Quasiparticle Interference Patterns in Vortex Cores of Type-II Superconductors

1.

ИФМ РАН
Quasiparticle interference patterns in vortex
cores of type-II superconductors
A.S.Mel’nikov
A.A.Bespalov
MIPT Center for Advanced Mesoscience and
Nanotechnology,
Laboratory of Spin Phenomena in
Superconducting Nanostructures and Devices,
Inst. Phys. Microstr. RAS
Inst. Phys. Microstr. RAS
D.Annenkov
MIPT
Landau Institute for
Theoretical Physics

2.

Electrons and holes in vortex core. CdGM spectral
branch.
Patterns of local density of states. How should they look
like?
What is seen in STM/STS experiments? Some recent
puzzles.
Influence of defects on anomalous spectral branches in
vortices.
Small angle quasiparticle scattering on smooth potential
in vortex core. Non - monotonic spectral branches.
DOS patterns for a vortex in the presence of smooth
potential +point defect
“Backscattering” for chiral spectral branches?
Outline

3.

Bogolubov – de Gennes theory.
Quantum mechanics of electrons and holes in vortex
core in type-II superconductors
2
ie
2
A
2 2
kF
c
U
u v u
2m
2m
2
2
ie A
2 2
kF
c
*
U
v
u v
2m
2m
| M r | e
iM

4.

Quasiclassical approximation. Andreev equations.
ie
i VF
A U u v u
c
ie
i VF
A U v *u v
c
k F
Coherence
length
kF
1
VF

5.

Andreev bound states
2
S
N
e
vx , k x
S
2
2e
h
d
x

6.

Trajectory passing through the vortex center
Superconducting phase difference = pi
s
0
ˆ
H i V ˆ z ˆ x Re ( s ) ˆ y Im ( s, b)
s
s
u
1
1
ˆ
0 exp
Re (t )dt
v i
V 0
(s )
energy=0
0
s

7.

Analogy with Josephson junction
d
Spectrum of subgap
states
E cos
2

8.

Trajectory with finite impact parameter
k k (cos p , sin p )
r
s
k
C.Caroli,
P.G.de Gennes,
J.Matricon
(1964)
p
b
Impact parameter
classical
trajectory
k k k
2
F
2
z
0
(k )
k
Fermi level
2

9.

Precession of classical trajectory
p
Deviation from exact
backscattering due to the
superflow around vortex
t
Vs
hole
electron
p
Vs
p ~
V
t ~
VF
p VsVF Vs k F
~
~
~ 0 ~
t
V
k k
Precession
frequency

10.

Precession of classical trajectory
p
Deviation from exact
backscattering due to the
superflow around vortex
t
Vs
hole
electron
p
Vs
p ~
V
t ~
VF
p VsVF Vs k F
~
~
~ 0 ~
t
V
k k
Precession
frequency

11.

Precession of classical trajectory
Deviation from exact
backscattering due to the
superflow around vortex
Vs
hole
electron
t ~
t
p
Vs
p ~
V
p
VF
p VsVF Vs k F
~
~
~ 0 ~
t
V
k k
Precession
frequency

12.

Quantum mechanics of precessing trajectories
[ p , ˆ ] i
Spectrum as a function of the momentum
component along the vortex axis
Quasiparticle subgap branch:
Bohr-Sommerfeld quantization
rule
0
0
EF
2
( )d 2 (n )
p
p
0
0 n
n (k z )
b
k
minigap
0
0 ~
kF
5 2
3 2
1 2
kz
kF
1

13.

LDOS patterns
m=1
spectrum
2
2
LDOS
profiles
Classical impact
parameter at a
fixed energy
0
1 0
p
p

14.

CdGM levels and density of states in clean
superconductors. STM/STS patterns
STM tip
2
1
En
n
EF
2
dI/dV (x,y)
H.F.Hess, PRL 64, 2711 (1990)

15.

Some recent STM experimental results

16.

Some recent STM experimental results

17.

18.

Why the local density of states can differ from the standard
CdGM picture?
Influence of defects?

19.

Modification of the subgap spectra due to
normal scattering by a defect
Impurity atom in the core
Columnar Defects
Andreev scattering
a
normal scattering
(point defect)
Larkin, Ovchinnikov 1998;
Koulakov, Larkin 1999
specular
reflection
A.S.M., A.V.Samokhvalov, Zubarev 2009

20.

Vortex pinned by columnar defect.
Analogy with Josephson junction
R F
E cos
| r | e
i
ˆ ( R) 0
Boundary condition:
E 0 cos cos 0
2
b2
E 0 sign(b) 1 2
R

21.

Vortex pinned by columnar defect.
Spectrum and LDOS. Growing
minigap.
r
| r | 0
r2 2
v
m
M=1
2
v 0
DOS

22.

Scattering of quasiparticles on smooth
potential
k F R 1
bVc (0)
1
b R: ~
FR
b R: 0
0
Impact
parameter
kF
classical
trajectory
b
Length scale of
defect potential
bVc (0) bVc (0)
Vc (0)
~ ~
~
~ CdGM
FR
R
k F R
R

23.

Transformation of anomalous spectral branch.
We can change the slope of the anomalous spectral branch.
Vc 0
2
kF R

24.

Transformation of anomalous spectral branch.
We can change the sign of the slope
Vc 0
2
kF R
Several modes with different
angular momenta and group
velocities at a given energy
We can not break the subgap branch, BUT CAN MAKE IT NON MONOTONIC !

25.

Koulakov-Larkin approach

26.

Density of states

27.

Angular modulation of wavefunctions for a vortex + smooth potential + point
impurity.
Necklace?

28.

Can we apply this reasoning to the problem of backscattering for edge modes in
topological insulators?
Edge modes with opposite
group velocities?
k||

29.

Smooth defect potential can result in
• non-monotonic dependence of subgap energy vs angular
momentum
• several modes at a given energy level
• “backscattering” of states at a chiral branch
• necklace like LDOS interference patterns
• soft minigap
• step – like profiles of minigap map
Григорий Ефимович, с Юбилеем Вас!

30.

Quasiclassical approximation for BdG equations
2
ie
2
A
2 2
kF
c
U
u v H u
2m
2m
2
2
ie
A
2 2
kF
c
*
U
v
u H v
2m
2m
ˆ u , v ˆe iS
Coherence
length
VF
| S | k F
1

31.

Quasiclassical approximation
1
i (| p| k ) s
ˆ
( p)
e
ˆ ( s, p )ds
k
ik z z 2
u
e
ik r cos( p )
ˆ
e
ˆ (r cos( p ), p )d p
v 2 0
k k F2 k z2
b k
k k cos p , sin p
0 v (r)e i
ˆ e
i p i ˆ z p 2
fˆ ( s )
Hˆ fˆ Efˆ
ˆ
ˆ
H i V z ˆ x Re ( s ) ˆ y Im ( s )
s

32.

Электронная структура вихревого состояния
Связанные состояния фермионных возбуждений в коре вихря
Профиль сверхпроводящей щели: потенциальная яма для электронов
0
kF
Оценка минищели в спектре возбуждений
r
z
2 0
0
2
min ~
~
~
2
m
m vF k F
C.Caroli, P.G.de Gennes,
J.Matricon (1964)
(k r )
k r
k r k F2 k z2

33.

STM наблюдения вихрей. DOS
2
LDOS u (r ) ( )
YBCO
NbSe2
360 nm
PRL, 101, 166407 (2008)
PRL, 75, 2754 (1995)

34.

More reference STM images of vortex cores.

35. STM experimental results

H. F. Hess et al PRL (1990)
I. Maggio-Aprile et al PRL (1995)
NbSe2
YBCO
Ch. Renner et al PRL (1991)
B. Sacepe et al PRL (2006)
Boron
doped
diamond
x
Nb1-xTaxSe2

36.

STM experimental results
I. Guillamon et al PRL (2008)
M.R. Eskildsen et al PRL (2002)
MgB2
360 nm
NbSe2
B. W. Hoogenboom et al PRB (2000)
BSCCO
theor ~ 0 H c 2 exp

37. STM experiment: Anomalous Large Vortex Core

N. C. Yeh et al EPL (2009)
YBCO

38.

Влияние беспорядка на электронную структуру вихрей.
От чистого к грязному пределу.
1. Один атом примеси в коре
Vortex line with an impurity
atom in the core
Larkin-Ovchinnikov
1998
superflow
2. Увеличение конентрации
примесей = размытие zero-bias
аномалии в центре вихря

39.

R F
Дефект конечного радиуса
Спектр квазичастиц на отражающихся траекториях.
Аналогия с джозефсоновским контактом.
E 0 cos cos 0
| r | e
i
b2
E 0 sign (b) 1 2
R

40.

Аномальные ветки спектра в многоквантовых вихрях
Влияние дефекта
спектры
M=1
2
M=2
DOS
Минищель растет
M=3