Ginzburg-Landau theory of superconducting surfaces under the influence of electric fields
P. Lipavský,1,2K. Morawetz,3 J. Koláček,2and T. J. Yang41Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague 2, Czech Republic 2Institute of Physics, Academy of Sciences, Cukrovarnická 10, 16253 Prague 6, Czech Republic
3Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
and Max-Planck-Institute for the Physics of Complex Systems, Noethnitzer Str. 38, 01187 Dresden, Germany
4Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan 共Received 13 October 2005; published 23 February 2006兲
A boundary condition for the Ginzburg-Landau wave function at surfaces biased by a strong electric field is derived within the de Gennes approach. This condition provides a simple theory of the field effect on the critical temperature of superconducting layers.
DOI:10.1103/PhysRevB.73.052505 PACS number共s兲: 74.62.Yb, 74.20.De, 74.25.Op, 74.78.⫺w
The critical temperature of a thin superconducting layer is increased or lowered by an electric field applied perpendicu-lar to the layer.1–5 Similarly to the conductivity of inverse layers in semiconductors, superconductivity of thin metallic layers can thus be controlled by a gate voltage, which makes these structures attractive for applications.
In this paper, we show that the phase transition in a thin metallic layer is conveniently described by the Ginzburg-Landau共GL兲 theory, where the electric field E enters the GL boundary condition as
冏
ⵜ 冏0
=冏
ⵜ⌬ ⌬冏
0 =1 b= 1 b0 + E Us . 共1兲Briefly, the logarithmic derivative of the GL function or the gap function⌬ at the surface is a sum of the zero-field part 1 / b0and the field induced correction E / Us.
The zero-field part has been derived by de Gennes6 from the BCS theory. A typical value b0⬃1 cm is large on the scale of the GL coherence length; therefore, this contribution is usually neglected. This approximation, 1 / b0⬇0, corre-sponds to the original GL conditionⵜ= 0.
Here we employ the de Gennes approach to derive the field induced correction E / Us. The correction becomes
im-portant for the above-mentioned experiments, where fields of the order of 107V / cm are realized. Small electric fields ap-pearing, e.g., in Josephson junctions, do not require such corrections.
We start from the condition 1 b= 1 2共0兲 1 N0V
冕
−⬁ ⬁ dx⌬共x兲 ⌬0冋
1 − N共x兲 N0册
共2兲 derived by de Gennes关Eq. 共7.62兲 in Ref. 6兴. Here, N0 is the density of states of a bulk material, V is the BCS interaction, and N共x兲 is the local density of states at position x. The actual gap function⌬共x兲 has a nontrivial profile close to the surface at x = 0, but it has only slow variation at distances exceeding the BCS coherence length0= 0.18បvF/ kBTc. Forx⬃0it is crudely linear⌬共x兲⬇⌬0共1+x/b兲, so that ⌬0is not the true surface value but the extrapolation of the gap func-tion toward the surface. In Eq. 共2兲, we have used the GL
coherence length at zero temperature共0兲=0.740 for pure metals.
In measurements of the field effect on the transition tem-perature, the zero-field term b0 is included in the reference zero-bias transition temperature. Accordingly, we can assume a model of the crystal for which 1 / b0= 0. The simplest model of this kind is a semi-infinite jellium, where for zero field the density of states is steplike, N共x兲=N0 for x⬎0 and N共x兲=0 elsewhere. Using that the gap function is restricted to the crystal,⌬共x兲=0 for x⬍0, one can check that from 共2兲 fol-lows 1 / b0= 0.
Now we include the electric field. According to the Anderson theorem,7 the electric field does not change the thermodynamical properties directly but only via the density of states. The change of the density of states is also indirect. The penetrating electric field induces a deviation ␦n of the
electron density. The density deviation changes the Fermi momentum. Since the density of states depends on the Fermi momentum, its value becomes modified. We express this complicated indirect effect approximatively via a local linear expansion
N共x兲 = N0+N0
n ␦n共x兲. 共3兲
The de Gennes condition共2兲 then reads
E Us = − 1 2共0兲 1 N02V N0 n
冕
0 ⬁ dx⌬共x兲 ⌬0 ␦n共x兲. 共4兲 The actual space profile of␦n in superconductors isun-known. In fact, some of recent measurements suggests that the electric field penetrates deep into superconductors.8 In-terpretation of these observations is not yet settled; therefore, we prefer to assume that the screening in superconductors is similar to the screening in normal metals so that␦n is
non-zero only on the scale of the Thomas-Fermi screening length. The typical Thomas-Fermi length is ⬍1 Å, while the gap function varies on a scale typical to the BCS kernel ⬃0. Accordingly, in the integral共4兲, we can take ⌬共x兲⬇⌬共0兲 and obtain
PHYSICAL REVIEW B 73, 052505共2006兲
1 Us = 1 2共0兲 1 N02V N0 n ⌬共0兲 ⌬0 ⑀0 e. 共5兲
In this rearrangement, we have used the surface charge de-termined by the applied field⑀0E = −e兰0⬁dx␦n共x兲.
The effective potential Us given by 共5兲 depends on bulk
material parameters0, N0V, andN0/n, and on the ratio of the gap at the surface to the bulk value
=⌬共0兲
⌬0 . 共6兲
According to de Gennes estimates,6 the surface ratiois of the order of unity. A heuristic derivation of the field-effect from the GL equation cannot cover this factor.9
It is advantageous to express the effective potential Usvia
the usual parameters of the GL theory. First we employ the BCS relation for the critical temperature kBTc
= 1.14បDexp共−1/N0V兲. The critical temperature depends
on the density of electrons. Comparing alloys with different impurity doping, it has been deduced that the dominant sity dependence enters the critical temperature via the den-sity of states.10 We can thus assume
D/n⬇0 andV /n
⬇0 with the help of which we express the derivative of the density of states via the logarithmic derivative of the critical temperature. Formula共5兲 then simplifies to
1 Us =2ln Tc ln n e mc2. 共7兲
Here, we have expressed the electron density via the London penetration depth2共0兲=m/共0ne2兲. Its ratio to the GL co-herence length defines the GL parameter=共0兲/共0兲.
Let us estimate the effective potential Usfor niobium. The
charge carriers are electrons, e = −兩e兩, with the mass close to the electron rest mass, m⬇1.2me. The GL parameter is on
the edge of the type-I and -II materials, = 0.78, and the logarithmic derivative is of moderate amplitude, ln Tc/ln n = 0.75 共see Ref. 11兲. Taking ⬇1, one finds
Us= −1.3⫻106V. As one can see, a large field E
⬃106V / cm is necessary to create a field-induced correction at least comparable to the commonly neglected zero-field value 1 / b0⬃1/cm.
The effective potential共7兲 is the major result of this paper. Now we use it in the boundary condition共1兲 to evaluate the transition temperature T* of a biased layer of a finite thick-ness L. General steps of our analysis parallel the theory of the Little-Parks effect.12 It is also in a close analogy to the theory of surface superconductivity in short coherence length materials.13
Let us assume that the electric field is applied only to the left surface at x = 0, whereas the right surface at x = L is free of the field. We take 1 / b0= 0 for simplicity, so that we use the boundary conditions
冏
ⵜ 冏
0 = E Us , 共8兲冏
ⵜ 冏
L = 0. 共9兲The GL function is given by the dimensionless GL equa-tion 2共T兲ⵜ2+−兩兩2= 0 共see Ref. 12兲. The transition point is characterized by an infinitesimally small GL func-tion,兩兩2→0. At the transition temperature T*, the nonlinear term in the GL equation thus vanishes2共T*兲ⵜ2+= 0. This equation is solved by共x兲⬀cos关共x−L兲/共T*兲兴, which satis-fies the right boundary condition共9兲 while the left boundary condition共8兲 demands L 共T*兲tan
冉
L 共T*兲冊
= EL Us . 共10兲When the superconductor has a coherence length that satisfies the condition 共10兲, the nonzero GL wave function nucleates and the system undergoes a transition to the super-conducting state. Since the coherence length is a function of temperature,
共T兲 = 共0兲
冑
1 − TTc
, 共11兲
one can find from共10兲 and 共11兲 the transition temperature T*. It reads T*= Tc− Tc 2共0兲 L2 g
冉
EL Us冊
, 共12兲where the function g共兲 is a root of
冑
g tan冑
g =. The func-tion g is plotted in Fig. 1.Although Eq.共12兲 is simple by itself, we find it useful to discuss its asymptotic solutions. Let us start with the experi-mentally most important limit. The effect of the electric field on the transition temperature is rather small and most conve-niently observed on very thin layers. In this case, 兩EL兩 Ⰶ兩Us兩 and one can use the linear approximation g共兲⬇
shown as the dashed line in Fig. 1. Within linear approxima-tion, the transition temperature共11兲 simplifies to
FIG. 1. The dimensionless shift of the transition temperature due to the electric field as given by Eq.共12兲. The exact solution of
冑
g tan冑
g = 共thick full line兲, the linear approximation g⬇ for thinlayers 共tangential dashed line兲, the constant approximation g ⬇2/ 4 for large suppressive fields 共dotted-dashed line兲, and the parabolic approximation g⬇−2for large supportive fields共thin full line兲. The inset shows a detail of the parabolic approximation.
BRIEF REPORTS PHYSICAL REVIEW B 73, 052505共2006兲
T*= Tc−
Tc
n
⑀0E
eL . 共13兲
Formula 共13兲 shows that surface ratio 共6兲 reduces field effect on the transition temperature T*. One can compare 共13兲 to a simple estimate that assumes that the induced charge is homogeneously distributed across the layer,
e␦n共x兲⬇−⑀0E / L. A bulk critical temperature modified by the
excess charge is then interpreted as the transition tempera-ture, T*= Tc+共Tc/n兲␦n. Apparently, the simple estimate
does not include the surface ratio.
Based on the simple estimate, one is tempted to say that Tc/n can be uniquely determined from experimental data
on T*. The present theory shows, however, that surface ratio obscures the observed value.
Among high-Tcsuperconductors there are many materials
of large . From Eq. 共7兲, one can see that these materials have much lower effective potential Us; therefore, they
re-veal a much stronger field effect on the transition tempera-ture. With these materials it is possible to achieve the oppo-site limit—the regime of thick layers兩EL兩Ⰷ兩Us兩.
A measurement in the regime of thick layers has been already performed by Matijasevic et al.14We will use param-eters of their sample to evaluate effects expected from for-mula 共12兲. The sample Sm0.7Ca0.3Cu3Oy is overdoped with
Tcreduced to 50 K. The carriers are holes, e =兩e兩, of the mass
m = 3.46meand the density n = 5.75⫻1021/ cm3. Based on the
authors claim that Tcin a monolayer would increase by 10 K
at the imposed voltage, one can deduce ln Tc/ln n
= −3.12. With a typical GL parameter= 100, one obtains the GL coherence length共0兲=1.3 nm and an effective potential
Us= −56.6 V. Note that this is by four orders of magnitude
smaller than the niobium value. The applied field is enhanced by a large dielectric function to an effective value E = 7.8 ⫻107V / cm. For a sample width of L = 50 nm and⬇1, we find 兩EL/Us兩=6.9, which confirms that these measurements
are in the thick layer limit.
In the thick layer limit, one has to distinguish whether the electric field supports or depresses the transition temperature. Let us first discuss the depression that appears for Tc/n
⫻⑀0E / e⬎0. In this case, EL/Us⬎0 and the function g
ap-proaches the constant asymptotic value g→2/ 4 = 2.47 shown as the dashed-dotted line in Fig. 1. Since g⬍2/ 4, relation共12兲 yields the upper estimate of the depression of the transition temperature
Tc− T*⬍ Tc
22共0兲
4L2 . 共14兲
One can see that the maximal depression is limited by the layer thickness. Within the adopted approximations, the ac-tual value of the electrostatic field does not matter once the asymptotic regime is reached. For parameters of Ref. 14, one finds from formula共14兲 the lower estimate Tc− T*⬍0.08 K.
Formula共12兲 gives a slightly smaller value, as one can guess from Fig. 1. For 兩EL/Us兩=6.9, the dimensionless shift is
g共6.9兲=1.89, i.e., T*− Tc= −0.064 K. Matijasevic et al.14
re-ported no suppression of the superconductivity. Our estimate
shows that the suppression is below the sensitivity of their method.
A different situation is met if the direction of the electric field is reversed. The electric field then supports the super-conductivity since 共Tc/n兲⑀0E / e⬍0. In this case, EL/Us
⬍0 and the function g approaches its quadratic asymptotics,
g→−2 shown as the thin full line in Fig. 1. Since g = L2/2共T*兲, the coherence length 共T*兲 is imaginary giving the GL function exponentially decaying from the biased sur-face. In this limit,
T*→ Tc+ Tc
E22共0兲
Us2
, 共15兲
the critical temperature does not depend on the width of sample and increases quadratically with the electric field. For parameters of Ref. 14, we obtain T*− T
c= 1.6 K in a
reason-able agreement with the reported shift by 1 K.
The increased critical temperature共15兲 is independent of the layer width L. This shows that in this limit the supercon-ductivity above Tcis stimulated in the same way as the
su-perconductivity on the surface of an infinite sample predicted by Shapiro.15 Formula 共15兲 differs from Shapiro’s formula 共9兲 only by a factor due to the impurity limited coherence length assumed in Ref. 15.
We note that the present discussion does not account for the charge reservoirs typical to layered CuO materials. For a microscopic study devoted to these materials, see Ref. 16. We also do not assume an eventual effect of the electric field on the chemical composition, e.g., due to oxygen motion as proposed in Refs. 17 and 18.
In summary, using the de Gennes approach we have de-rived the GL boundary condition for a superconducting sur-face exposed to the electric field. This boundary condition allows one to conveniently evaluate the field effect on sur-face sensitive phenomena from the GL theory. Its implemen-tation is demonstrated for the field effect on the supercon-ducting phase transition in metallic layers. Our approach recovers known features, in particular, that for thin layers the transition temperature can be linearly enhanced or sup-pressed depending on the orientation of the applied field. We have found, however, that compared to former theories the linear coefficient is modified by the value of the gap at the surface. In the limit of thick layers, we obtain a field-induced surface superconductivity with the shift of the critical tem-perature depending on the square of the electric field. We also obtain the upper limit on the suppression of the critical temperature being independent of the field and inversely pro-portional to the square of the layer width. These features agree with the experimental data.
The authors are grateful to P. Martinoli who brought this problem to their attention. This work was supported by Grant No. GAČR 202/04/0585, No. GACR 202/05/0173, and No. GAAV A1010312 by the National Science Council of Taiwan, with Grant No. NSC 94-2112-M-009-001, and by DAAD Project No. D/03/44436. The European ESF program AQDJJ is also acknowledged.
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