of the insulating layer of order 1000 ˚A subjected to
tonian (3)–(7) as the hopping of correlated electrons (as voltages of order ∼ 20V could display superconductiv-
in the t−J model) where Coulombinteractions have al- ity, if an optimal material can be found with a suffi-
readybeenaccountedfor. Inthiscase,however,onemust ciently large low frequency dielectric constant of order
implicitly assume that doubly occupied sites have been 30-100 and strong quasilocalized electronic modes of en-
projectedout,sothatnon-retardeds-wavesuperconduc- ergy ∆ /4t ∼ 2.5 [29]. With a 20 V gate field and a
sp
tivity is impossible. Modelling superconductivity within typical 18 nF capacitance of a 0.03 cm2 gate dielectric
this framework in higher-angular momentum pair chan- (1000˚A), this wouldcorrespondto asurfacechargeden-
nels will require including interactions among the two- sityofroughly12µC/cm2. Itisnowroutinetofabricate
levelsystemsattheinterface,soastoproduceanonlocal epitaxialhighǫoxidedielectrics,suchasSrTiO withpo-
3
pair potential. While we have not calculated these ef- larizations in the range of 10-40 µC/cm2 [8]. Interface-
fects explicitly, as they are technically significantly more mediated 2D superconductivity therefore seems to us to
difficult, our expectation is that the creation and even- be plausibly within reach if good interfaces can be man-
tual suppression of the paired state by the electric field ufactured.
will be qualitatively similar to what we have calculated. Acknowledgements. This work is supported by NSF-
Thisexpectationissupportedbytheobservationthatan INT-0340536, DAAD D/03/36760, NSF DMR-9974396,
increase in T with doping has to result trivially from BMBF 13N6918A, by a grant from the A. v. Hum-
c
an enhanced carrier density. Moreover, the buildup of a boldt foundation, by the Deutsche Forschungsgemein-
field-dependentrepulsiveinteractionV hastotakeplace schaft through SFB 484, by the ESF THIOX pro-
z
due to the eventual saturation of the dipole moments gramme,and by the Texas Center for Superconductivity
withelectricfield,evenwhenthetwo-levelsystemsinter- at the University of Houston. The authors are grate-
act. Investigations along these lines are in progress. ful to Y. Barash, A. Hebard, P. Kumar, G. Logvenov,
In addition, our considerations can in principle be ap- D. Maslov, K.A. Mu¨ller, T.S. Nunner, N. Pavlenko,
plied to other, more weakly correlated systems than the C.W. Schneider and D. Tanner for useful conversations.
copperoxides. Ourmodelmaybeusedtotreatsituations We thank A. Herrnberger for his help with the prepara-
inwhichtheDS-channelcontainsanattractives-wavein- tion of Fig. 1.
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