Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
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)
[email protected]/~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