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Localized excitations in a type-I superconductor
Solid State Communications,
Vol. 8, pp. 21-22,
1970.
Pergamon Press.
Printed in Great Britain
LOCALIZED EXCITATIONS IN A TYPE1 SUPERCONDUCTOR* Martin S. Fullenbaum Department of Physics,
University of Virginia, Charlottesville,
(Received
Virginia 22903
27 October 1969 by R.H. Silsbee)
The excitation spectra for localized surface states in a type-1 superconductor are found by a series expansion method using the Bogoliubov equations. Results for the two lowest eigenvalues as a function of magnetic field are given.
The Bogoliubov equations are written as’
WE REPORT here the results of a calculation of the localized excitations found in a type-1 superconductor and first discussed by Pincus.’ In the presence of a static magnetic field parallel to the surface of a superconductor, screening currents are set up which damp out the applied field in some characteristic distance, X, called the penetration depth. Those electrons travelling parallel to the current form excited states whose energy is less than A, the gap parameter, and are localized in the surface region. Experimental studies of the surface impedance in aluminium by Budzinski and Garfunkel,2 and in indium by MakIonado and Koch3 are consistent with the existence of these surface states. (We should point out that Bud&ski and Garfunke12 attribute the anomalous low frequency behavior in the surface impedance to a p.v. correction to the BCS density of states.
a.&) = K, u(r) + A(r) v(r),
(1)
TV
(2)
= -Hz v(r) + A* (r) u(r),
where
K, =
2 - E,,
with EF being the Fermi energy. The gap function A (r) may be taken to be real in the London .gauge (VA = 0) and, since A(r) varies over a coherence length, 6, may be assumed constant because 5‘>> X. The semi-infinite geometry to be considere is that of a superconductor in the half space z > 0 with a static magnetic field in the y ditectio H,(z) = H exp (-z/X), A,(z) = -H exp (- z/X). We write u(r) and v(r) as u(z)exp (ik,x + ik,y) and v(z)exp (ik,x + ik,y), respectively. Then, upon discarding the A2 term, (1) and (2) can be written as
Our method consists of finding the eigenvalues, E, of the Bogoliubov equations such that E < A. Our basis assumption is that the magnetic field varies as exp (-z/X) measured from the surface of the superconductor. (This is in contrast to Pincus who approximated the exponential by a square well of width X). As in reference 1, the A2 term will be discarded.
where
*Supported%y
Here a is a parameter measuring the perpendicular kinetic energy,
NSF Grant GU-1531.
21
EU(Z) = L,u(z) + Av(z)
(3)
EV(Z) = L,v(z) + Au(z)
(4)
L,
= -@2/2m)
(dz/dzz) - a+ V(z),
L,
= +(h2/2m) (d2/dz2) + a+ V(z).
LOCALIZED
22
EXCITATIONS
IN A TYPE-I
a= EF - E,, with
Vol. 8, No. 1
SUPERCONDUCTOR
2, u(v)
= 4Av(v),
(5)
s,v(y)
= -4Au(~),
(6)
with V(z)
= -V
exp(-z/h)
with V = H h ep,/mc, and V > 0 for electrons travelling parallel to the current, i.e., p, > 0.
2,
= y2(d 2/dy”) + y(d/dy)
2,
= yZ(dz/dyz)+
y(d/dy)+
+ 4(o+ 4(u-
E) + y2, E)--22;
all energies are dimensionless in units of B. The operators 2, and 9, now admit series solutions of the form u(y) = Zua,yn+Cand vb) = Z~,,Y~+~; substitution into (5) and (6) then yields the coefficients a, and b,, as well as the leading exponent, c. The solutions can be written as
C/A
~01) = Au, (Y) + Bu,01),
(7)
vcV) = Av, Cv) + Bv,Cy),
(8)
with A and B as constants. The subscripts on the n’s and v’s refer to a particular value of c. More details of this solution will be given elsewhere.
FIG. 1. The energy E as a function of V, the field parameter for states in the energy gap, with a= 0. The solid line is the result of our calculation while the dashed line is Pincus’s result.
A more succinct form is achieved tution y = 2\I(V/B) x exp(-z/2A), B = h2/2m X2. The result is
By imposing the appropriate boundary conditions for u and v at x = 0, for y = 2\/V, one obtains a transcendental equation which must be solved numerically for E. (For comparison with Pincus we have required the derivative of u and v to vanish on the surface of the superconductor. Parameters appropriate to Aluminium near T = O°K have been used, A = 5OOA, A = 2.105’K,
Acknowledgements - We wish to acknowledge helpful discussions with Professors Beck, Fowle and Nam. Thanks are also due to Professor J. Fred Koch who stimulated our thinking on this problem.
by the substiwith
REFERENCES 1.
PINCUS P., Phys.
2.
BUDZINSKI W.V. and GARFUNKEL M., Phys. Rev. Lett. GARFUNKEL M., Phys. Rev. 173, 516 (1968).
Rev. 158, 346 (1967).
3.
MALDONADO J. and KOCH J. FRED,
4.
DE GENNES P.G.,
Superconducting
16, 1100 (1966); 17, 24 (1966);
to be published.
Metals
and Alloys,
W.A. Benjamin.
New York (1966).
Partant des equations de Bogoliubov, le spectre d’excitation des etats de surface localises dans un supraccnducteur de type-1 est dtudie par une methode de diveloppement en serie. Les deux valeurs propres infdrieures sont calculees en fonction de champ magnetique.
Vol. 8, pp. 21-22,
1970.
Pergamon Press.
Printed in Great Britain
LOCALIZED EXCITATIONS IN A TYPE1 SUPERCONDUCTOR* Martin S. Fullenbaum Department of Physics,
University of Virginia, Charlottesville,
(Received
Virginia 22903
27 October 1969 by R.H. Silsbee)
The excitation spectra for localized surface states in a type-1 superconductor are found by a series expansion method using the Bogoliubov equations. Results for the two lowest eigenvalues as a function of magnetic field are given.
The Bogoliubov equations are written as’
WE REPORT here the results of a calculation of the localized excitations found in a type-1 superconductor and first discussed by Pincus.’ In the presence of a static magnetic field parallel to the surface of a superconductor, screening currents are set up which damp out the applied field in some characteristic distance, X, called the penetration depth. Those electrons travelling parallel to the current form excited states whose energy is less than A, the gap parameter, and are localized in the surface region. Experimental studies of the surface impedance in aluminium by Budzinski and Garfunkel,2 and in indium by MakIonado and Koch3 are consistent with the existence of these surface states. (We should point out that Bud&ski and Garfunke12 attribute the anomalous low frequency behavior in the surface impedance to a p.v. correction to the BCS density of states.
a.&) = K, u(r) + A(r) v(r),
(1)
TV
(2)
= -Hz v(r) + A* (r) u(r),
where
K, =
2 - E,,
with EF being the Fermi energy. The gap function A (r) may be taken to be real in the London .gauge (VA = 0) and, since A(r) varies over a coherence length, 6, may be assumed constant because 5‘>> X. The semi-infinite geometry to be considere is that of a superconductor in the half space z > 0 with a static magnetic field in the y ditectio H,(z) = H exp (-z/X), A,(z) = -H exp (- z/X). We write u(r) and v(r) as u(z)exp (ik,x + ik,y) and v(z)exp (ik,x + ik,y), respectively. Then, upon discarding the A2 term, (1) and (2) can be written as
Our method consists of finding the eigenvalues, E, of the Bogoliubov equations such that E < A. Our basis assumption is that the magnetic field varies as exp (-z/X) measured from the surface of the superconductor. (This is in contrast to Pincus who approximated the exponential by a square well of width X). As in reference 1, the A2 term will be discarded.
where
*Supported%y
Here a is a parameter measuring the perpendicular kinetic energy,
NSF Grant GU-1531.
21
EU(Z) = L,u(z) + Av(z)
(3)
EV(Z) = L,v(z) + Au(z)
(4)
L,
= -@2/2m)
(dz/dzz) - a+ V(z),
L,
= +(h2/2m) (d2/dz2) + a+ V(z).
LOCALIZED
22
EXCITATIONS
IN A TYPE-I
a= EF - E,, with
Vol. 8, No. 1
SUPERCONDUCTOR
2, u(v)
= 4Av(v),
(5)
s,v(y)
= -4Au(~),
(6)
with V(z)
= -V
exp(-z/h)
with V = H h ep,/mc, and V > 0 for electrons travelling parallel to the current, i.e., p, > 0.
2,
= y2(d 2/dy”) + y(d/dy)
2,
= yZ(dz/dyz)+
y(d/dy)+
+ 4(o+ 4(u-
E) + y2, E)--22;
all energies are dimensionless in units of B. The operators 2, and 9, now admit series solutions of the form u(y) = Zua,yn+Cand vb) = Z~,,Y~+~; substitution into (5) and (6) then yields the coefficients a, and b,, as well as the leading exponent, c. The solutions can be written as
C/A
~01) = Au, (Y) + Bu,01),
(7)
vcV) = Av, Cv) + Bv,Cy),
(8)
with A and B as constants. The subscripts on the n’s and v’s refer to a particular value of c. More details of this solution will be given elsewhere.
FIG. 1. The energy E as a function of V, the field parameter for states in the energy gap, with a= 0. The solid line is the result of our calculation while the dashed line is Pincus’s result.
A more succinct form is achieved tution y = 2\I(V/B) x exp(-z/2A), B = h2/2m X2. The result is
By imposing the appropriate boundary conditions for u and v at x = 0, for y = 2\/V, one obtains a transcendental equation which must be solved numerically for E. (For comparison with Pincus we have required the derivative of u and v to vanish on the surface of the superconductor. Parameters appropriate to Aluminium near T = O°K have been used, A = 5OOA, A = 2.105’K,
Acknowledgements - We wish to acknowledge helpful discussions with Professors Beck, Fowle and Nam. Thanks are also due to Professor J. Fred Koch who stimulated our thinking on this problem.
by the substiwith
REFERENCES 1.
PINCUS P., Phys.
2.
BUDZINSKI W.V. and GARFUNKEL M., Phys. Rev. Lett. GARFUNKEL M., Phys. Rev. 173, 516 (1968).
Rev. 158, 346 (1967).
3.
MALDONADO J. and KOCH J. FRED,
4.
DE GENNES P.G.,
Superconducting
16, 1100 (1966); 17, 24 (1966);
to be published.
Metals
and Alloys,
W.A. Benjamin.
New York (1966).
Partant des equations de Bogoliubov, le spectre d’excitation des etats de surface localises dans un supraccnducteur de type-1 est dtudie par une methode de diveloppement en serie. Les deux valeurs propres infdrieures sont calculees en fonction de champ magnetique.