Electric field effect on superconductivity at complex oxide interfaces

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by  Jason T. Haraldsen

Electric field effect on superconductivity at complex oxide interfaces
Jason T. Haraldsen1,2, Peter W¨olfle3, and Alexander V. Balatsky1,2
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and
3Institute for Condensed Matter Theory and Institute for Nanotechnology,
Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
We examine the enhancement of the interfacial superconductivity between LaAlO3 and SrTiO3
byaneffectiveelectricfield. Through thebreakingofinversion symmetryat theinterface, weshow
2
thatatermcouplingthesuperfluiddensityandanelectricfieldcanaugmentthesuperconductivity
1
transition temperature. Microscopically, we show that an electric field can also produce changes
0
in the carrier density by relating the measured capacitance to the density of states. Through the
2
electron-phonon induced interaction in bulk SrTiO3, we estimate thetransition temperature.
n
a
J I. INTRODUCTION electric field.18 These studies show that T can be in-
c
4 ducedand/orenhancedbythe creationofcarrierspulled
from a source (i.e. LAO, oxide gel, or doping).15,16,18
Superconductivity is typically thought to be produced
]
n by the pairing of electrons below a critical temperature The nature of the superconducting state at the in-
o T .1 The overall pairing interaction is dependent on a terface is a subject of ongoing discussion. One pos-
c
c subtle balance of the repulsive Coulomb interaction and sibility is that the superconducting state is conven-
-
r any attractive interaction supplied by the exchange of tional electron-phonon driven, but there is also a grow-
p collective excitations in the system.1,2 That balance de- ing list of possible unconventional states including
u pends,inacomplexway,onthenumberofcarrierswithin magnetism driven ones.19,20 Indeed recent data from
s
. the material.2–4 This effect becomes all the more impor- scanned probes point to a complicated spatial pattern
t
a tantinthecaseofspatialinhomogeneity,e.g. ataninter- of coexisting superconductivity and magnetism at the
m face, where the spatialdistribution of the carrierdensity interfaces.21 Early reports on ferromagnetic order coex-
- will be important.5,6 Therefore, it is of great interest to istingwithsuperconductivity22 havebeenindependently
d have a tuning parameter like an external electric field confirmed.23 For an assessment of the present situation
n
producing a change to the carrier distribution and effec- see Ref. [24].
o
c tively maximizing the transition temperature Tc. For the following, it will be important to know the
[ In recent years, oxide heterostructures have attracted thickness of the mobile electrons layer at the interface.

2 muchattentionduetoarichspectrumofunexpectedbe- Early estimates of the 2-D density of mobile carriers
v havior discovered at interfaces.7,8 Specifically, the emer- have yielded values much below the theoretical predic-
6 gence of a 2-D electrongas, that becomes superconduct- tion of 0.5 electrons per unit cell.25 Apparently, most of
6 ing ataround200mK,atthe epitaxially growninterface the added charge is not mobile and is trapped within a
4 of the insulator SrTiO (STO) and a few atomic lay- few unit cells of the surface, as observed by high-energy
5 ers of LaAlO (LAO) (i3llustrated in Fig. 1) has sparked photoemission experiments,26 while the mobile electrons
. 3
2 an intense search for the mechanism of superconductiv- may cover a region of thickness 10 nm or more.
1 ity aswell as for improvedmaterials.7–11 Studies investi- Therecentexperiments21suggestthatthespatialsepa-
1
gating the electron mobility have demonstrated that in- rationofsuperconductingandferromagneticregionsmay
1
trinsic doping may be key to understanding the under- allow spin singlet pairing to coexist with ferromagnetic
:
v lying mechanisms.12 While the superconductivity occurs domains. The spatial extension of the density of mobile
i
X at much lower temperatures than in high-Tc supercon- carriersnormal to the interface over severaltens or even
ductors, this finding shows the importance of investigat- hundreds of unit cells speaks againsta prominent role of
r
a ing interfacial physics in heterostructures. Even though the on-site Coulomb interaction and therefore a pairing
there are limitations due to lattice matching and strain mechanisminduced by magnetic fluctuations. All ofthis
withinthematerials,13,14thecouplingbetweenthesema- points to a conventional s-wave superconducting state.
terialsmayprovideinformationthatiscriticaltotheun- In this study, we investigate i) the effect of an elec-
derstanding of these heterostructures. tric field applied externally on the 2-D and 3-D carrier
Further studies of the LAO/STO system have shown density. Wecalculatethevariationofthe2-Dcarrierden-
that T can be increasedby an applied electric field.15,16 sity, capacitance, and the 3-D carrier density depth pro-
c
Bulk STO has been shown to become superconduct- file with increasing electric potential for LAO/STO het-
ing with doping by oxygen vacancies and by niobium erostructures. Fromchangesoftheoverallcarrierdensity
(Nb) at about the same temperature as observed in the attheinterface,wededucethatthenonlinearcapacitance
LAO/STO system.17 Furthermore, STO has been found of the layer at the interface implies an energy depen-
to exhibit interfacial superconductivity when brought in dent Density of States (DOS); ii) using modifications of
connect with an oxide gel in the presence of an external charge carrier density and DOS as a tunable parameter,
2
FIG. 1: An illustration of LaAlO3/SrTiO3 interface, which
detailstheinterfaceregion. Theinterfacial(lightercolor)area
is asmall region from −δz to δz. Thebulk (darkercolor) re-
gionisthatvolumethatisessentiallyunaffectedbytheinter-
face interactions. For the purpose of this study, we examine
theeffectofanappliedelectricfieldproducedthroughoutthe
sample. Therefore, all calculations are to mimic this general
setup.
we calculate changes in superconducting transition tem-
perature T . We assume the pairing to be conventional
c
s-wave pairing as identified in detailed studies of doped
bulk STO.17 Using this modelassumption,we showthat
the T for the band insulator STO can be modified by
c
changes in charge density, and we find that T does ex-
c
hibit a maximum ata certaincarrierdensity determined
bythepeakinthecapacitance. Weshowaplausiblecon-
nectionbetweentheincreaseincarrierdensityandTcdue FIG. 2: Model calculation for the shifting electrostatic dif-
toanappliedelectricfield. Thisobservationcouldleadto ference potential (EDP) due to an applied electric potentials
a possible mechanism for a tunable functional response: ranging from 0 V to 10 V using a semi-empirical, extended
dramaticchangesinthesuperconductingpropertieswith Hu¨ckel model for the LAO/STO interface.30,31 The model
the application of external fields. shows the diffusion of the potential (approximately 1 unit
cell per 2V) from the LAO region into the STO region (the
arrows are to help guide the eye). In thesimulation, thelast
two unit cells are used as the electrodes (LAO- cathode and
II. FIELD EFFECT ON THE
STO-anode)producinganoverallelectricpotentialoverthe
GINZBURG-LANDAU FREE ENERGY
sample region.
The breaking of inversion symmetry at the interface
of heterostructures introduces the interface normal vec- boundaryconditionsandstrain(asshowninFig. 1), the
tor as a preferred direction of the system.27 The normal standard heterostructure free energy can be broken into
component of the electric field F is therefore compat- twomaincomponentsconsistingofbulk(B)andinterfa-
z
ible with the symmetries of the free energy, allowing a cial (I) regions. The effect of the interface decays off as
coupling term of electric field and superconducting den- one moves into the bulk. In the case of the LAO/STO
sity, F ψ 2, with the superconducting order parameter interface, the total free energy FE is then given by
z
| |
ψ. Thisisasimpleconsequenceofscreeningeffectonthe
δz 0
charge density within the superconductor.28 The charge FE = − F(z) dz+ F(z) dz
tot BSTO ISTO
distribution at the interface would be dependent on the Z−L Z−δz (1)
distancefrominterface,typicallyontheorderofnm. On
δz L
theotherhandtherelevantsuperconductinglengthscale
+ F(z) dz+ F(z) dz,
is set by a fairly large coherence length ξ/2, typically on ILAO BLAO
Z0 Zδz
the scale of tens to hundreds of nm.3 Therefore, we may
where the depth-dependent free energy for the bulk and
use a Ginzburg-Landau(GL) functional in our approach
interface are the given by
when we address the coupling between SC and electric
degrees of freedom.2,4 β
Since the interface has the additional constraints of FE(z)B =αB ψ(z)B 2+ B ψ(z)B 4 (2)
| | 2 | |
3
as
FE(z)ψI2 =αI|ψ(z)I|2+λFz|ψ(z)I|2+γη2|ψ(z)I|2
1.2 0.0
=α˜ ψ(z) 2.
-0.5 | I|
1.0 (4)
F) 0.8 --11..502Dn ( wsuhpeerrecoα˜n=duαctIi+ngλFstza+teγ,ηw2h=ena˜(α˜T<−T0c02–)4a.ndThdeersecfroibree,stthhee
n 1 shifted critical temperature is given by
( 0
C 0.6 --22..50 cm 13 T =T0 λcFz +(γη2)c (5)
0.4 ) -2 c c − a˜
(cid:18) (cid:19)
-3.0
0.2 -3.5 where λc and (γη2)c are taken at T =Tc0. From this we
conclude that the electric field has to compete with the
0.0 -4.0 lattice strain. Even a large lattice strain may be over-
-200 -150 -100 -50 0 50 100 150 200
compensated by a reasonably strong electric field, which
V (V) therefore can still effect the superconducting order pa-
rameter. However, if the lattice matching is close, then
FIG.3: Simulatedcapacitanceand2-Dcarrierdensityn2d as the strain can be assumed to have a negligible effect on
afunction ofelectric potentialfortheLAO/STOinterfacein the superconducting state. Recent measurements indi-
Ref. [15]. It should be noted that the change in curvature cate the various degrees of lattice mismatch may even
denotes an effective change in electron or hole doping. produce an increase in the superconducting T .29
c
Thus, the addition of extra quadratic terms to the
free energy modifies the critical temperature. While the
straintermsusuallyproduceanegativeshiftofT ,asuit-
c
and ably directed electric field could gives rise to a positive
shift. Thismaybeinterpretedasbeingduetoanincrease
in the carrier density of the superconductor. While this
FE(z)I =αI|ψ(z)I|2+ β2I|ψ(z)I|4+g|▽ψ(z)I|2 is shown here only in the phenomenological GL model,
(3) we will now turn to a modeling of the charge density in-
+λFz ψ(z)I 2+γη2 ψ(z)I 2, duced by an applied electric field and show how that is
| | | |
correlated with T .
c
whereα andβ arethestandardGLenergyparame-
B,I B,I
ters. InthestandardGLformalism,theparameterα = III. SEMI-EMPRICAL MODELING OF THE
B
a (T T0),witha>0,determinesthenormal(α >0) ELECTROSTATIC DIFFERENCE POTENTIAL
B − c B
andsuperconductingstates(α <0),whileβ isalways
B B
positive and helps stabilize the ground state. To examine the change in the electron density at an
LAO/STO interface, we calculate the electrostatic dif-
For the interface region, the gradient and strain η
ference potential (EDP) φ using a semi-empirical, ex-
terms are produced by the interface and are typically ∗
tended Hu¨ckel model. The EDP is described as the dif-
positive, where the gradientproduces the boundary con-
ference between the electrostatic potential φ of the self-
dition for the interface and can be assumed to be in the
consistent valence charge density and the electrostatic
small wavevector limit. For the field interaction, we use
F = (0,0,F ). The coefficient λ will in general depend potential from atomic valence densities.30,31 Therefore,
z the EDP is determined by solving the Poisson equation
onthe electricfield strength. Therefore,the localcarrier
2φ = ρ /ǫ ,whereisρ istheelectrondifferenceden-
densitycanincreaseordecreasedependingonpositiveor ∇ ∗ − ∗ r ∗
sity.
negative F , respectively. It should be noted that while
z Figure2showsthe calculatedEDPforvariousapplied
the electric field is global, only the superconductivity at
electricpotentials. Thiscalculationisproducedbybuild-
theinterfacewillincreaseduetotheaccumulationofcar-
ing a three dimensional interface of LAO and STO that
riers from the carrier source. This may be different for
is3x3x 8unitcellsforeachside. Theelectricpotential
a high temperature superconductor,but giventhat STO
is produced by simulating a device where positive and
is an insulator, the distribution of carriers is lessened.
negativeelectrodesareplaceontheendsoftheinterface.
Focusing on the effects of the additional quadratic For this simulation, the LAO electrode is negative and
terms, it can be easily shown that the standard GL def- the STO electrode is positive. As the electric potential
initions are changed. The inclusion of the electric-field is increasedfrom0Vto 10V,the EDP is observedto dif-
interaction and strain terms induces a shift in T . Con- fuse fromLAO into STO. This details the shifting of the
c
centrating on the ψ2 terms, we re-write the free energy electron charge density through the interface. To gain
4
a more detailed view of the change in the charge and gularity and is discontinuous at F = A/B. The singu-
−
carrierdensities,weexaminethegeneralelectrostaticsof laritycanberemovedbyexpandingoutthedenominator
the interface and compare to known experimental data. to aquadraticterm,producing acontinuousfunction for
all F,
1
IV. CARRIER DENSITY AND CAPACITANCE ǫ (F)STO = (7)
r A C +C BF +C B 2F2
1 2A 3 A
The basic connection of the charge density and an ap- (cid:16) (cid:0) (cid:1) (cid:17)
where C (i = 1, 2, and 3) are determined from experi-
plied electric field is givenby integrating Gauss’s law for i
mental fits. From this, we find that
a dielectric medium, keeping the nonlinear dependence
ofthedielectricfunctionǫ(F)=ǫ ǫ (F)onelectricfield.
This function may be determined0thrrough the measured n (F)= 4ǫ0 tan−1 (2C3AB√FC+4C2A) −tan−1 √CC24
capacitance at the LAO/STO interface. The change in 2d (cid:16) (cid:16) eB√C (cid:17) (cid:16) (cid:17)(cid:17)
4
the 2-D carrierdensity n2d in a semiconducting channel, (8)
induced by an electric field is then given by where C =4C C C 2.
4 1 3 2
−
Toexaminethemodulationofthecarrierdensityalong
2ǫ0 F the LAO/STO interface, we compare to data provided
n (F)= ǫ (F)dF (6)
2d e r by Ref. [15]. Here, a gate voltage is used to obtain
Z0
an electric field, where F=F + V/d, V is the electric
0
where F is the effective electric field within the sample, potential and d is the sample thickness (0.5mm). The 2-
ǫ0 is the permittivity of free space, ǫr(F) is the electric D carrierdensity is also relatedto the capacitanceC(V)
field dependent relative permittivity, and e is the basic by
unit charge. The field dependence of the relative per-
mittivity for STO has been shown to be ǫ (F)STO = ∂δn2d
r C(V)=eS , (9)
1/A 1+ BF , where A and B are temperature depen- ∂V
A (cid:18) (cid:19)
dent variables determined experimentally (4.097x10 5
and(cid:0)4.907x10(cid:1)−10 m/V for STO at 4.2K, respectively−). nwhe(r2e00SVi)s(tmheeaasruerainogfVcaipnacVitooltrs)a.ndTδhne2cdro=ssn-s2edc(tVio)n−al
This howeverproduces a carrierdensity which has a sin- 2d
areaofthe capacitorisdeterminedby adjustingthe sim-
ulated peak capacitance to match the the observed ca-
pacitance. Using this method, the cross-sectionalarea is
determined to be about 5 mm2. Therefore, the capaci-
tance will be

1020 meV)1102EF(meV)=4.30+0.0 410V-5.05x10-5V2 C(V)= Ad C1+C2BA(F0+ 2Vdǫ)0S+C3 BA 2 F0+ Vd 2 .
E(F8 (cid:16) (cid:0) (cid:1) (cid:0) ((cid:1)10(cid:17))
) 1019 6 DQautaadratic Fit Comparing to the data in Ref [15] and adjusting only
-3 m 40 50 100 150 200 the Ci parameters, Figure 3 shows the capacitance and
V (V)
(c 2D carrierdensity as a function of gate voltagefor STO.
n 3d1018 012 00V00VV Ttehnrdoeudghpafirtatminegteorfst(hCe1,δnC22d, daantdaCfr3o)marReedf.e[t1e5r]m, itnheedetxo-
be4.25,-0.37,and0.29,respectivelywithavalueofF =
0
1.2x105 V/m. Here, we find that the change in the 2-D
carrierdensityshiftsfromelectrontoholedopingaround
1017 30 V. This produces the corresponding peak in the ca-
0 5 10 15 20 25 30 35 40
pacitance. Further analysis in Section V will correlate
z (nm)
the capacitance to the change in T .
c
To understand the accumulation of charge at the
LAO/STO interface as a gate voltage is applied, we ex-
FIG. 4: Simulated 3-D carrier density n3d as a function of amine the 3D carrier density as function of depth. The
depthfor various electric potentials (0V - solid black, 100V - depthprofile ofthe chargeaccumulationinSTO is given
dashed red, and 200 - dotted blue) for the LAO/STO inter- by summing over the density profiles ξ (z)2 of all oc-
i,j
face. Thisdescribestheincreaseintheoverallcarrierdensity | |
cupied states
at the interface as the electric potential is increased. As this
occurs, the relative interfacial region reduces in depth. The
g m
einlescettrischopwotsenthtieale.ffAectfiitvteinFgeormf tiheisnedragtyaEtoFaaqsuaadfruantcictiopnolyo-f n3d(F,z)= 2jπ~2∗j (∆Ei,j)Θ(∆Ei,j)|ξi,j(z)|2!,
j=l,h i
nomial providesa general functional form. X X (11)
5
where ∆E = E E , and the sum over j = l,h is
i,j F i,j
−
over the light and heavy electronic bands in STO. The
inner sum is over the occupied states labelled i,j for the
energy of eigenstate E and Fermi energy E , where 0.40
i,j F 0.4
the Θ function assures that all states E < E are
i,j F 0.35 K)0.3
included. m∗j is the effective mass (m∗l = 1.2m0 and T (c0.2
m∗h = 4.8m0), m0 is the electron mass, and gj = 1,2 is 0.30 0.1
rtheperbesaenndteddegbeynaertarciya.n1g6,u32laTrshheacpoenafinndintghpeoctoernrteisaploinsdwinelgl K) 0.25 0.-0200 -100 0V (V1)00 200 300
(
eigenstates are therefore normalized Airy functions: T c0.20
E 0.15
i,j
ξ (z)=n Ai α z , (12)
i,j c j
− eF 0.10
(cid:20) (cid:18) (cid:19)(cid:21)
where n is the normalization constant in units of 0.05
c
(length)−1, Ei,j = eαFj 32π i− 14 2/3 and αj = 0.00-200 -150 -100 -50 0 50 100 150 200
2m∗jeF/~2 1/3. (cid:0) (cid:0) (cid:1)(cid:1) V (V)
Figure 4 shows the n electron density as a function
3d
(cid:0) (cid:1)
of z. Here, the the Fermi energy E (V) is determined
F
self-consistently by calculating
FIG. 5: Calculated superconducting transition temperature
EF(V) Tc as a function of electric potential for the LAO/STO in-
n (V)= n (z)dz = N (E,V)dE (13) terface. Here, it is shown that due to the increase in carrier
2d 3d 2d
Z Z0 density, there is a distinct change in Tc. Eventually, at some
critical potential,theCoulomb repulsion takesoverandTc is
where N2d(E,V) is the density of states reduced. Inset shows the experimentally observed change in
Tc from Ref. [16] (dashed blue) and Ref. [15] (dotted red).
gjm∗j Thedifferenceinpeakwidthandlevelcanvarydependingon
N (E,V)= Θ(∆E ) , (14)
2d 2π~2 i,j the pair interaction’s dependence on V as well as the exact
!
jX=l,h Xi determination of ∂EF(V)/ ∂V.
andcomparetothevaluefromEq. 8. Wecanalsodeter-
minetheaveragen3dbyhn3di= n23ddz/ n3ddzandthe EF(meV)=4.30+0.0410V-5.05x10−5V2. Therefore,we
average thickness d = n zdz/ n dz. To compare
av 3d 3d can used the derivative to find an approximation to ∂V
R R
the DOS to the measured capacitance, we consider
/ ∂E (V). The integralover N (E,V) is dependent on
R R F 2d
the electric-field dependence of the effective masses and
δn (V) δE (V)
2d F
δV =N2d(EF(V),V) δV Ei,j. However, it is assumed that this contribution is
small within the confines of the interface.
(15)
Ef(V) δ
+ N (E,V)dE
2d
δV
Z0 V. EFFECT ON SUPERCONDUCTING
Here, it is shown that as the gate potential is increased, TRANSITION TEMPERATURE Tc
the carriers in the system are increased at the interface.
This increases the overalldensity of carriesand shortens To explore the consequences of the change in carrier
the interfacial depth. By using the above determined density induced by an applied electric field on the su-
Fermi energy E (V), the density of states in 2d may be perconducting transition temperature. We exploit the
F
relatedtothederivativeofn (V)withrespecttoV and fact that the change in the DOS will modify the (di-
2d
hence to the capacitance as mensionless)pair coupling. We conjecture that the pair-
ing is similar to the one observed in bulk doped STO.17
N2d(EF(V),V)= ThereweakcouplingBCStheoryprovidesaquantitative
description of the observed transition temperature as a
∂n Ef(V) δ ∂V function of doping. Within weak coupling theory, the
2d
N (E,V)dE =
" ∂V −Z0 δV 2d #∂EF(V) (16) csuriptiecraclontedmucpteirnagtuisregiTvcenbeblyow which a material becomes
1 Ef(V) δ ∂V 1
C(V) N2d(E,V)dE Tc =TDe−λpN(EF), (17)
"eS −Z0 δV #∂EF(V)
whereT istheDebyetemperature(513KforLAO/STO
D
The inset of Fig. 4 shows E as a function of V. Us- multilayers), λ is the effective interaction , and N(E )
F p F
ing a quadratic polynomial fit to the data, we find that is the electronic density of states at the Fermi level in
6
d=3 dimensions. In the case of bulk doped STO, a de- where R is the sheet resistance and R = ~/e2
sheet Q
≈
tailed theoretical study of the transition temperature in 4.12kΩisthe resistancequantum. Herethe functionf is
theframeworkofEliashberg’stheoryhasbeenperformed defined in terms of the gap parameter ∆(T) as
Ref. [33]. The conclusion was that inter- and intra-
valley electron-phonon processes as well as Coulomb re- T ∆(T) ∆(T)
f = tanh (20)
pulsioncontribute todetermining T . These calculations
c T ∆(0) 2T
accountedquantitativelyfortheobservedmaximumofTc In the limit of(cid:18)weca(cid:19)k disorder, TKT(cid:20)B T(cid:21)c. At strong
asa functionofcarrierdensity. The initialincreaseofTc disorder, when Rsheet >> 2RQ one≈gets a suppres-
at low density is connected with the increase in the den- sion of the KTB transition temperature as T =
KTB
sity of states, while the decrease beyond the maximum 2.18T R /R .
c Q sheet
may be tracedto the weakeningof the attractioncaused
by enhanced screening. Comparing the values of density
n found above with the density at the maximum of
3d
T in Ref. [33], nm = 1020cm 3 we conclude that our VI. CONCLUSION
c 3d −
systems are on the low density side, where density de-
pendent screening effects of the pair interaction are less In conclusion, we examined the effect of an electric
relevant. We therefore assume that the pair interaction potential on the carrier densities and superconducting
λp may be taken as a constant, which we deduce from transitiontemperatureforthe conducting LAO/STOin-
the dimensionless coupling, λpN(EF) = 0.13 and the terfaces. We find that the carrier density and supercon-
maseaλspured6.D5OS1N0−(4E9FJm)=3.2H×ow10e4v7eJr−,1tmhe−3paaitrdinetnesriatyctniom3nd dexutcetrinnaglteelmecpterircatfiuerled.caRnecbeenstilgy,nitfihcearnethlyavmeobdeiefinedexbpyearin-
≈ ×
may also have a dependence on the applied potential.As mental observations that have explored the relationship
shownabovethe2delectronicdensityofstatescanbedi- between an applied potential on STO heterostructures
rectlymeasuredthroughthecapacitance,andthedepen- and the increased carrier density or capacitance. Here,
dence of the Fermi energy on applied voltage, while the wetakethisrelationshipfurtherandassumeitisthefield
latter can be inferred from the observed density-voltage effect that, through nonlinear capacitance, changes car-
relation. Therefore, the 3d DOS can be approximately rier density. That in turn affects the superconducting
expressed as coupling and ultimately T . We present a mechanism of
c
this phenomenon assuming a conventional pairing state
1 ∂V
N(E (V),V)= C(V) , (18) at the interface, which shows a possible route to tunable
F
Sedavγ ∂EF(V) material properties and suggests an exciting perspective
on the way to tunable superconductivity.
where d is the average thickness of the electron layer.
av
Thereisasecondtermthatwediscussinthesupplemen-
tary material, but as discussed above, this term is small
and can be ignored. Hence, a change in the capacitance VII. ACKNOWLEDGEMENTS
with applied electric field provides a direct effect on T
c
withinthissystem. Figure5showsthecalculatedT asa
c We wouldlike to acknowledgementhelpful discussions
function of V for the LAO/STO heterostructure, where
with Q. Jia, D. Yarotski, Y. Liu, J-X. Zhu, S. A. Trug-
γ 0.0045 has been adjusted to place the peak value of
≈ man, and Th. Kopp. This work was supported, in part,
T atthe experimentallydetermined0.37K.This details
c by the Center for Integrated Nanotechnologies, a U.S.
the subtle significance of a gate voltage on the super-
Department of Energy, Office of Basic Energy Sciences
conducting state. Inset of Fig. 5 details and compares
user facility and in part by the LDRD and, in part, by
the experimentally observed change in T from Ref. [16]
c UCOPTR-027. Los AlamosNationalLaboratory,anaf-
(dashed blue) and Ref. [15] (dotted red).
firmative actionequalopportunity employer,is operated
The observed transition into the superconductive
by Los Alamos National Security, LLC, for the National
state is most likely of the Kosterlitz-Thouless-Berezinski
Nuclear Security Administration of the U.S. Depart-
(KTB)type. IngeneralthetransitiontemperatureT
KTB ment of Energy under contract DE-AC52-06NA25396.
is related to the bulk transition temperature T by34
c PW gratefully acknowledges the award of a Carl Schurz
Memorial professorship at the University of Wisconsin,
T T R
KTB KTB Q
f =2.18 (19) Madison.
T T R
c . (cid:18) c (cid:19) sheet
1 J. Bardeen, L.H. Cooper, and J.R. Schrieffer,Phys. Rev. 1969).
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