Boundary condition for Ginzburg-Landau theory of superconducting layers

0.16 MB·English

by  Jan Kolacek

Boundary condition for Ginzburg-Landau theory of superconducting layers
Jan Kol´aˇcek1 , Pavel Lipavsky´1,2 , Klaus Morawetz3,4 and Ernst Helmut Brandt5
1Institute of Physics, Academy of Sciences, Cukrovarnick´a 10, 16253 Prague 6, Czech Republic
2 Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague 2, Czech Republic
3Forschungszentrum Dresden-Rossendorf, PF 51 01 19, 01314 Dresden, Germany
4 International Center for Condensed Matter Physics, 70904-910, Bras´ılia-DF, Brazil and
5Max Planck Institute for Metals Research, D-70506 Stuttgart, Germany
9 Electrostatic charging changes the critical temperature of superconducting thin layers. To un-
0 derstandthebasicmechanism,itispossible tousetheGinzburg-Landautheorywith theboundary
0 condition derived by de Gennes from the BCS theory. Here we show that a similar boundary con-
2 dition can be obtained from the principle of minimum free energy. We compare the two boundary
conditions and use theBudd-Vannimenustheorem as a test of approximations.
n
a
PACSnumbers: 74.20.De
J

7
2 I. INTRODUCTION II. FREE ENERGY IN THE

SUPERCONDUCTOR
]
n
o We start with the free energy
c Much experimental effort is devoted to find supercon-
- ducting materials with critical temperatures as high as
r = (f +f +f +f )dr, (1)
p possible. It is well known that the critical temperature F Z TF GC GL elst
u depends on the charge carrier density. The charge car-
s rierdensitycanbechangedbydopingandtosomeextend where we include only the terms most relevant for the
t. it can also be changed by electrostatic charging. Conse- above specified problem.
a
m quentlyitisanattractivetasktodetermine,howelectro- The first term fTF is the Thomas-Fermi internal en-
ergy, for which we use the LDA (local-density approxi-
staticchargingevokedbyanappliedelectricfieldchanges
d- the critical temperature of superconductors. Generally mation)
n the experiments revealed that it is easier to increase T
o than to decrease it1,2. The Ginzburg-Landau (GL) thec- fTF = 3 3π2 23 ~2 n35. (2)
c 5 2m
ory with the de Gennes boundary condition can be used (cid:0) (cid:1)
[
to understand this behavior3. Thesecondtermf representsthecondensationenergy
GC
1 for which we use the formula following from the Gorter-
v Casimir two fluid model5,6
4 Theaimofthispaperistoshowhowasuperconductor
4 screenstheexternalelectricfieldandtowhichextentthe 1 n T2 n
2 boundary condition derived from the minimum free en- fGC = 4γTc2(cid:18) ns +2T2r1− ns(cid:19). (3)
4 ergyprincipleiscompatiblewiththedeGennesboundary c
.
1 condition. The electrostatic energy density term reads
0
1
9 f = ǫ E2+eϕδn, (4)
0 Charges at a solid surface partially leak out of the elst 2 0
: surface. This creates a surface dipole. The Budd-
v Vannimenus theorem4 describes the step in the surface in the form suitable for performing variations. For sim-
i plicity we exclude the magnetic field and it’s related ki-
X potential due to this surface dipole as a simple expres-
neticenergyofthescreeningcurrent. Wetakethevector
ar ssiuointedofttohteesbtultkhefraepepernoexrigmyadtieonnssituys.eTdhinertehfoisrepiatpiesr.well potential A to be zero and write the GL gradient term
as
~2
Inthefirstchapterweexplainthemodelandtheparts f = ψ2 ψ 2. (5)
considered in the free energy of the superconductor and GL 2m∗ n|∇ |
solvetheEuler-LagrangeequationsfortheGLandcharge
HerewehavechosentheGL wavefunctionψ normalized
carrierwavefunction andthe surface potential. Inchap-
with respect to the total charge carrier density n. In
ter III the corresponding equations outside the super-
thespiritoftheThomas-Fermiapproximationthecharge
conductor are solved and in chapter IV the continuity carriersaredescribedbyawavefunctionψ withn=ψ2
n n
requirements determine the remaining constants. Chap-
andthesuperconductingfluiddensityusedintheformula
ter V presents the numerical values which are compared
for the condensation energy (3) reads
with the de Gennes boundary condition in chapter VI.
Finally we conclude in chapter VII. n =2ψ2ψ2. (6)
s n
2
In short, the free energy is expressed by three inde-
pendentvariables: thescalarpotentialϕdeterminingthe
∂lnT
electric field E= ϕ, the GL wavefunction ψ and the 1 t4 ξ2 2δψ+t4δψ+4t4 cδψ =0, (16)
−∇ − t∇ ∂lnn
charge carrier wave function ψn. We assume that the (cid:0) (cid:1)
materialparameters,the criticaltemperature T andthe
c
Sommerfeld parameter γ depend on the charge carrier’s
∂lnT ∂lnT ∂lnγ π2a2ξ2
density n, by using the approximations 2t4 c c +2 B t δψ
(cid:18) ∂lnn (cid:18) ∂lnn ∂lnn(cid:19)− 12λ4 (cid:19) n
TF
γ(n) = γ0(cid:18)1+ n−nlantlat∂∂llnnnγ(cid:19), (7) +2t4∂∂llnnTncδψ− π126aλ2B4TξFt2δϕ=0. (17)
n n ∂lnT
Tc(n) = Tc0 1+ − lat c , (8) Close to the planar surface we can assume exponential
(cid:18) n ∂lnn (cid:19)
lat dependenciesofthedeviationsandfromtheELequations
(15-17) we get a second order equation for the square of
where n is the crystal lattice density. In the following
lat the expected penetration depth λ. Two solutions arise
we shall use three characteristic length:
(i) the Thomas-Fermi screening length λ2TF = 32nǫe02EF, ouOt obfseirtv.ing that λTF,aB ξ0, we find a first approxi-
(ii) the Bohr radius aB = 4πmǫe02~2 mate solution λ in the form≪of the coherence-like length
(iii) and the coherence length ξ02 = 4γ~02Tncl20atm. ξ =ξ 2t2 . (18)
From the charge neutrality requirement we know that t 0√1 t4
−
ψn∞ =√nlat (here and in the following the subscript
∞ In this solution the scalarpotential is constantand local
denotes the magnitude far from the surface). To keep
charge neutrality is preserved. Only the deviation of the
things simple, we use the approximations
wave function δψ is nonzero (C will denote its magni-
ξ
ψ = √n (1+δψ ) (9) tude).
n lat n
A very small penetration depth characterizes the sec-
ψ = ψ∞(1+δψ) (10)
ond solution such that this solution can be simplified.
eϕ = EF(ϕ∞+δϕ) (11) Using the same approximation as above, we find that
the second penetration depth equals the Thomas-Fermi
and suppose that the deviations of the three indepen-
screening length λ . In this solution the scalar poten-
TF
dent variables from the optimum values are small. In
tial displays a sharp step (C will denote its magni-
TF
a homogeneous superconductor far from the surface all
tude)andfromthePoissonequation(15)followsthatthe
these deviations have a zero value and the derivatives of
charge carrier’s density changes accordingly. The sharp
the free energy formula (1) with respect to them must
step on the GL wave function ψ is negligibly small due
be alsozero. Fromthese requirementswegetthe magni- to the factor λ2 /ξ2 which enters the resulting formula.
TF t
tudes of the optimum superfluid fraction
It corresponds to the well known fact that the GL wave
function ψ cannot abruptly change.
ψ∞
= 1 t4 (12) The general solution can be written as a sum of the
ψn∞ p − two above described solutions:
and the optimum magnitude of the scalar potential x
δϕ = CTFexp − (19)
(cid:18)λ (cid:19)
TF
2λ4 ∂lnT ∂lnγ
ϕ∞ = 1+ TF 2 1 t4 c + 1+t4 . 3 x
− π2a2Bξ02 (cid:18) (cid:0) − (cid:1) ∂lnn (cid:0) (cid:1)∂lnn(cid:19) δψn = −4CTFexp(cid:18)λ− (cid:19) (20)
(13) TF
TthhueseelqeucatrlsostthaetiFceprmotienentiearlgyenergy of the charge carrier δψ = (13λ2TtF4t)4ξ2∂∂llnnTncCTFexp(cid:18)λ−x(cid:19)
− t TF
x
E = ~2 3π2n 2/3, (14) +Cξexp(cid:18)−ξ (cid:19). (21)
F t
2m
(cid:0) (cid:1)
Here C describes the step of the scalar potential in
TF
with a small (lower than gap) correction represented by
unitsofE /eaccordingto(11). Using(19)-(21)wecan
F
the second term in (13).
calculate the free energy
Using the second order expansion of the free energy
(1),fromthevariationwegetthreelinearEuler-Lagrange 3 2(1+t4)λ4 3
= + TF dr λ C2
(EL) equations for the three independent variables, F Z (cid:18)5 π2a2ξ2 (cid:19) − 4 TF TF
B 0
43λ2TF∇2δϕ+δψn =0, (15) +8λ4TπFa(12Bξ−t t4)Cξ2+ 4π82ta42Bλ5Tξt2F ∂∂llnnTncCTFCξ. (22)
3
Forasemi-infinitemediumthefirsttermgivesaninfinite Thedensityofthetunnelingchargecarriersquicklydrops
contribution which is not influenced by the surface con- to zero. Using an analogous notation as above we write
ditions, sowe do notneedto dealwithit. The lastthree
ψ =√n δψ (27)
terms correspond to the surface energy, which according n lat n
to the principle of minimum free energy should take an
supposedly that δψ is small. For the wave function ψ
n
extremum. The minimum of the free energy is obtained
we use the approximation
for
ψ =ψ∞+δψ, (28)
3λ ξ ∂lnT
TF t c
C = C , (23)
ξ − 4ξ02 ∂lnn TF where ψ∞ represents the seuperfluid fraction in the vac-
uum far from the surface. Let us remind that ψ is nor-
in which case the derivative of the wave function ψ at e
malizedtothe chargecarrierdensity,see(6),sothatψ∞
the surface is zero. For lead at t = 0.9 we get C =
ξ does not need to be zero. The free energy density in the
−0.00044CTF. Asexpected,thedeviationoftheGLwave vacuum thus reads e
functionδψ is muchsmallercomparedtothe sharpsteps
onthescalarpotentialδϕandonthechargecarrierwave f = 8λ4TF 2( δψ )2+δψ2( ψ)2
function δψn. out π2a2B (cid:16) ∇ n n ∇ (cid:17)
We see that the principle of minimum free energy en-
3
tails the GL boundary condition. Towards the surface −4λ2TF(∇δϕ)2+(ϕW+δϕ)δψn2 (29)
theGLwavefunctionδψdisplaysasmallgradualchange,
onlyveryclosetothesurfaceitsderivativejumpstozero. and we can write the Euler-Lagrangeequations.
The solution is complete, if the parameter C is deter- Thevariationwithrespecttothewavefunctionψgives
TF
mined. It can be derived from the requirement of con- the condition
tinuity with a solution minimizing the total free energy δψ2 2δψ =0. (30)
including the one of the vacuum outside. n∇
We see that δψ remains constant or changes linearly.
The proximity effects indicate that the correlated
III. FREE ENERGY OUTSIDE THE SLAB charge carriers can remain correlated even if they are
tunneling. Forsimplicitywesuppose,thatthesuperfluid
Now we approximate the free energy density outside fraction of the charge carriers tunneling outside the ma-
the superconductor by terial does not change, so we take δψ = ψ(0). The two
other Euler-Lagrangeequations read
= (fW+fGL+felst)dr. (24) 32λ4
F Z 2ϕ δψ TF 2δψ =0 (31)
W n− π2a2 ∇ n
B
We include the electrostatic term, the GL gradient cor-
rectionandthe vonWeizs¨ackerkinetic energyfunctional
3
δψ2+ λ2 2δϕ=0. (32)
~2 n 2 TF∇
2
f = ψ . (25)
W 2m|∇ n| In the same way as above we can try the exponential
solution
In the limit of rapidly varying densities this kinetic en-
ergytermis dominantandwhendescribing chargecarri- x
δψ =K exp (33)
n n
ers tunneling outside the superconductor this term can- (cid:18)λW(cid:19)
not be omitted. We havenot included this term into the 2x
formula(5)describingthe freeenergyinside. Thereason δϕ=Kϕexp , (34)
(cid:18)λ (cid:19)
W
is that inside the superconductor the Thomas-Fermi in-
ternal energy plays the dominant role and moreover, as where λW denotes the tunneling length which follows
it is shown e.g. in the book of Dreizler and Gross7, in from the Euler-Lagrangeequation (32) as
thelimitofnearlyhomogeneoussystemsthesecondorder
6K
termofthegradientexpansionprovidesabetterapprox- ϕ
λ = − . (35)
imation. IthasthesamestructureasthevonWeizs¨acker W pKn
kinetic energyfunctional, but its coefficientis nine times
The work function can be determined from (31) as
lower. Wesupposethatfortheroughestimatespresented
here this relatively small correctioncan be neglected. 16λ4
ϕ = TF . (36)
Inthevacuumfarfromthesurfacethescalarpotential W π2a2λ2
B W
reaches the magnitude of the work function ϕ , so that
W
we can approximate In this waywe havean approximatesolutionoutside the
superconductor, which should be linked to the solution
eϕ=E (ϕ +δϕ). (26) inside.
F W
4
IV. CONTINUITY REQUIREMENTS Here fel denotes the spatial density of the electronic
free energy, which can be roughly approximated by the
At the surface the continuity of the wave function ψn Thomas-Fermi internal energy fTF defined in (2). Then
and the continuity of the scalar potential ϕ with its the Budd-Vannimenus theorem (43) predicts a sharp
derivative (continuity of the electric field) must be en- step, CTF = 52, of the scalar potential in units of EF/e.
sured. We get three conditions The numerical solution (42) gives a comparable result
whatstronglysupportsthe applicabilityofthe hereused
2λ4TF 2 1 t4 ∂lnTc + 1+t4 ∂lnγ approximations.
π2a2ξ2 (cid:18) − ∂lnn ∂lnn(cid:19) We saw that the numerical values of the measurable
B 0 (cid:0) (cid:1) (cid:0) (cid:1)
8λ2 K2 quantities are reasonable. In Fig. 1 the scalar po-
−1+CTF =Kϕ− 3π2TaF2 Kn (37) tential is plotted. As expected, inside the supercon-
B ϕ ductor the scalar potential acquires the Fermi energy
value, while in the vacuum outside it reaches the work
C = 2KϕKn +E (38) function value ϕw. The dashed and dotted lines corre-
TF a
− 6K spond to the experimentally accessible applied electric
ϕ
p− field Ea = ±0.01eEλTFF ∼= ±1.7×107 V/cm. As it is seen
3
1 C =K (39)
TF n
− 4
where the term E representing the applied electric field
a
isincludedintotheconditionofcontinuityfortheelectric
field. Fromthecontinuityrequirements(37-39)weobtain 0
the equation
j
F
2λ4 ∂lnT ∂lnγ E
TF 2 1 t4 c + 1+t4 /
π2a2ξ2 (cid:18) − ∂lnn ∂lnn(cid:19) j
B 0 (cid:0) (cid:1) (cid:0) (cid:1) e
λ2 ( 4+3C )4 24(C E )2
TF − TF + TF− a
−144π2a2 (C E )2 ( 4+3C )2
B TF− a − TF
1+C =0, (40)
TF
−
determining the step of the scalar potential C .
TF
-1
-0.5 0.0 0.5
V. NUMERICAL VALUES x[nm]
FIG. 1: The electrostatic potential ϕ calculated for lead at
The sixth order equation (40) can be numerically
t=0.9K. The dashed and dotted lines correspond to applied
solved. For small applied electric fields the linear ex- electric field Ea =±0.01
pansion
in Fig. 2, the external electric field is screened on the
C =C +ζE (41)
TF TF0 a
Thomas-Fermi screening length. In the figure 3 devia-
is applicable and for lead at temperature t=0.9 we get tions of the chargecarrierdensities from the equilibrium
the numerical solution values are plotted. We can see that the magnitudes of
these deviations are small. Close to the surface the su-
C =0.457 0.53E . (42)
TF a perfluid density n decreases and this decrease is com-
− s
pensated by an increase of the normal fluid density n .
The numerical estimate for the tunneling length follows n
The total charge carrier density n shows no change on
to be from (35) λ = 3.17 λ and the work function
W TF
the scale of the coherence length.
accordingto(36)ϕ =1.43eV.Takingintoaccounthow
W
many simplifications we have used, it is surprising that
the obtained results seem to be quite reasonable. The
VI. COMPARISONS WITH DE GENNES
estimated magnitude of the work function is comparable
FORMULA
with the experimentally determined value of ϕ = 4.25
W
eV.8
Now we compare the GL boundary condition follow-
Thesharpstepofthescalarpotentialcanbeestimated
fromthemodifiedBudd-Vannimenustheorem4according ing from the minimum free energy principle with the de
Gennes boundary condition9
to which
e(ϕ∞−ϕ0)=(cid:18)∂∂fnel − fnel(cid:19). (43) ∇ψψ(cid:12)(cid:12)0 = ∇∆∆(cid:12)(cid:12)0 = 1b = b10 + EUas, (44)
(cid:12) (cid:12)
(cid:12) (cid:12)
5
200 x[mm]
0 1 2
0
Y
r d dy
*
0 40 x[nm]
1 0.2 0.4
-1
r
-200 -2
-0.5 0.0 0.5
x[nm]
-3
FIG.2: Charge densityρ. Tomakethescreening visible, the
dashed and dotted lines correspond to the applied electric
field Ea =±0.3
FIG. 4: Deviation δψ of the wave function ψ from the equi-
librium value. Close to the surface its derivative approaches
zero (see the insert). The dashed and dotted lines indicate
how theextrapolation length changes with applied field.
2
n
d
n to be close to one. For lead the formula (45) gives Us =
n 1.35 107 V. From the minimum free energy we know,
o ×
ati dn however,thatthederivativeatthesurfaceshouldbezero.
vi 0 InFig.4we seehow the deviationofthe wavefunctionψ
e
d at the surface decreases with the derivative determined
*
bytheparameterC . Fromthiswegettheextrapolation
40 ξ
1 length b0 2.8 mm, a value comparable with the one
dn estimated≈by de Gennes. Only very close to the surface
-2 s (on the distance of Thomas-Fermi screening length) the
derivative of the wave function ψ approaches zero (see
insert of the Fig. 4).
0 1 2 In figure 4 we can also observe how the extrapolation
x[mm] length changes if an electric field E is applied. By sub-
a
stituting b = ξ /C into (44) and using (23) with the
t ξ
−
approximation (41) we get a simple expression for the
FIG. 3: Deviations of charge carrier densities. Toensure vis- effective potential U
s
ibility,thedashedanddottedlinescorrespond totheapplied
field Ea =±0.1. 1 3ζλ2 ∂lnT e
= TF c . (47)
U 4ξ2 ∂lnn E
s 0 F
accordingtowhichthederivativeofthegapatthesurface
This formula is similar to the de Gennes formula (45).
is not exactly zero even without external electric field.
We should notice, however, that in this formula the ex-
The zero field extrapolation length b is around 1 cm
0
trapolationparameterζ/2 of (41) appears instead of the
(almostinfinityfromthemicroscopicpointofview). The
surface ratio η which enters de Gennes formula (45).
effective potential U
s
1 3ηλ2 ∂lnT e
= TF c (45)
U 2ξ2 ∂lnn E VII. CONCLUSIONS
s 0 F
determines how the extrapolation length b changes if an
It was shown in this paper that the minimum free en-
external electric field E is applied3. De Gennes esti-
a ergy principle entails a zero derivative of the wave func-
mated the surface ratio
tion ψ at the surface of the superconductor. On the
∆(0) scale of the coherence length, however, even if no ex-
η (46)
≡ ∆ ternal electric field is applied, the derivative is nonzero
0
6
and its magnitude corresponds to the de Gennes esti- Acknowledgments
mate. Only on the Thomas-Fermi screening length scale
it approaches zero. In the presence of an external elec-
tric field the extrapolation length changes according to This work was supported by the Czech research
equation(44),withthe effectivepotentialgivenbyequa- plans MSM 0021620834 and No. AVOZ10100521, by
tion (47). This formula is similar to formula (45) follow- grantsGACˇR202/07/0597and202/08/0326andGAAV
ing from the de Gennes theory. The agreement with the IAA100100712aswellasGermanPPPprojectofDAAD
Budd-Vannimenus theorem and the numerical estimates and the BMBF. The financial support by the Brazilian
support the applicability of the proposed approach. Ministry of Science and Technology is acknowledged.
1 D. Matthey, S. Gariglio, and J. M. Triscone, Appl. Phys. 6 C.J.GorterandH.B.G.Casimir,Z.Techn.Phys.15,539
Lett. 83, 3758 (2003). (1934).
2 V. C. Matijasevic, S. Bogers, N. Y. Chen, H. M. Appel- 7 R.M.DreizlerandE.K.U.Gross,DensityFunctionalThe-
boom, P. Hadley, and J. E. Mooij, Physica C 235, 2097 ory (Springer-Verlag,Berlin, 1990).
(1994). 8 H. B. Michaelson, J. Appl. Phys 48, 4729 (1977), ISSN
3 P. Lipavsky´, K. Morawetz, J. Kol´aˇcek, and T. J. Yang, 0021-8979.
Phys.Rev.B 73, 052505 (2006). 9 P. G. de Gennes, Superconductivity of Metals and Alloys
4 P. Lipavsky´, K. Morawetz, J. Kol´aˇcek, J. J. Mareˇs, E. H. (Benjamin, New York,1966).
Brandt,andM.Schreiber,Phys.Rev.B70,104518(2004).
5 C.J.GorterandH.B.G.Casimir,Phys.Z.35,963(1934).