PHYSICAL REVIEW B
VOLUME 57, NUMBER 13
1 APRIL 1998-I
Moving vortex line: Electronic structure, Andreev scattering, and Magnus force
S. Hofmann* and R. Kümmel
Institut für Theoretische Physik der Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
~Received 12 June 1997!
The wave functions of quasiparticles in a vortex line, moving with velocity vW L relative to the lattice when a
transport current with drift velocity vW T is applied, are calculated by solving the time-dependent Bogoliubov–de
Gennes equations for a high-k superconductor in contact with a reservoir of chemical potential m . Far away
from the vortex core the pair potential has the constant modulus D ` . Comparison with the wave functions of
a vortex at rest shows that vortex motion modifies the amplitudes, the radial wave numbers of the states with
energy E.D ` , and the penetration lengths of states with energy E,D ` by a term 6« v cosQ. Here Q is the
azimuthal angle of cylinder coordinates with the z direction parallel to the vortex axis, and « v 5\k r v ; v
5 u vW T 2 vW L u and k r 5 A(2m/\ 2 ) m 2k 2z , with k z being the wave number of propagation in the z direction. If one
neglects terms of the order of « 2v in the spectrum of the bound states, one obtains the same eigenvalues as for
the vortex at rest. The supercurrent force on the corresponding quasiparticles, caused by Andreev scattering at
the core boundary, is calculated with the v -modified wave functions. It transfers half of the Magnus force from
the moving condensate to the unpaired quasiparticles in the vortex core. @S0163-1829~98!05113-3#
I. INTRODUCTION
The motion of vortex lines in type-II superconductors dissipates energy, because the unpaired electrons in the vortex
core are subject to frictional forces from the lattice. This
limits the technological applications of conventional and especially high-temperature superconductors ~HTSC’s!. Discovery of the HTSC’s has stimulated anew intensive research
in vortices and their motion, documented in recent
reviews.1–3 While being disadvantageous from a technological point of view, vortex motion has challenged experimental
and theoretical physicists with a wealth of interesting problems. One of the oldest is the question, which force acting on
W
a moving vortex line is in balance with the frictional force F
from the lattice? Different answers have been given by the
theories of Bardeen and Stephen ~BS!, on the one hand,4 and
Nozières, Vinen, and Warren ~NVW!,5,6 on the other hand.
While NVW show that this force should be the Magnus
force,
W 5FW Magnus5n s eL ~ vW T 2 vW L ! 3F
W,
2F
~1!
which has been confirmed recently by Berry-phase
considerations,7–9 BS have instead
W;
2FW 52n s eL vW L 3F
~2!
here n s is the density of the superconducting electrons,
e52 u e u is the charge of an electron, L is the length of the
vortex line which is assumed to be parallel to the z axis, vW T
is the drift velocity of the applied transport supercurrent, vW L
W 5eW z F 0 , with F 0
is the velocity of the vortex line, and F
5h/2u e u being the flux quantum.
The basic difference between the two theories consists in
an ‘‘interface force,’’ introduced phenomenologically in the
NVW theory by momentum-balance arguments and absent in
the BS theory. It is required for the complete transfer of the
Magnus force from the superconducting condensate outside
0163-1829/98/57~13!/7904~12!/$15.00
57
the core to the unpaired electrons inside the core which are in
equilibrium with the lattice. Nozières and Vinen state, ‘‘It is
essential to know, how the Magnus force is shared between
the bulk of the core and the ‘interface’ with the superfluid.’’5
The bulk share turned out to be half of the Magnus force.5,6
It originates from the electrostatic field in the core of radius
rc :6
EW c 5
1
2 p r 2c
W5
~ vW L 22 vW T 1 vW nc ! 3F
1
2 p r 2c
W.
~ vW L 2 vW T ! 3F
~3!
The second equality holds if no pinning centers inhibit the
motion of the vortex so that the drift velocity inside the core
vW nc is equal to vW T . ( vW T is vW s1 in Ref. 6.! The interface force,
however, which has to provide the other half of the Magnus
force,6 has remained unclear in its physical nature until recently. In a previous Letter10 we have shown by a semiclassical calculation that this interface force is the result of momentum transfer from the Cooper pairs in the condensate to
the core electrons by Andreev scattering at the core boundary. It is a special and important manifestation of a second
force, called supercurrent force Wf D2 ,11,12 involved in Andreev
scattering.13 This force is the fourth member of the full set of
forces acting on quasiparticles in an inhomogeneous,
current-carrying superconductor with pair potential D(rW )
W
W , and scalar potential V.
5 u D(rW ) u e i w (r ) , vector potential A
These forces are obtained from the time-dependent
Bogoliubov–de Gennes equations ~TdBdGE’s! ~Refs. 14 and
15! and the resulting generalized Ehrenfest theorem for quasiparticles with electron component u(rW ,t) and hole component v (rW ,t).11 Accordingly, the total force on a quasiparticle,
defined as
Wf [
7904
d
dt
E
d 3 r c † ~ rW ,t ! pŴ c ~ rW ,t ! ,
~4!
© 1998 The American Physical Society
57
MOVING VORTEX LINE: ELECTRONIC STRUCTURE, . . .
with
c ~ rW ,t ! 5
S
S D
u ~ rW ,t !
v~ rW ,t !
~5!
being a solution of the TdBdGE’s, and
pŴ [
S D
pW e
0
0
pW h
[
F
\
W 2eAW ~ rW ,t !
¹
i
G
0
0
F
\
W 1eAW ~ rW ,t !
¹
i
G
D
being the quasiparticle momentum operator, is the sum of the
four forces:
Wf 5 Wf e 1 Wf h 1 Wf D1 1 Wf D2 .
~7!
It consists of the usual diagonal electromagnetic forces on
electrons (e) and holes (h),
Wf e 5
Wf h 5
E
E
d 3 ru *
d 3r v *
F
F
G
G
e
]
W 2BW 3pW e ! 2e AW 2¹
W V u, ~8!
~ pW e 3B
2m
]t
e
]
W 2BW 3pW h ! 1e AW 1¹
W V v,
~ pW 3B
2m h
]t
~9!
and two off-diagonal pair potential forces
E
E
Wf D1 522Re
Wf D2 54
m
\
W 3AW , and
Here BW 5¹
vW s 5
W uDu !e iw,
d 3 ru * v~ ¹
~10!
d 3 r vW s Im~ u * v D ! .
~11!
F
~12!
\
2e
W w 2 AW
¹
2m
\
G
force ‘‘obtained in a form that involves virtual transitions
between the core levels around the Fermi level’’ is due to the
same physical mechanisms, i.e., Andreev scattering and supercurrent force, which give rise to
2
Wf D [ Wf D1 1 Wf D2 5 Im
\
~6!
is the gauge-invariant Cooper pair velocity. The off-diagonal
force Wf D1 is responsible for the change of quasiparticle momentum from a value above to a value below the Fermi
surface ~or vice versa! in usual Andreev scattering. The second off-diagonal force Wf D2 , which is present if there is a
supercurrent flowing with Cooper pair velocity vW s , is the
new supercurrent force. In this paper we present a complete
quantum mechanical calculation of this supercurrent force by
solving the time-dependent Bogoliubov–de Gennes equations for an isolated moving vortex line and show that the
screening current around the vortex line and the modification
of the quasiparticle wave functions by the motion of the
vortex relative to the superconducting condensate result indeed in half of the Magnus force.
Šimánek16 has calculated the force on the center of a
moving vortex line in the adiabatic limit with the help of the
instantaneous eigenstates of the superfluid and the Bogoliubov transformation of the many-body Hamiltonian. The
wave functions in this transformation are the solutions of the
stationary Bogoliubov–de Gennes equations. The way they
enter Šimánek’s equations ~5!, ~10!, and ~15! shows that his
7905
E
d 3 ru * v
S
D
\
W 22eAW D.
¹
i
~13!
The sum of Wf D over all occupied quasiparticle states yields
the total off-diagonal force on the vortex. Because of symmetry reasons, only Wf D2 has a component perpendicular to
vW T 2 vW L and may therefore be involved in the Magnus-force
transfer across the core boundary. Therefore, it is sufficient
to calculate Wf D2 .
A fringe benefit of our going beyond the adiabatic limit in
solving the TdBdGE’s is the discovery of a new sort of quasiparticle states. They may be called ‘‘angular bound states,’’
because in two angular ranges their wave functions decay
exponentially outside the core, and in the two complementary angular ranges they spread as undamped scattering
states.
The paper is organized as follows. In Sec. II, by a Galilei
transformation, the TdBdGE’s in the lattice frame of reference ~LFR! are changed into stationary BdGE’s in the vortex
frame of reference ~VFR!. In Sec. III we study the
asymptotic behavior of the quasiparticle wave functions of
the vortex system and discuss the three types of quasiparticle
states: bound, scattering, and angular bound states. Section
IV describes how matching of the solutions outside the core
to the ones inside the core is achieved with the help of a
Fourier transformation of the angle-dependent matching conditions. The complete, velocity-dependent wave functions
for the case of a moving vortex with a square-well pair potential are presented. With these wave functions we calculate
the supercurrent-force contribution to the Magnus force in
Sec. V and find that it is equal to one-half of the Magnus
force, if the normal vortex core has a diameter of about one
coherence length. A summary and outlook concludes the paper.
II. GALILEI TRANSFORMATION OF THE TDBDGE’s
We consider a low-T c , high-k type-II superconductor. In
the absence of external fields its pair potential is homogeneous, isotropic, and small compared to the Fermi energy. In
this superconductor vortices are induced by an applied magnetic field B ~parallel to the z direction! which satisfies B c1
,B!B c2 , so that vortex-vortex interactions can be ignored,
and it is sufficient to consider an isolated vortex. This vortex
is driven by an applied supercurrent with constant drift velocity vW T . It moves with constant velocity vW L . Both velocities lie in the x-y plane:
vW T 5 v Tx eW x 1 v Ty eW y ,
~14!
vW L 5 v Lx eW x 1 v Ly eW y .
~15!
The system is described by the mean-field TdBdGE’s,14,15
with a pair potential which in the lattice frame of reference
and the corresponding coordinates rW8 and t 8 is given by
S. HOFMANN AND R. KÜMMEL
7906
57
Thus, in the LFR the TdBdGE’s of an isolated moving vortex line are
i\
]
]t8
ˆ ~ rW 8 ,t 8 ! C̃ ~ rW 8 ,t 8 ! .
C̃ ~ rW 8 ,t 8 ! 5H̃
~21!
C̃(rW 8 ,t 8 ) is the spinor quasiparticle wave function ~with
electron component ũ and hole component ṽ ) in the LFR,
and the matrix Hamiltonian
FIG. 1. Frames of reference: the coordinates (rW 8 ,t 8 ) with the
Cartesian components (x 8 ,y 8 ) and the angle Q 8 correspond to the
lattice frame of reference ~LFR! while the coordinates (rW ,t) with
the Cartesian components (x,y) and the angle Q correspond to the
vortex frame of reference ~VFR! which moves with velocity vW L
relative to the LFR.
D̃ ~ rW 8 ,t 8 ! 5D ~ rW 8 2 vW L t 8 ! .
ˆ (rW 8 ,t 8 )5
H̃
H̃ e ~ rW 8 ! 5
D ~ rW ! 5D ~ r,Q ! 5D 0 ~ r ! e 2iQ
~17!
is the pair potential of a vortex at rest; i.e., the coordinates
(rW ,t) are those of a frame of reference fixed to the vortex.
The radial part D 0 (r) of the pair potential vanishes at the
center of the vortex and assumes the constant value D ` far
from the core. The phase of the pair potential furnishes the
superfluid velocity
\
eW
2mr Q
~18!
of the screening current.
With Eqs. ~16! and ~17! and Fig. 1 the pair potential in the
lattice frame of reference has the form
D̃(rW 8 ,t 8 )5D 0 @ A~ r 8 cos Q 8 2 v Lx t 8 ! 2 1 ~ r 8 sin Q 8 2 v Ly t 8 ! 2 #
S
3exp 2i arctan
r 8 cos Q 8 2 v Lx t 8
D
.
D̃ * ~ rW 8 ,t 8 !
W
2H̃ *
e ~r8!
D
~22!
,
1
W 8 2eÃW ! 2 2« F
~ 2i\¹
2m
~20!
1
W 8 1m vW T ! 2 2« F .
~ 2i\¹
2m
~23!
In order to avoid the rather complicated structure ~19! of
the pair potential in the LFR we now switch over to the VFR
by means of the Galilei transformation
rW 8 5rW 1 vW L t,
t 8 5t.
~24!
This transformation shifts the pair potential D̃(rW 8 ,t 8 ) back to
the pair potential D(rW ) of the vortex at rest. The wave functions C(rW ,t) in the VFR follow from the LFR wave functions C̃(rW 8 ,t 8 ) by the same transformation:
C ~ rW ,t ! [C̃ ~ rW 1 vW L t,t ! .
~25!
Applying the Galilei transformation to the differential operators,
W 8 5¹
W,
¹
~19!
The magnetic field of the vortex may be neglected5,17,18 as
well as the electric field induced by the motion of the
vortex.5,6,19 The chemical potential m in the TdBdGE’s is
that of the energy and particle reservoir to which the superconductor is coupled. We neglect all influences of entropy
production associated with vortex motion on the chemical
potential, because the number of degrees of freedom of the
reservoir is assumed to be very much larger than that of the
superconductor.15 Then m is the same as in the case of no
vortex motion and equal to the Fermi energy « F of the superconductor. The vector potential in the LFR is determined
by considering a region far away from the vortex core. There
W w in Eq. ~12!, and
vW s0 of Eq. ~18! vanishes, and so does ¹
vW s 5 vW T . With that Eq. ~12! turns into
m
AW 52 vW T [ÃW .
e
D̃ ~ rW 8 ,t 8 !
~16!
5
r 8 sinQ 8 2 v Ly t 8
H̃ e ~ rW 8 !
contains the pair potential of Eq. ~19! and the single-electron
Hamiltonian
Here
vW s0 52
S
]
]t8
5
~26!
]
]
]
2 v Lx 2 v Ly ,
]t
]x
]y
we transform the TdBdGE ~21! to the VFR:
S
H 0e ~ rW ! 1H 1 ~ rW !
]
W
i\ C ~ r ,t ! 5
]t
D * ~ rW !
D ~ rW !
2H 0e ~ rW ! 1H 1 ~ rW !
~27!
D
C ~ rW ,t ! .
~28!
H 0e
Here
and H 1 are, appropriate to the symmetry of our
problem, written in cylinder coordinates (r,Q,z),
H 0e ~ r,Q,z ! 52
S
1 ]2
]2
\2 ]2
1 ]
1
1
1
2m ] r 2 r ] r r 2 ] Q 2 ] z 2
S
1
2 « F 2 m v 2T
2
and
D
D
~29!
57
MOVING VORTEX LINE: ELECTRONIC STRUCTURE, . . .
F
H 1 ~ r,Q,z ! 52i\ ~ 2 v x sin Q1 v y cos Q !
1 ~ v x cos Q1 v y sin Q !
G
1 ]
r ]Q
]
,
]r
~30!
vW [ vW T 2 vW L ,
v x [ v Tx 2 v Lx ,
v [ u vW u ,
~31a!
v y [ v Ty 2 v Ly
S
~31b!
DS
H 0e ~ r,Q ! 1H 1 ~ r,Q !
u ~ r,Q !
E
5
v~ r,Q !
D * ~ r,Q !
Here the pair potential is given by Eq. ~17!, and
H 0e ~ r,Q ! 52
S
D
1 ]2
\2 ]2
1 ]
1
1
2« r , ~34!
2m ] r 2 r ] r r 2 ] Q 2
\ 2 k 2z
1
,
« r [« F 2 m v 2T 2
2
2m
F
H 1 ~ r,Q ! 52i\ v 2sin Q
have been introduced. From now on we assume, without loss
of generality, v y 50 and therefore v 5 v x .
Since the Hamiltonian in Eq. ~28! does not depend explicitly on time t and the z coordinate, the ansatz
C ~ rW ,t ! 5
where the definitions
~35!
G
]
1 ]
1cos Q
.
r ]Q
]r
~36!
Apart from the small term 21 m v 2T , which can be disregarded, H 0e (r,Q) is the same as for a vortex at rest. The
effects of the applied supercurrent as well as of the motion of
the vortex are contained in the term H 1 (r,Q). Note that this
term depends only on the relative velocity vW 5 vW T 2 vW L between vortex and superconducting condensate, exactly as the
Magnus force, Eq. ~1!, does. For a vortex at rest and not
immersed in an applied supercurrent the term H 1 (r,Q) vanishes as well as the Magnus force. The same is true for a
vortex drifting along with the condensate ( vW L 5 vW T ).
If the vortex is immersed in an applied supercurrent but
does not drift along with it ( vW L Þ vW T ), H 1 (r,Q) must be
taken into account. In this case, the wave functions will not
be the same as those of the vortex at rest. Moreover, the term
H 1 (r,Q) breaks the symmetry of the problem: The wave
functions can no longer be separated into radial and angular
functions. This drastically complicates the problem of solving the BdGE’s. Fortunately, the solutions in the asymptotic
limit r→` open up a way which will lead us in a sufficiently
good approximation to the full solutions to be used in Eq.
~11!. The two steps on this way, taken in Secs. III and IV, are
guided by the following considerations.
The dynamic, symmetry-breaking quantities which generate the supercurrent force are the superfluid velocity vW s0 of
the screening current, Eq. ~18!, and the vortex velocity vW
relative to the condensate, Eq. ~31!. In the asymptotic limit
vW s0 becomes vanishingly small so that vW dominates in the
asymptotic wave functions. These are calculated in Sec. III.
7907
S
D
u ~ r,Q ! 2iEt/\ ik z
e
e z
v~ r,Q !
~32!
leads to the stationary Bogoliubov–de Gennes equations for
the electron and hole components u(r,Q) and v (r,Q) of the
radial and angular wave functions:
D ~ r,Q !
2H 0e ~ r,Q ! 1H 1 ~ r,Q !
DS
D
u ~ r,Q !
.
v~ r,Q !
~33!
Then, in Sec. IV, the influence of vW s0 is taken into account
by functions which are independent of vW and multiply the
vW -dependent asymptotic wave functions. Each such product
represents a solution of the BdGE’s outside the vortex core.
Appropriate matching to the solutions inside the core will
finally provide the energy eigenvalues and wave functions of
the bound states needed for the calculation of the supercurrent force.
III. ASYMPTOTIC WAVE FUNCTIONS
A. Nonlinear differential equations and proper solutions
The electron and hole components u A (r,Q) and v A (r,Q)
of the asymptotic wave functions are solutions of the
asymptotic BdGE’s which result from Eq. ~33! for r→`:
r→`
~ H 0e 1H 1 2E ! u A ~ r,Q ! 1D ` e 2iQ v A ~ r,Q ! ——→ 0,
~37!
r→`
~ 2H 0e 1H 1 2E !v A ~ r,Q ! 1D ` e iQ u A ~ r,Q ! ——→ 0,
~38!
where D ` [D 0 (r→`). In Eqs. ~37! and ~38! only those
terms are kept which do not vanish for r→`.
Inspired by the asymptotic solutions of the BdGE’s for a
vortex at rest,17,18 we start with the ansatz
u An ,n ~ r,Q ! 5e inQ e ik n ~ Q ! r ,
~39!
v An ,n ~ r,Q ! 5c n ~ Q ! e i ~ n11 ! Q e ik n ~ Q ! r .
~40!
Here n is an integer and n will be explained later. The functions u An ,n (r,Q) and v An ,n (r,Q) differ from the solutions for a
vortex at rest in so far, as they include the angular-dependent
functions c n (Q) and k n (Q) in places, where the latter ones
have constants. Finding the functions c n (Q) and k n (Q) is
equivalent to finding the asymptotic wave functions
u An ,n (r,Q) and v An ,n (r,Q). We are looking for those solutions
which in the limit v →0 turn into the known asymptotic solutions of the vortex at rest.
7908
S. HOFMANN AND R. KÜMMEL
Inserting the ansatz of Eqs. ~39! and ~40! into the
asymptotic BdGE’s ~37! and ~38! we find after some algebra
the coupled nonlinear differential equations for k n (Q) and
c n (Q),
05
\2 2
$ k ~ Q ! 1 @ k 8n ~ Q !# 2 % 2« r 2\ v@ k 8n ~ Q ! sin Q
2m n
2k n ~ Q ! cos Q # 2E1D ` c n ~ Q ! ,
~41!
Inserting these k n and c n into Eqs. ~39! and ~40! yields the
four well-known17,18 asymptotic solutions for the vortex at
rest.
In the second place, there are solutions of the type
k n (Q)5k n cos(Q2Q0), where the k n are those of Eq. ~47!
and the Q 0 are arbitrary. Inserting these solutions into Eqs.
~39! and ~40! delivers functions containing plane waves
propagating in the x and y directions:
u An ~ r,Q ! 5e inQ e ik n r cos~ Q2Q 0 ! 5e inQ e i ~ k x x1k y y ! ,
\
052
$ k 2 ~ Q ! 1 @ k n8 ~ Q !# 2 % 1« r 2\ v@ k n8 ~ Q ! sin Q
2m n
2
2k n ~ Q ! cos Q # 2E1D ` c 21
n ~ Q !,
d
k ~ Q !.
dQ n
~43!
In order to decouple the system of differential equations ~41!
and ~42! we rewrite Eq. ~41!,
D ` c n ~ Q ! 52
\2 2
$ k ~ Q ! 1 @ k 8n ~ Q !# 2 % 1« r
2m n
5c n e i ~ n11 ! Q e i ~ k x x1k y y ! ,
\2 2
$ k ~ Q ! 1 @ k n8 ~ Q !# 2 % 2« r
2m n
1q 1 A$ E1\ v@ k 8n ~ Q ! sin Q2k n ~ Q ! cos Q # % 2 2D 2` ,
~51!
~44!
multiply Eq. ~42! by D ` c n (Q), and insert Eq. ~44!:
F
G
2
\2 2
$ k n ~ Q ! 1 @ k n8 ~ Q !# 2 % 2« r 2 $ E1\ v@ k n8 ~ Q ! sin Q
2m
2k n ~ Q ! cos Q # %
2
1D 2` .
~45!
We have to solve Eq. ~45! for k n (Q), and then insert k n (Q)
into Eq. ~44! in order to obtain c n (Q).
Before doing so, let us see how k n (Q) and c n (Q) are
related to the corresponding quantities in the asymptotic
wave functions of the vortex at rest. This will help us to
select the physically appropriate solutions. In this case v 50,
and Eq. ~45! reduces to
\2 2
$ k ~ Q ! 1 @ k n8 ~ Q !# 2 % 2« r 56 AE 2 2D 2` .
2m n
~46!
There are two types of solutions.
In the first place, k n (Q) might be k n , independent of Q.
Then Eq. ~46! is easily solved for k n , resulting in
k n 56
A
2m
\2
~ « r 6 AE 2 2D 2` ! ,
~47!
where n P(1,2,3,4) counts the four solutions due to the four
possible combinations of the 1 and 2 signs. From Eq. ~44!
with v 50 and Eq. ~47! one finds
c n5
E
7
D`
A
E2
D 2`
21.
~50!
where k x 5k n cos Q0, k y 5k n sin Q0, and c n is given again by
Eq. ~48!. These asymptotic solutions are not the cylindrical
plane waves required for a vortex at rest. Thus, they are
dismissed. The corresponding solutions of Eq. ~45! with v
Þ0 will be dismissed, too.
Equation ~45! for k n (Q) is rewritten as
05
1\ v@ k n8 ~ Q ! sin Q2k n ~ Q ! cos Q # 1E,
05
~49!
v An ~ r,Q ! 5c n e i ~ n11 ! Q e ik n r cos~ Q2Q 0 !
~42!
where the slash denotes differentiation with respect to Q:
k 8n ~ Q ! [
57
~48!
where q 1 P(21,11) denotes the two sheets of the square
root. In order to solve this nonlinear differential equation we
introduce the substitution
« n ~ Q ! [ A$ E1\ v@ k 8n ~ Q ! sin Q2k n ~ Q ! cos Q # % 2 2D 2` .
~52!
The function « n (Q) has no immediate physical meaning, but
for the vortex at rest it becomes
« n ~ Q ! ——→ AE 2 2D 2` ,
v →0
~53!
which is just the term giving the dependence of the radial
wave number k n , Eq. ~47!, on the quasiparticle energy E.
Subsequently we will express k n and k 8n by « n (Q). This
will lead us to an algebraic equation for « n (Q) which can be
solved approximately.
Inserting Eq. ~52! into ~51! results in
05
\2 2
$ k ~ Q ! 1 @ k 8n ~ Q !# 2 % 2« r 1q 1 « n ~ Q ! .
2m n
~54!
In order to eliminate k 8n (Q) we multiply Eq. ~54! by
(2m/\ 2 )sin2Q,
05k 2n ~ Q ! sin2 Q1 @ k 8n ~ Q !# 2 sin2 Q
S
2 k 2r 2q 1
2m
\2
D
« n ~ Q ! sin2 Q,
~55!
define k r [ A2m« r /\, and insert k 8n (Q)sin Q, as obtained
from Eq. ~52!,
57
MOVING VORTEX LINE: ELECTRONIC STRUCTURE, . . .
k 8n ~ Q ! sin Q5
q 2 A« 2n ~ Q ! 1D 2` 2E
1k n ~ Q ! cos Q, ~56!
\v
Equation ~59! can be satisfied in two ways: first, by
« n (Q)5 const so that
where q 2 P(21,11), and arrive at the quadratic equation
for k n (Q):
q2
05k 2n ~ Q ! 12k n ~ Q !
1
S
A« 2n ~ Q ! 1D 2` 2E
\v
q 2 A« 2n ~ Q ! 1D 2` 2E
\v
D S
cos Q
2
2 k 2r 2q 1
2m
\2
D
« n ~ Q ! sin2 Q.
~57!
The eight solutions of Eq. ~57! are
k n ~ Q ! 52
q 2 A« 2n ~ Q ! 1D 2` 2E
\v
F
1q 3 k 2r 2q 1
2
S
2m
\2
S
F
\2
1 $ q 1 « n ~ Q !@ q 2 A« 2n ~ Q ! 1D 2` 2E #
« n~ Q !
q 2 A« 2n ~ Q ! 1D 2` 2E
\v
3 k 2r 2q 1
2
S
1q 2
2m
\2
DG
sin Q,
~58!
~ \ v ! 2 A« 2n ~ Q ! 1D 2`
D
« n~ Q !
\v
DG
1« m q 2 A« 2n ~ Q ! 1D 2` % sin Q.
2 1/2
« n ~ Q !@ q 2 A« 2n ~ Q ! 1D 2` 2E #
q 2 A« 2n ~ Q ! 1D 2` 2E
This gives solutions u An (r,Q) and v An (r,Q) which for v →0
become functions of the kind given in Eqs. ~49! and ~50!.19
As we already discussed, these are not the appropriate solutions for the vortex at rest and therefore are to be dismissed.
Consequently, here we dismiss the solutions with « n (Q)
5const as well. Second, Eq. ~59! is satisfied if the expression
in the exterior square brackets vanishes. Thus, writing the
two terms in this expression on one common denominator,
one obtains
2 @ q 2 A« 2n ~ Q ! 1D 2` 2E # 2 % 1/2cos Q
F
1q 3 q 1
2 21/2
sin Q
G
~59!
q 1 « n ~ Q ! 1q 2 A
« 2n ~ Q ! 1D 2`
D`
.
~60!
Therefore, the problem of solving the asymptotic BdGE’s
~37! and ~38! has now been reduced to solving Eq. ~59!.
k n ~ Q ! 52
q 2 A« 2n ~ Q ! 1D 2` 2E
\v
cos Q1q 3
~62!
Here we have introduced the definitions
« m [m v 2
~63!
« v [\k r v .
~64!
and
« m is the kinetic energy of an electron pair moving with
velocity v . The energy « v can formally be interpreted as a
Doppler term which shifts the energy of a quasiparticle moving with momentum \k r parallel to the superconducting condensate drifting with velocity v .20
By squaring out the roots Eq. ~62! could be transformed
into an algebraic equation of eighth degree which might easily be solved numerically. This would be a convenient
method of solving exactly the system of coupled nonlinear
differential equations ~41! and ~42!. However, we are interested in analytical approximations for k n (Q). These are obtained by the following reasoning.
The energies « m and « v are connected by the relation
for « n (Q). This equation is simpler than Eq. ~51!. Its solutions determine k n (Q) according to Eq. ~58!. Furthermore,
by inserting Eqs. ~54! and ~56! into Eq. ~44! we can express
c n (Q) by « n (Q):
c n~ Q ! 5
~61!
05q 1 « n ~ Q ! q 3 $ « 2v 22« m q 1 « n ~ Q !
q 2« n~ Q !
d« n ~ Q !
cos Q
dQ \ v A« 2n ~ Q ! 1D 2`
m
d« n ~ Q !
50.
dQ
cos Q
where q 3 P(21,11), like q 1 and q 2 , denotes the sheets of
the square root.
Finally, we eliminate k n (Q) and k 8n (Q) from Eq. ~51! by
inserting Eq. ~58! and its derivative. This results in the equation
05
7909
« m5
« 2v
2« r
~65!
.
With the exception of a negligible number of quasiparticles
traveling nearly parallel to the vortex axis, « r is of the order
of the Fermi energy. Thus it follows from Eq. ~65! that « m is
much smaller than « v or D ` , and we can expand the righthand side ~rhs! of Eq. ~62! with respect to « m . Equation ~58!,
written in terms of « m and « v ,
$ « 2v 22« m q 1 « n ~ Q ! 2 @ q 2 A« 2n ~ Q ! 1D 2` 2E # 2 % 1/2
\v
sin Q,
~66!
S. HOFMANN AND R. KÜMMEL
7910
can be expanded in terms of « m as well. Carefully executing
these expansions and comparing with the results for the vortex at rest, we find that—in order to get a k n (Q) different
from k n of Eq. ~47! and thus have an effect of H 1 (r,Q), i.e.,
vortex motion, on wave propagation—we have to expand up
to first order in « m . In c n (Q), on the other hand, one may
neglect « m altogether and yet get a dependence on « v . After
a straightforward but rather lengthy calculation, presented
elsewhere,19 we find four solutions for k n (Q):
k n ~ Q ! 5 a k r 2 ab
kr
A~ E2 a « v cos Q ! 2 2D 2` ,
2« r
where k r /2« r 5« m /\ v « v , and c n (Q) becomes
c n~ Q ! 5
S
E2 a « v cos Q
~ E2 a « v cos Q ! 2
1b
21
D`
D 2`
D
~67!
1/2
.
~68!
The coefficients a 561 and b 561 will be associated
with the index n in the way given in Table I.
Inserting Eqs. ~67! and ~68! into Eqs. ~39! and ~40! yields
the four asymptotic wave functions of the quasiparticles in a
superconductor with a moving vortex line. As for v →0, the
k n (Q) and c n (Q) approach the expressions ~47! and ~48!,
and the wave functions for the moving vortex smoothly
change over to the wave functions for the vortex at rest.
The deviation of the function k n (Q), Eq. ~67!, from the
wave number k n , Eq. ~47!, as well as the deviation of
c n (Q), Eq. ~68!, from c n , Eq. ~48!, is caused by « v . If one
wants to neglect the « v term and thereby replace the wave
functions for the moving vortex by those for the vortex at
rest, one has to demand that « v be small compared to the
lowest-energy eigenvalues of the bound quasiparticles in a
vortex at rest. These are of the order D 2` /« F , 17 so that « v
,D 2` /« F is required. Replacing k r in Eq. ~64! by its maximum value k F and using j '\ 2 k F /( p mD ` ) for the coherence length j , the last equation can be rewritten as v , v c
[(D ` /\k F )(1/k F j ), where typically k F j '103 . This corresponds to Šimánek’s condition for the validity of the adiabatic approximation.21,16
Subsequently, we will not make the adiabatic approximation but rather assume that v is only about an order of magnitude smaller than D ` /\k F .
B. Angular bound states
Vortex motion creates a new type of quasiparticle states
which are hybrides between bound and scattering states. This
can be seen from the energy and angle dependence of the
k n (Q) of Eq. ~67! in the asymptotic solutions ~39! and ~40!.
TABLE I. Labeling of the different combinations of a and b
values in Eqs. ~67! and ~68!, and thereafter, by the index n .
57
As is well known, in the case of a vortex at rest one has
bound states for u E u ,D ` and scattering states for
u E u >D ` . 17,18 This is because the wave numbers k n in Eq.
~47! are complex for bound states and real for scattering
states. In the case of a moving vortex the functions k n (Q)
depend not only upon E and D ` but also on « v . Furthermore, they are functions of the angular coordinate Q and
thus not quantum numbers. According to Eq. ~67! there are
scattering states with real k n (Q) for all Q only if u E u >D `
1« v . Likewise, the k n (Q) will be complex for all Q only if
u E u ,D ` 2« v . Then the wave functions with n 51 and n
52 vanish for r→` and are those of bound quasiparticles;
the wave functions with n 53 and n 54 are to be dismissed,
because they diverge for r→`.
Wave functions with energies D ` 2« v < u E u ,D ` 1« v
cannot be classified as belonging to bound or scattering
states. For energies in this range it depends on the angular
coordinate Q whether k n (Q) is real or complex. For complex k n (Q) the wave functions with n 53 and n 54 increase
exponentially with r and have to be dismissed as in the case
of the bound quasiparticles, whereas the wave functions with
n 51 and n 52 decrease exponentially. Let us look into the
latter ones in some more detail for E.0 and « v ,D.
First, we consider the wave functions with n 51. From
Eq. ~67! it follows, that k 1 (Q) will be real for
cos Q<
E2D `
.
«v
~69!
With the definition
Q v [ arccos
E2D `
,
«v
~70!
where 0<Q v < p , it follows from Eq. ~69! that k 1 (Q) is real
in the interval
QP @ Q v ,2 p 2Q v #
~71!
and complex in the interval
QP]2Q v ,Q v @ .
~72!
Consequently, the wave functions behave like those of scattering states in the interval ~71! and like those of bound
states in the interval ~72!.
Second, we make the same considerations for the wave
functions with n 52 and find that k 2 (Q) is real for
cos Q>2
E2D `
,
«v
~73!
i.e., for
QP @ Q v 2 p , p 2Q v # ,
~74!
~75!
and complex for
n
a
b
QP] p 2Q v , p 1Q v @ .
1
2
3
4
11
21
11
21
21
11
11
21
Therefore, the wave functions with n 52 behave like those
of scattering states in the interval ~74! and like those of
bound states in the interval ~75!.
The real part of k n (Q) is always approximately given by
a k r . Thus, the wave functions with n 51 have positive ra-
57
MOVING VORTEX LINE: ELECTRONIC STRUCTURE, . . .
dial momentum and describe electrons moving away from
and holes moving towards the vortex center in the angular
range defined by Eq. ~71!, while the wave functions with n
52, having negative radial momentum, describe electrons
moving towards and holes moving away from the vortex
center in the angular range defined by Eq. ~74!.
In the angular range defined by Eq. ~72! the wave functions with n 51 are exponentially damped. The same is true
for the wave functions with n 52 in the angular range defined by Eq. ~75!. Therefore, we call these states tentatively
‘‘angular bound states.’’ In these two angular ranges the current contributions from the outgoing and the incoming waves
do not cancel. Their sum yields a net current flow in the
2x direction, opposite to the flow of the condensate. This
quasiparticle countercurrent, stimulated by the condensate
flow, corresponds to the quasiparticle countercurrent in
superconducting-normal-superconducting ~SNS! junctions
which is responsible for the oscillations of the Josephson
current.22
The width of the energy interval @ D ` 2« v ,D ` 1« v # , for
which there is only a limited range of directions in which a
quasiparticle can move freely, is 2« v . As v →0 this interval
vanishes and there are no angular bound states in a vortex at
rest. For finite vortex velocities v . v c the angular bound
states may be neglected if « v !D ` . In the following we will
assume that this condition holds so that there are essentially
only bound and scattering states. Of these only the bound
states are important for Cooper pair destruction and creation
in Andreev scattering and the resulting supercurrent force.
IV. VORTEX CORE AND ITS VICINITY
We use the model of Nozières, Vinen, and Warren, 5,6 in
which the vortex has a normal core of radius r c with a superfluid of uniform density outside the core. Thus, in the pair
potential of Eq. ~17!, D 0 (r)5D ` for r.r c and D 0 (r)50 for
r,r c , with r c ' j . In Ref. 18 it has been shown that the
bound states calculated with this model do not deviate significantly from those obtained with a spatial variation of the
pair potential one finds from the Ginzburg-Landau equations.
Furthermore, in Ref. 23 ~and for the example of superconducting multilayers! it has been shown that, and how, one
may replace self-consistent pair potentials by equivalent
square-well pair potentials of appropriate heights and widths.
By this method one could determine D ` and r c with the help
of a self-consistent pair potential, if one were interested in,
e.g., very accurate energy spectra. For our purpose, however,
it is sufficient to consider the two quantities as free parameters and see if for physically reasonable values of these
parameters the supercurrent force may be equal to one-half
of the Magnus force. As it will turn out in Sec. V, only the
ratio r c / j ~i.e., the product r c D ` ) matters and is reasonable,
indeed.
Inside the normal core, for r,r c , the wave functions
with
u Nn ~ r,Q ! 5e 2i ~ « v /2« r ! k r r cos Q e inQ J n ~ k e r ! ,
~76!
v Nn ~ r,Q ! 5e i ~ « v /2« r ! k r r cos Q e inQ J n ~ k h r ! ,
~77!
k e5
k h5
S D
2m
\
S D
\
A
1/2
« r 1E n 1
2
2m
2
7911
1/2
A
« r 2E n 1
k r[
A2m
\
S
D
S
D
«m
kr
«m
'k r 1
E n1
,
2
2« r
2
«m
kr
«m
'k r 2
E 2
,
2
2« r n 2
A
« F2
\ 2 k 2z
2m
~78!
~79!
~80!
,
are exact solutions of the BdGE’s ~33! with D 0 (r)50 and
E[E n ; here J n ( z ) is the Bessel function of the first kind
and order n.
The quasiparticle wave functions outside the vortex core,
r.r c , which solve the BdGE’s ~33! with the pair potential,
Eq. ~17! have been calculated in Ref. 19. We do not reproduce the rather lengthy calculations here, but indicate only
the principal steps and approximations. We restrict the analysis to bound states with u E n u ,D ` 2« v and n!k r r c , and to
low velocities, so that « v / p D!1.
For the solutions in r.r c , we make the ansatz
u Sn ,n ~ r,Q ! 5 f n ,n ~ r,Q ! u An ,n ~ r,Q ! ,
~81!
v Sn ,n ~ r,Q ! 5g n ,n ~ r,Q !v An ,n ~ r,Q ! ,
~82!
where the deviations of the wave functions from the
asymptotic wave functions of Eqs. ~39! and ~40! are given by
the functions f n ,n (r,Q) and g n ,n (r,Q) and are due to the
screening current around the vortex core.
Since the deviations from the asymptotic wave functions
are small ~in the sense that they vary slowly in space, as one
knows from the vortex at rest18!, one may neglect the small v
in f n ,n (r,Q) and g n ,n (r,Q). As a consequence these functions become independent of Q and are found to be
f n ,n ~ r,Q ! 5 f̂ n ,n ~ r ! 5 j n ,n ~ r ! e F̂ n ,n ~ r ! ,
~83!
g n ,n ~ r,Q ! 5ĝ n ,n ~ r ! 5 j n ,n ~ r ! e Ĝ n ,n ~ r ! ,
~84!
with
j n ,n ~ r ! 5
F̂ n ,n ~ r ! 52
Ĝ n ,n ~ r ! 5
Ar
exp
S D
in 2
,
2K n r
D ` ~ n1 21 !
i
2
ĉ n e q̂r E 1 ~ q̂r ! ,
8K n r
2« r
i ~ n1 21 !
K nr
1
2
~85!
~86!
D ` ~ n1 21 ! 21
i
2
ĉ n e q̂r E 1 ~ q̂r ! ,
8K n r
2« r
~87!
where
S
D
En
ĉ n 5exp ~ 21 ! n i arccos
,
D`
q̂5
kr
AD 2` 2E 2n ,
«r
~88!
~89!
7912
S. HOFMANN AND R. KÜMMEL
K n 52 ~ 21 ! n k r 1i
E 1 ~ q̂r ! 5
E
kr
AD 2` 2E 2n ,
2« r
` e 2q̂t
r
t
~90!
~91!
dt.
In r c the quasiparticle wave functions u N (r,Q), v N (r,Q)
of the bound states for r,r c must smoothly join the wave
functions u S (r,Q), v S (r,Q) for r.r c which have the correct asymptotic properties discussed in the preceding section.
In view of the angle-dependent matching conditions
u N ~ r c ,Q ! 5u S ~ r c ,Q ! ,
~92!
v N ~ r c ,Q ! 5 v S ~ r c ,Q ! ,
~93!
] N
u ~ r,Q !
]r
U
] N
v ~ r,Q !
]r
U
5
r5r c
5
r5r c
] S
u ~ r,Q !
]r
U
] S
v ~ r,Q !
]r
U
,
~94!
,
~95!
r5r c
This is the same eigenvalue equation as that for a vortex
at rest. For r c ' j and not too small k r solutions with E n
,D ` exist only for N50. Thus, the energy eigenvalues of
the bound states depend on the angular momentum quantum
number n and the wave number k z of propagation parallel to
the direction of the magnetic field. ~For the sake of brevity
we designate the eigenvalues E n,k z just by E n .)
The vortex velocity v does not influence the energy eigenvalues within our approximation, because in the Fouriertransformed matching conditions only the arguments z of the
Bessel functions
J n2n 8 ~ z ! 'J 0 ~ z ! d n,n 8
r5r c
C n ~ rW ,t ! 5
`
u ~ r,Q ! 5Q ~ r c 2r !
2
3
v n ~ r,Q ! 5Q ~ r c 2r ! B n11 v Nn11 ~ r,Q ! 1Q ~ r2r c !
3@ D 1,n v S1,n ~ r,Q ! 1D 2,n v S2,n ~ r,Q !# .
`
( (
D n ,n v Sn ,n ~ r,Q !
~97!
are formed, and the coefficients A n and B n , which appear in
the u N (r,Q) and v N (r,Q), and the D 1,n and D 2,n , which
appear in the u S (r,Q), v S (r,Q), have to be determined in
such a way that the matching conditions are satisfied and the
wave functions are normalized.
After a Fourier transformation with respect to Q the
matching conditions turn into a system of four equations, in
which products of the coefficients A n , B n , and D n ,n with the
Fourier transforms of the u ( n ,)n and v ( n ,)n at r c are summed
over all n from 2` to 1`. Each of these products contains
a Bessel function J n2n 8 ( z ). Since z is of the order of
« v / p D!1, these Bessel functions may be replaced by
J 0 ( z ) d n,n 8 . Thus, only one term is significant in each of the
sums over n, and one can write down explicitly the energyeigenvalue equation of the bound states for n!k r r c . If one
neglects terms of second and higher order in v , this eigenvalue equation becomes
AD 2` 2E 2n
E n k rr cE n p
1 1N p 2
arccos 5
D`
«r
2
«r
S D
3 n1
1 q̂r
e c E 1 ~ q̂r c ! .
2
~101!
and
( B n v Nn ~ r,Q ! 1Q ~ r2r c !
n52`
n 51 n52`
~100!
1D 2,n u S2,n ~ r,Q !# ,
~96!
`
2
D
u n ~ r,Q ! 2iE t/\ ik z
e n e z,
v n ~ r,Q !
u n ~ r,Q ! 5Q ~ r c 2r ! A n u Nn ~ r,Q ! 1Q ~ r2r c !@ D 1,n u S1,n ~ r,Q !
`
v~ r,Q ! 5Q ~ r c 2r !
S
with
( A n u Nn ~ r,Q ! 1Q ~ r2r c !
n52`
( ( D n ,n u Sn ,n~ r,Q ! ,
n 51 n52`
~99!
depend upon v . Expansion of these Bessel functions with
respect to v provides only quadratic or higher-order contributions of v to the eigenvalue equation. These have to be
neglected within our linear approximation.
Because of Eq. ~99!, only the coefficients A n , B n , and
D n ,n which belong to a given n and E n are nonzero in Eqs.
~96! and ~97!. Thus, the eigenfunctions of the bound states
with energy E n are
the superpositions
3
57
~98!
~102!
For the absolute squares of the coefficents we find within
our approximations
u A n u 2 5 u B n11 u 2 5 u D 1,n u 2 2 p k r e 2r c /l 0 ,
~103!
u D 2,n u 2 5 u D 1,n u 2 ,
~104!
where from normalization follows
u D 1,n u 2 5 ~ 8 p r c Le 2r c /l 0 1D 0 ! 21 ,
~105!
with
D 0 58L
E
p
0
S
dQl 2 ~ Q ! exp 2
rc
l 2~ Q !
D
~106!
and
l 05
\ 2k r
Re~ D 2` 2E 2n ! 21/2.
2m
~107!
The velocity- and angle-dependent quasiparticle decay
length l 2 (Q) is defined in Eq. ~113!.
With the help of these functions we will show in the next
section that the supercurrent force on the quasiparticles localized in the vortex core by Andreev scattering transfers
half of the Magnus force from the Cooper pair condensate to
the unpaired core electrons.
57
MOVING VORTEX LINE: ELECTRONIC STRUCTURE, . . .
V. SUPERCURRENT FORCE AND MAGNUS FORCE
We insert the wave functions of Eqs. ~101! and ~102! into
Eq. ~11!. This yields the supercurrent force on a bound quasiparticle in the state characterized by the quantum numbers
n and k z for angular momentum around and momentum
along the z axis; the radial quantum number N is 0. The
superfluid velocity vW s 5 vW s (r) is the sum
vW s ~ r ! 5 vW s0 ~ r ! 1 vW
u S2,n ~ r,Q ! 5 j 2,n ~ r ! e inQ e 2ik r r e 2r/2l 2 ~ Q ! ,
v S1,n ~ r,Q ! 5 j 1,n ~ r ! c 1 ~ Q ! e i ~ n11 ! Q e ik r r e 2r/2l 1 ~ Q ! ,
~109!
Wf D2 ~ k z ,n ! 5
\ Ak 2F 2k 2z eF 0 L
pm
3
E
2p
0
dQeW Q
~111!
where the c n (Q) are given by Eq. ~68! and the l n (Q) are
quasiparticle decay lengths derived from Eq. ~67!,
l n ~ Q ! 5 @ 2 Im k n ~ Q !# 21
\ 2k r
Re$ D 2` 2 @ E n 1 ~ 21 ! n « v cos Q # 2 % 21/2.
2m
~113!
Inserting the superfluid velocity ~18!, the wave functions
u Sn ~ r,Q ! 5D 1,n u S1,n ~ r,Q ! 1D 2,n u S2,n ~ r,Q ! ,
~114!
v Sn ~ r,Q ! 5D 1,n v S1,n ~ r,Q ! 1D 2,n v S2,n ~ r,Q ! ,
~115!
and
u D 1,n u 2
E S
D
e 2r/l 2 ~ Q ! e 2r/l 1 ~ Q !
2
,
r l 2~ Q !
l 1~ Q !
` dr
rc
~116!
where we neglected the integrals over the rapidly oscillating
products @ u S1,n (r,Q) # * v S2,n (r,Q) and @ u S2,n (r,Q) # * v S1,n (r,Q)
and made use of the relation
D ` Im@ c n ~ Q !# 5 ~ 21 ! n
\ 2 Ak 2F 2k 2z
2m
l 21
n ~ Q !,
~117!
which follows from Eq. ~68! with Eq. ~113! for u E n u <D
2« v .
In order to evaluate further Eq. ~116! we make a crude
approximation: We replace r 21 in the integral by its maximum value r 21
c . Thus, all the following expressions for the
supercurrent force would have to be multiplied by a numerical factor smaller than 1 in order to get quantities which
would correspond to the exact integral. This way we can
carry out the integration over r and obtain an approximate
analytical expression for Wf D2 . In order to perform the integration over Q we write the angular unit vector eW Q in terms
of Cartesian unit vectors eW Q 52eW x sin Q1eW ycos Q. Using furthermore the relation l 1 (Q)5l 2 ( p 2Q), which follows directly from Eq. ~113!, and observing that both l n (Q) depend
on the angular coordinate Q only via cos Q, we find
Wf D2 ~ k z ,n ! 5eW y
3
~110!
v S2,n ~ r,Q ! 5 j 2,n ~ r ! c 2 ~ Q ! e i ~ n11 ! Q e 2ik r r e 2r/2l 2 ~ Q ! ,
~112!
5
and the pair potential D(r,Q)[0 for r,r c
D(r,Q)5D ` e 2iQ for r>r c into Eq. ~11! we find
~108!
of the screening current velocity of Eq. ~18! and the relative
velocity of Eq. ~31!. Since v !D ` /(\k F ), the relative velocity vW is small compared to the screening current velocity
vW s0 ( j ) in the vicinity of the core and will be neglected in
vW s (r). The pair potential D5D(r,Q) is given by Eq. ~17!.
As in Sec. IV we adopt the local model for the considered
high-k superconductor: The pair potential depression near
the vortex center is replaced by a normal core of radius r c
' j , i.e., D(r,r c ,Q)[0, while outside of this core the
modulus of the pair potential is replaced by the value D ` .
We stay within the approximations introduced in Sec. IV.
According to Eq. ~11! contributions to the supercurrent
force Wf D2 come from the regions of finite D, i.e., r.r c .
Since the supercurrent force acts only on quasiparticles
which create or destroy Cooper pairs in Andreev
scattering,10–12 the u and v functions in Eq. ~11! are the
exponentially decaying wave functions of the bound states,
given by Eqs. ~101! and ~102! for r.r c .
Within our approximations we may replace the functions
f̂ n ,n (r) and ĝ n ,n (r) by j n ,n (r) @the small functions F̂ n ,n (r)
and Ĝ n ,n (r) are only important for the derivation of the eigenvalue equation ~98!#, and write the wave functions of the
bound states in the superconducting region outside the core
as
u S1,n ~ r,Q ! 5 j 1,n ~ r ! e inQ e ik r r e 2r/2l 1 ~ Q ! ,
7913
4\ Ak 2F 2k 2z eF 0 L
p mr c
E
p
0
u D 1,n u 2
S
dQ cos Q exp 2
D
rc
.
l 2~ Q !
~118!
The supercurrent force on a quasiparticle has only a component in the y direction, i.e., perpendicular to the relative velocity vW 5 vW T 2 vW L , just like the Magnus force of Eq. ~1!.
Numerical integration of Eq. ~118! yields in the supercurrent force as a function of E[E n,k z . This is shown in Fig. 2
for k z 50 ~solid line!.
The supercurrent force depends nearly linearly on the
quasiparticle energy for u E u ,D ` 2« v and vanishes for u E u
.D ` 2« v . Note that, in Fig. 2, « v 50.1D ` , which is rather
large. For smaller values of « v the supercurrent force may
well be approximated by a linear function in E for u E u
,D ` and by zero for u E u .D ` , as shown by the dashed line
in Fig. 2.
Analytically a linearized expression for the supercurrent
force can be obtained by a first-order Taylor expansion
around E n 50. With the definition
k z [k F cos v ,
this yields
~119!
S. HOFMANN AND R. KÜMMEL
7914
FIG. 2. The magnitude eW y • Wf D2 of the supercurrent force Wf D2 on
quasiparticles in a moving vortex line as a function of the quasiparticle energy E, in arbitrary units. Here quasiparticles with wave
number k z 50 parallel to the vortex axis are considered. Solid line,
Wf D2 from numerical integration of Eq. ~118!; dashed line, Wf D2 linearized according to Eq. ~120!.
Wf D2 ~ v ,n ! 5eW y
e v F 0 k F E n sin v
,
2 p D ` @ r c 1l̂ 0 ~ v !#
FIG. 3. The function I(r c / j ) as defined by Eq. ~130!.
The term ( ] E/ ] n) v21 can be obtained from the eigenvalue
equation ~98!. Approximating in it e x E 1 (x)'1/x turns it into
n1 2
k F r c E sin v p
E
5 1N p 2
,
arccos 2
D`
«r
2
k rr c
1
~121!
In deriving Eq. ~120! we have kept only terms linear in v .
W D2 acting on the core because of Andreev
The total force F
scattering is the sum of all supercurrent forces acting on the
bound quasiparticles. In the sum of Wf D2 , approximated by Eq.
~120!, over all occupied quasiparticle states
FW D2 52
(k (n Wf D2~ v ,n ! f 0~ E n ! ,
S D
]E
]n
~122!
the spin degeneracy is taken into account by a factor of 2 and
f 0 (E n ) is the Fermi distribution function. For the sake of
simplicity we limit the calculation to T50 K so that only the
quasiparticle states with negative energy—which make up
the ground state of the normal core—are occupied: f 0 (E n )
51 for E n ,0 and f 0 (E n )50 for E n .0.
The sum over k z is transformed in an integral over v :
(
k
z
E
Lk F
dk z 5
2p
2k F
kF
E
p
0
d v sin v .
~123!
For fixed v the energetic separation between two energy
eigenvalues differing in n by 1 is very small.18 Therefore, we
can transform the sum over n in an integral over E n [E:
(n → E2D
0
dE
`
S D
]E
]n
21
~124!
;
v
the lower integration limit has been approximated by
2D ` , in the spirit of the linear approximation of Fig. 2 and
Eq. ~120!. Thus the total supercurrent force is
FW D2 5
Lk F
p
E
p
0
d v sin v
E
0
2D `
dE
S D
]E
]n
21
v
21
5
v
k F r c sin v
AD 2` 2E 2
1
Wf D2 ~ v ,E ! .
~125!
2mr 2c
\2
~127!
.
We insert Eqs. ~120! and ~127! into Eq. ~125!, assume that
the spectrum and the density of all bound states can be approximately described by Eqs. ~126! and ~127!, and carry out
the integration over E:
W D2 5eW y
F
e v F 0 Lk 2F
2 p 2D `
F
E
p
0
dv
sin2 v
r c 1l̂ 0 ~ v !
3 2D ` k F r c sin v 2
z
L
→
2p
~126!
with N50 for not too small v . From Eq. ~126! one finds
~120!
where
\ 2 k F sin v
l̂ 0 ~ v ! 5
.
2mD `
57
D 2` mr 2c
\2
G
.
~128!
With l̂ 0 ( v ) according to Eq. ~121!, the coherence length j
5\ 2 k F /( p mD ` ), and the electron density n s 5k 3F /(3 p 2 ),
Eq. ~128! results in
SD
r
1
W D2 5 FW MagnusI c ,
F
2
j
~129!
with FW Magnus given by Eq. ~1! and
I
SD E
rc
3r c
5
j
pj
p
0
d v sin2 v
r c / j 1 p sin v
.
r c / j 1 ~ p /2! sin v
~130!
The function I(r c / j ) is plotted in Fig. 3. It assumes the
value 1 for r c '0.4j . Considering the approximations made
on the way to Eq. ~129! this is reasonable. Without the approximation of replacing 1/r by 1/r c in the integral ~116!,
1
W D2 5 FW Magnus ,
F
2
~131!
would result for a somewhat larger value of r c .
VI. SUMMARY AND OUTLOOK
The motion of a vortex line relative to an applied supercurrent causes an angular asymmetry in the wave functions
of the unpaired quasiparticles bound in the vortex core: Outside the core, in a given direction characterized by the azi-
57
MOVING VORTEX LINE: ELECTRONIC STRUCTURE, . . .
7915
muthal angle Q, the damping of the wave functions which, at
the core boundary, match to radially outgoing electron—and
radially ingoing hole—wave functions is different from the
damping of the wave functions which match to radially outgoing hole—and radially ingoing electron—wave functions.
The different penetration lengths l 1 (Q) and l 2 (Q) in Eq.
~116! are responsible for the supercurrent force and the resulting half of the Magnus force. This angle-dependent different damping of outgoing and ingoing waves corresponds
to the semiclassically computed10 different rates of Cooper
pair formation and destruction— and the associated different
momentum transfers from the circulating condensate to the
core electrons—in electron→hole and hole→electron scattering processes at the core boundary. The detailed quantum
mechanical calculations presented in this paper confirm our
earlier semiclassical considerations: The sum of all supercurrent forces, which originate from Cooper pair momentum
transfers to the core electrons by Andreev scattering, is equal
to one-half of the Magnus force. It explains microscopically
the ‘‘interface force’’ in the Nozières-Vinen-Warren theory
of vortex motion.5,6
Recently Stone24 showed by a quasiclassical geometric
optics model that in a moving vortex spectral flow is
thwarted by an analog of Bloch oscillations originating from
the discrete nature of the core state spectrum. This is a consequence of the fact that due to the supercurrent force the
Andreev reflection fails to be perfectly retroreflective and
causes the core bound states to precess in a sense opposite to
that of the superflow. Therefore, momentum transferred to
the vortex core can only escape via relaxation processes. We
have implicitly taken into account such processes by assuming a constant drift velocity vW of the core quasiparticles rela-
tive to the condensate. That means that all momentum gains
from the supercurrent force and the electrostatic field in the
core are assumed to be dissipated right away to the lattice.
@This is also the condition for the validity of Eq. ~1!.# This
way the nonequilibrium effects relevant in our context have
been incorporated. The constant drift velocity causes the
asymmetry in the wave functions, discussed above, which
gives rise to FW D2 .
Since we have worked with the mean-field TdBdGE’s,
our analysis is, strictly speaking, only valid for conventional
type-II superconductors. In order to extend it to the strongly
correlated high-temperature superconductors one must use
the time-dependent density-functional Bogoliubov–de
Gennes equations ~TdDFBdGE’s!.25 The integral equations
defining their vector and pair potentials in terms of exchange
correlation functionals of the gauge-invariant current density
W and the anomalous density D IP ~which measures offdiagonal long-range order! are difficult to solve, if one takes
into account all electromagnetic fields. If, however, one neglects the corresponding scalar and vector potentials—as one
does when calculating the electronic structure of vortices in
conventional superconductors—things become simpler.
Then, at T50 K the TdDFBdGE’s are essentially given by
Eqs. ~21!–~23! with a pair potential which is the sum of the
mean-field pair potential and the ~negative! variational derivative of the exchange-correlation functional Q xc @ W ,D IP#
* .15,26 Thus, the calculation of an appropriwith respect to D IP
ate exchange-correlation functional is crucial for the analysis
of the influence of Andreev scattering on vortex motion in
high-temperature superconductors. This is a task for further
research.
*
15
Present address: Fakultät für Elektrotechnik, Ruhr Universität Bochum, D-44780 Bochum, Germany.
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