Quantal Andreev Billiards and their Spectral Geometry

Inanc Adagideli & Paul M. GoldbartDepartment of Physics and

Seitz Materials Research Laboratory

University of Illinois at Urbana-Champaign

Phys. Rev. B 65 (2002) 201306(R)

Int. J. Mod. Phys. B 16, 1381 (2002)

goldbart@uiuc.eduw3.physics.uiuc.edu/~goldbart

Outline of the talk

•What is spectral geometry?

•What is Andreev reflection?

•Schrödinger billiards; Andreev billiards

•Andreev billiards: Classical properties

•Andreev billiards: Quantal properties

•Applications to high-Tc superconductors

Planck’s BB radiation spectrum

ω

T3)/( TcV κh>>

U

•cubic oven filled with electromag. radiation

•thermal equilibrium •volume •temperature •2 polariz. states, periodic boundary cond’s

⇒ spectral energy density at frequency

1)/exp()(

3

32 −=

TcVU

κωω

πω

h

What is spectral geometry?

•Planck •Lorentz •Hilbert •Weyl

What is spectral geometry?

•Lorentz (Wolfskehl Lect. Göttingen1910): suggests short-wave DOS (henceBBRS) depends only on oven volume not shape

•Planck (1900): introduces quanta to derive black body radiation spectrum

•Hilbert (1910): “…will not be proven in my lifetime…”

•Weyl (1911): Lorentz correct; via Hilbert’s integral equation theory

Spectral geometry:2D Laplacian -- “Can one hear the shape of a drum?”

•study eigenvalues of Laplacian: with on the boundary

•origin: energies of Schrödinger particle in box; or normal mode freq’s of scalar wave equation

•collect eigenvalues in a distribution (DOS)

•study connection of with shape, especially at large

Ψ=Ψ∇− E2 0=ΨE

E

)(Eρ)(Eρ

•K

ac

Spectral geometry: 3D Laplacian-- e.g.: smoothed DOS at large energy --

with B.C.s

•volume , surface

•local principal radii of curvature and

ππρ

164)( 2

SEVE m≈

Ψ=Ψ∇− E2

( ) L+++ −−

Σ∫ E

RRdS 121

121 1

21

12π

V Σ

1R 2RV

ΣDirichlet (-)Neumann (+)

Spectral geometry: 3D Laplacian-- e.g.: DOS oscillations --

•also gives long-range clustering of levels (semiclass. remnant of quant. shell structure)

•evaluate MRE in saddle-point approximation (specular reflection, closed paths,…)

• dominated by classical periodic orbits (cpo), w/ tracings

•trace formulae for DOS osc’s(classical ingredients only)

)(Eρ

)/)((sin)(.trac

2/1.o.p.c

osc jjp

ljp

j pEpS

AE

jαρ +≈ ∑∑ + h

h

Potential Theory I

•View as due to charge-layer or dipole-layer on boundary

– solve

– with on boundary (Dirichlet), or

– with on boundary (Neumann)

•BVP of classical electrostatics

0)(2 =∇ rr ϕ)()( αψαϕ =)()(n αψαϕα =∇⋅

)(rϕ

∫Σ

∇⋅= )( ),(n )( 0 βµβσϕ ββ rr Gdd

∫Σ

= )( ),()( 0 βνβσϕ β rr Gdc

)(βν)(βµ

0G is familiar Coulomb potential

apply Neumann BC here

++

++

++

+

Potential Theory II–charge-layer potential continuous

(but electric field discontinuous)–similarly dipole-layer potential discontinuous

ϕzzE

z ν21

∫Σ

∇⋅+−=∇⋅ )( ),(n )(21)r(n lim 0 βνβασανϕ ααβα Gdcrα→r

z

)(αψ

ϕzE

– solve F/II integral equation for by iterationν

S.G. for Laplacian: Balian-Bloch scheme•seek Green function

•obeys

plus for on boundary

•use rep. of from classical potential theory

•obtain via Multiple Reflection Expansion

),,( zG rr ′( )rrrrr ′−=′−−∇ δ),,()( 2 zGz

0),,( =′ zG rr r

GG

= + + …+

•evaluate terms via large - asymptotics

•gives DOS via)(Eρ 01 |TrIm)( iEzGE +=−= πρ

E

Some further settings for S.G.

•acoustics and elasticity

•thermodynamics (e.g. electronic, magnetic and vibrational properties of granular matter)

•superfluid films, Casimir effects

•nucleation

•nuclei, atomic clusters, nanoparticles,…

What is Andreev reflection?[Andreev, Sov. Phys. JETP 19 (1964) 1228]

incoming electronwhat happens?

•low-energy electron quasiparticle approaches superconductor from normal region

progressively more superconductivity

E

Fp− Fp

electronhole

holeelectron branch

pFermisea

electron

quasiparticle excitation spectrum

Charge-reversing retro-reflection•incident low-energy electron quasiparticle

retro-reflected as hole quasiparticle

progressively more superconductivity

electron

retro-reflected hole

(and vice versa)incomingsupercond.

pair potential

)(r∆

E

Fp− Fp

electronhole

holeelectron branch

p Fermisea

electronhole

quasiparticle excitation spectrum

Andreev reflection: semiclassics•gap excludes single-particle excitations•velocity must be reversed•but momentum cannot be…

4

FFFFFF

10/// −≈∆

≈∆

≈∆

≈≈ ∫∫ Evpdt

pdt

pdtdp

pp ξξξδ

•e/h conversion with retro-reflection•e acquires mate ⇒ Cooper-pair + hole

•Snell’s law: from action with reversed momentum

Schrödinger billiards

•shape is the only “parameter”•classical mechanics is “geometry”•DOS oscillations related to “polygons”

via trace formula•no separation of periods•quantum implications of classical chaos

Ψ=Ψ∇− )/2( 22 hmEwith on boundary 0=Ψ

•quantum mechanics of finite systems

Andreev billiards

•normal region surrounded by superconductor

•1-quasiparticle energy gap; low-energy states confined

•mechanism: Andreev reflection from boundary

•shape-spectrum connection– DOS oscillations related to chordal orbits and

creeping orbits via trace formula– separation of periods

What are they?

Basic issues

[Kosztin, Maslov & PMG, Phys. Rev. Lett. 75 (1995) 1735]

Andreev billiards: Model system

⇒ Bogoliubov-de Gennes eigenproblem

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛⎟⎠

⎞⎜⎝

⎛+∇∆−∆−−∇−

)()(

)()(

)()(22*

22

rr

rr

rr

vuEv

uκ

κ

10

10

),(ˆ),(

),(ˆ),(

ΦΨΦ=

ΦΨΦ=+↓

↑

ttv

ttu

rr

rr generic 1-qp state (Heisenberg rep.)

ground state (Heis. rep.)

remove spin-up; add spin-down (Heisenberg rep.)

e/h-qp wave functions •one-e/h-qp states:

( ) ⎟⎠⎞

⎜⎝⎛=∆ potential

pairenergy

excitationenergyFermi,,

22

2

Em

κhwhere

Andreev billiards: BB scheme•seek e/h Green function via

Balian-Bloch type scheme (potential theory plus MSE)

•integrate out propagation in superconductor; yields eff. Mult. Reflection Expansion

L

•evaluate via two asymptotic schemes

(A)

(B)

•sets which classical reflections rules hold

).const/(0/, 2 κκκ ∆→∆∞→ LL.const/, 2κκ ∆∞→L

Andreev billiards: Effective reflection•integrate out propagation in superconductor:

• Separate propagation (a) short range (b) long range

• sum all s. r. propagationeffective MRE with renormalized Green function

• l.r. propagation in superconductor vanishes• e/h interconversion dominant process

Normal reflection Andreev reflection

Andreev billiards: Scheme A

•limits:•essentially Andreev•classical rule:

perfect retro-reflection•ladder of states on chords

).const/(0/, 2 κκκ ∆→∆∞→ LL

( ) ( ) nEE πκ 2cos2 1 =∆−− −βα αβ

chord lengthe/h momentum difference

reflection phase-shiftinteger

•leads to the scheme-A DOS:

βαθαβθ

( ) ( ) ( )∫Σ

− ∆−−−≈ EiEiE 1cos2exp1coscosRe)( κθθρ βααβ βα

).const/(0/, 2 κκκ ∆→∆∞→ LLVan Hove-type singularities

•scheme A:

− sharp spectral features− signature of bunching of

energy levels − line-shape “quality” from

signature of Hessian − quantitative DOS via stationary

lengths and end-point curvatures

•role of stationary chords

•can “hear” lengths of the stationary chords

“Hearing” the geometry

•several stationary chords•yield distinct features•some examples…

chord length islocal maximum

ρ

E ρ

Echord length islocal minimum

•scheme B:•retro-refl. no longer perfect•now class. periodic orbits are

– stationary chords (coarse)– creeping orbits (fine)

•separation of period scales

S.G. of Andreev billiards: Scheme B.const/, 2κκ ∆∞→L

results for a circular Andreev billiard

What’s missing?•scheme B:•make perfect charge-conversion model •compare w/ periodic orbit results ⇒ better fit

.const/, 2κκ ∆∞→L

⇒ charge-preservingprocesses

are missing

Normal reflection in Andreev Billiards

•scheme B:•retro-refl. no longer perfect•charge interconv. no longer perfect

.const/, 2κκ ∆∞→L

Some conclusions•Quantal properties of Andreev billiards

– discussed in terms of spectral geometry

•Basic structures– Andreev’s approximation (all chords) – fine structure needs creeping orbits etc.

•“Can “hear” novel geometrical features – distribution of chord lengths – stationary chords, degeneracies and curvatures

•Still to do…– self-consistent superconductivity– soft and ray-splitting Schrödinger billiards – making an Andreev billiard!

A few references• Andreev, Sov. Phys. JETP 19 (1964) 1228

• Abrikosov, Fundamentals of the theory of metals• Kac, Am. Math. Monthly 73 (1966) 1

• Balian & Bloch, Ann. Phys. (NY) 60 (1970) 401

• Balian & Bloch, Ann. Phys. (NY) 69 (1972) 76

• Baltes & Hilf, Spectra of finite systems (1976)

• Guenther & Lee, PDEs of mathematical physics and integral equations (Dover, 1996; Sec. 8-7)

• Berry, in Chaotic behaviour of deterministic systems (Les Houches 1981; ed. by Iooss et al.)

• Kosztin, Maslov & PMG, Phys. Rev. Lett. 75 (1995) 1735

• Adagideli & PMG, Phys. Rev. B 65 (2002) 201306(R); Int. J. Mod. Phys. B (2002, in press)

Side-products I: Impurity states in d-wave supercondutor

•along trajectory determined by classical position and velocity:

•semiclassical motion in presence of impurity

•quantum-mechanical electron-hole scattering along classical trajectory

⎟⎠⎞⎜

⎝⎛=⎟

⎠⎞⎜

⎝⎛⎟⎠⎞

⎜⎝⎛

∂∆∆∂−

vuEv

uik

iks

s

F*

F

22

)()( cc xx Vs −∇=&&µ

∆

• nonmagnetic impurity

)( )();( cFc sks xx &∆∆ ≡

Side-products I: Impurity states in

d-wave supercondutor

•a realization of Witten’s supersymmetricquantum mechanics

•focus on E=0: 0)2( F =∆±∂ mϕsk( )F2/)(exp ksds∫ ∆±=⇒ ±ϕ

•normalizability ⇒ low-energy states for trajectories on which changes sign∆

+

-

-

•gives state at E=0•degeneracy proportional to

linear extent of impurity•no dep. on details of ∆

• does not give state at E=0• shape of DOS depends on

details of ∆

changesign no if ,0

in changesign if ,1

ies trajectors/c allover )(

∆

∑+= RkE FBG δρρ

0

0.4

0.8

1.2

1.6

2

2.4

2.8 Center10 Å20 Å

-0.1 -0.05 0 0.05 0.1dI

/dV

[10-8

Ω−1

]

Voltage [V]

Local Density of States Near an Atomic Scale Defect

+

-

-

Impurity states in d-wave supercon.Yazdani et al.; Hudson et al. (1999)

Side products I: Impurity in d-wave

•tunneling through the impurity

•diffraction effects in scattering

•transition between zero energy states

•splitting of E=0 peak

Side products II: Imp. in pseudogap

Pseudogap

Superconductor

LDOS

E

Thermal ∆E?

Superconducting correlations?

E

Quantum hΩ?

E

Phase Diagram

The

rmal

Flu

ctua

tion

s

DopingQuantum Fluctuations