Metals Mesoscopic SystemsExpected:A(ω)
ωth ω
A(ω)
ωth ω
Observed:
core level
{ε}
EF
EC
Absorptionh ν
Fermi Edge Singularities in the Mesoscopic X-Ray Edge Problem
Martina Hentschel,Denis Ullmo, and Harold U. Baranger
Duke University
• finite number of electrons on discrete level
• coherent, chaotic geometry
• fluctuations
{λ}
NIRT program
A(ω)
ωth ω
?
e.g. Peaked Edge L2,3-edgesimple metals like Al, Mg, (Na)
from K. Othaka, Y. Tanabe, RMP 62 2929 (1990):GaAs-AlxGa1-xAs Quantum well, Lee et al. (1987)
A(ω)I(ω)
The ‘classical’ X-Ray Edge ProblemSingularities at the Fermi edge threshold in
X-Ray Emission or Absorption Spectra of, e.g., metals
although <ψi | ϕi> ∼ 1:<Ψ|Φ> → 0 (as N →∞)
+
1 of 1023 electrons
Many-body ground state |Φ>made of single particle wf. |ϕi>
What happens when a core electron is excited?
~1023 cond. electrons respond Kondo Problem
V
Sudden perturbation
Anderson Orthogonality Catastrophe
Anderson Orthogonality Catastrophe (AOC)
P. W. Anderson, Phys. Rev. Lett. 18 1049 (1967)
0~~ ε−
ΦΨ
N
Important in:• Fermi edge singularities of x-ray and photoluminescence spectra• Kondo physics• Tunneling (e.g. in double quantum dots)• Similar phenomenon in particle physics
ground stateinitially
ground stateunder perturbation V
Orthogonality Block
screeningdipole selection rules
Orthogonality blockdue to AOC
Peaked or rounded edge ?
Many-body effect“Mahan’s enhancement”
Competition
acts universal
finite Nfinite N
chaotic geometry relative strength ?
Mesoscopic effects
Sample-to-sample fluctuations
Peaked or rounded edge ?
∑
∑
+=
++−=
−∝
l
l
lo
Fl
l
l
lo
th
lZ
l
El
l
A
πδ
δ
δ
πδ
πδ
β
ωωω β
)12(2 :rule sum sFriedel'
channel excitedoptically ....
energy Fermiat taken , mom. ang.for shift phase .... where
)12(22 with
)()( ption Photoabsor
0
2
Anderson orthogonality catastrophe(all δl )
counteracting(Mahan) many-body process (δlo only)
Citrin, PRB (1979)Tanabe and Othaka (1990)
I. Introduction
II. Mesoscopic Anderson Orthogonality Catastrophe
III. X-Ray Photoabsorption Spectra: Mesoscopic vs. Bulk-like
IV. Conclusion, Experimental Realizations
Outline of talk
• Model, numerical method, results
• Fermi golden rule approach, role of dipole matrix elements
AOC for a rank-1 perturbation VTanabe and Othaka, RMP (1990)Aleiner and Matveev, PRL (1998)
Ψ→+
Φ→
...},{ˆˆ :perturbed
...},{ˆ :dunperturbe
κκ ψλ
φε
VH
H kk
∏ ∏= +=
−−
−−=ΦΨ≡∆
M
i
N
Mj ijij
ijij
filled0
empty
1
22
))((
))((
εελλ
λεελ= f (eigenvalues only)
00 rrVN−=
e.g. core hole left behind at r0
è overlap between perturbed and unperturbed ground states:
• unperturbed level εk: equidistant (“picket fence”, “bulk-like”)
• perturbed level λκ: Schrödinger equation
Example for a rank-1 perturbation
Vk k
11: =
−∀ ∑ ελ
κκ
λ0 λ1 λ2 λ3 λ4 λ5 V big
V small
dV
d kkbπ
ελπ
δ arctan)(:) (Nshift phase =−≡∞→
∼δ
-4 -2 0 2 4 6 8λ-50
-25
0
25
50
y
6 level, attractive pertubation V
ε0 ε1 ε2 ε3 ε4 ε5
∆d
• Assumptions:{εk} à GOE / GUE distribution
{|φk(r0)|2} à Porter-Thomas distribution
Motivation: Random matrix theory
chaotic systems: quantum dots, nanoparticles
• Joint probability distribution
N → ∞ : νi = const. = V-1 β = 1 (2) for GOE (GUE)
(Aleiner/Matveev, PRL 1998)
−−
−
−−∝ ∑
∏∏
−>
iii
ji ji
ji jiji
ii iP )(
2exp
))((}){},({
,
2/1 ελνβ
λε
λλεελε β
Rank-1 perturbation in the mesoscopic case
VNr
k k
ok 1|)(| 2=
−∑ ελφ
κFluctuations: εk, φk(ro) à λκ::
Boundary effects
• run-away level
• “pressure” from far away level à level-dependent potential and phase shift
1/0−
−=Vde
dNλ
−+=
iiN
dViln
111ν
-4 -2 0 2 4 6 8λ-50
-25
0
25
50
y
ε0 ε1 ε2 ε3 ε4 ε5
∆
λ0 λ1 λ2 λ3 λ4 λ5 V big
d
Workhorse: Metropolis algorithm on the circle
• Start: picket fence (N+1 level ε, N+1 level λ, mean level spacing d=2π/Ν, shift δb)
• Random number in (0, 2N+1) àlevel εi or λi shifted within interval given by neighboring levels
• Every third step: move pair (εi, λi)• Memory lost after ~ N steps
• Metropolis step: accept / reject change with PM=min(1, P({εi},{λi}))
εi
εi+1
εi+2
εi-1λi
λi+2
λi+1
λi-1
ε
λ
δ
∆
εNλ0λN
ε0
M of N level filled
à generate many ensembles [{εk},{λk} ⇒ |∆|2 ]à distribution of overlaps |∆|2
Circle:constant DOS
d
Results:
1. Ground state overlap distribution P(|∆|2)
a) as perturbation V ~ vc increases b) as particle number N increases
0.2 0.4 0.6 0.8 1
overlap |∆|20
5
10
15
20
25
P(|∆
|2 )
vc/d = -0.1vc/d = -0.25vc/d = -1vc/d = -10
Onset of AOC
0 0.2 0.4 0.6 0.8 1overlap |∆|2
0
1
2
3
4
5
P(|∆
|2 )
N=1000N=500N=250N=100N=50N=10
bulk values
22
21
22
||
−
−
∝∆ π
δπδ bb
eN
b
|Vc|↑
N ↑
è P(|∆|2) determined by phase shift δF at Fermi energy(as in metallic x-ray edge problem)
0 0.2 0.4 0.6 0.8 1|∆|20
1
2
3
4
5
P(|∆
|2 )
GOE: N, M, vc /d250, 78, -1100, 50, -5100, 31, -1100, 14, -0.5 50, 16, -1
|∆b|2
0 1 2 3|∆|2/ |∆b|20
0.2
0.4
0.6
0.8
1
1.2
P(|∆
|2 )
0 1 2 30
0.5
1 GUE
Scaling and role of phase shift δF at Fermi energy
δF ≈ −π/2
1. P(|∆|2) cont.
Results:
2. Origin of Fluctuations in P(|∆|2)
∏ ∏= +=
−−
−−=∆
M
i
N
Mj ijij
ijij
filled0
empty 1
2
))((
))((
:Reminder
εελλ
λεελReference |∆b|2 - evaluate |∆|2
starting at the Fermi edge EF :
{ε} {λ}
EF
λM+1
εM
bulk case
bulk case
λM
range 2
EF
λM+1
εM
bulk case
bulk case
εM+1
λM
λM+2
εM-1
range 1
emptyj
filledi
2
1
12
)1r()()(
bbMM
MMd
d∆
−−−
=∆+
+δλλ
ελ
2. Fluctuations in P(|∆|2) cont.
0 0.2 0.4 0.6 0.8 10
2
4
6
P(|∆
|2 )
vc/d = -1
0 0.2 0.4 0.6 0.8 10
2
4
6
P(|∆
|2 )
vc/d = -0.25
0 0.2 0.4 0.6 0.8 1overlap |∆|2
0
2
4
6P
(|∆|2 )
vc/d = -10
GUE
0 0.2 0.4 0.6 0.8 10
2
4
6
P(|∆
|2 )
GOE
vc/d = -1
0 0.2 0.4 0.6 0.8 10
2
4
6
P(|∆
|2 )
vc/d = -0.25
0 0.2 0.4 0.6 0.8 1overlap |∆|2
0
2
4
6
P(|∆
|2 )
vc/d = -10 N=100, M=50overlap |∆|2
range 1range 2anal. range 1
10 20 30range n
0
0.1
DK
S
è deviation of range-n result from P(|∆|2):
vc /d-0.25-10
2. Fluctuations in P(|∆|2) cont.
è analytically understandingof overlap fluctuations:
• consider two level i = 0,1 around EF
in the mean field of other level
• s = (ε1−ε0) à Wigner surmise• |u0|2 , |u1|2 à Porter-Thomas • i ≥2: εi , |ui|2, λi à one random
variable
RMTRMTjustified !justified !
0 0.2 0.4 0.6 0.8 10
2
4
6
P(|∆
|2 )
vc/d = -1
0 0.2 0.4 0.6 0.8 10
2
4
6
P(|∆
|2 )
vc/d = -0.25
0 0.2 0.4 0.6 0.8 1overlap |∆|2
0
2
4
6
P(|∆
|2 )vc/d = -10
GUE
N=100, M=50overlap |∆|2
range 1range 2anal. range 1
Summary part II
AOC in mesoscopic systems
bulk-like mesoscopic chaotic
• {ε} equidistant{ε},{λ} fix
• single value |∆|2 ≡ |∆b|2
• bulk: N → ∞, |∆b|2 → 0
• {ε},{λ} fluctuating (GOE/GUE)
• RMT treatment justified
• broad distribution P(|∆|2)
• fluctuations dominated by levels around EF
• analytic treatment of range-1 approximation
AOC in disordered systems: Gefen et al. PRB 2002AOC in parametric random matrices: Vallejos et al. PRB 2002
Approaching the Mesoscopic X-Ray Edge Problem
Fermi edge singularities in x-ray spectra of metals
EF
EC
Absorption
Emission
h___ ω
AbsorptionA(ω)
ωΤ ω
EmissionI(ω)
ωΤ ω
A(ω)I(ω)
core
Misses many-body effects of core hole potentialon cond. e- : AOC and Mahan’s enhancement
• diagrammatic perturbation theory (Mahan, Nozieres,…)
• Fermi golden rule approach (Tanabe/Othaka)
bare
{ε}
Model: Fermi golden rule approach
( )CF
EEWA
FcF ΦΨ −−×ΦΨ= ∑ ωδπω
2ˆ2)(
] h.c.[ˆ :operator dipole
0~...~~ ),...,1( ~ : perturbed
0 :electron core
0... ),...,1( : dunperturbe
1
010
010
+=•
=Ψ=→•
=→•
=Φ=→•
∑=
+
++−
++
++
++−
++
N
kckkc
MM
cccc
MMkk
ccwW
cccNc
cc
cccNkc
κψ
φφ
φ
κκ
cF W ΦΨ ˆelement matrix Dipole
Tanabe and Othaka, RMP 1990
Model: Fermi golden rule approach
( )CF
EEWA
FcF ΦΨ −−×ΦΨ= ∑ ωδπω
2ˆ2)(
direct process replacement shake-updirect
EF
core o
j
M
i
0
{ε}
δF
h___ ω
{λ}
~ |wjc|2 |∆| 2 ~Σµ |wµc|2 |∆µγ|2
µ
γ
Dipole matrix element wjc
jE
cjcrayxr
w ψφ−∂
∂= rr
ufufw ccjc ′+′= σσ
)( rujknnonlocj
rr×= σψ∼δ(ro)
s-likebulk-like mesoscopic
θσ il
ll
rkierkJe j ∑== )(
rrrr
∑=
=
||'||,'
''
jjj
j
j
kkk
rkik ea
rrr
rrrσ
là orbital channel, partial wave decomp. là not conserved in chaotic systems
ϕc = s-like: ~J0’(ro) = 0 = 0 à wjc =0 at K-edge à rounded
ϕc = p-like: = 0 ~J0(ro) ≠ 0 à wjc ≠ 0 at L-edge à peaked
bulk-like
V= δ(r-r0) ↔ l=0: s-like cond. el.
mesosc. ϕc = s-like: wjo ~ σ’(r0) ~ ψj` à peaked or rounded K-edge (ψj`, ψj indep.)
ϕc = p-like: wjo ~ σ (r0) ~ ψj à stronger correlations at L-edge
Results:
1. Average Photoabsorption K-edgea) Contributions from the various processes
T
total (direct+repl.+shake up)naive bare (norm.)
direct process
M
{ε} {λ}
shake-up
direct process replacement
M
{ε} {λ}
direct + replacement
0 1 2 3 4 5 6 7 8(ω − ωth) / d
0
0.5
1
1.5
2
⟨Α(ω
)⟩
{ε}
bare
~|wjc|2
~|wjc|2
|∆|2
vc = -10 d, K-edgeN = 100, M = 50, GOE
• peaked edge • replacement processes near EF dominate • one-pair shake-up processes dominate
Results:1. Average Photoabsorption K-edgeb) Taking spin into account
vc = -10 d, K-edgeN = 100, M = 50, GOE
spectator spin
EF
à width of |Φ0⟩in basis of perturbedfinal states |ΨF⟩
active spin
EF
active
spectator
full spin
0 1 2 3 4 5 6 7 8(ω − ωth) / d
0
0.5
1
1.5
2
⟨Α(ω
)⟩
0 1 2 3 4 5 6 7 8(ω − ωth) / d
0.1
0.3
1
3
⟨Α(ω
)⟩
Comparison with bulk-like case
vc = -10 d, GOE
è Rounded edge goes into a (slightly) peaked edge as the system becomes coherent
M.H., D.Ullmo, H.U. Baranger, cond-mat/0402207, subm. to PRL
K-edge bulk-like
K-edge mesoscopic
L-edge bulk-like
Dependence on the number of electrons
bareN=24, M=12N=50, M=25N=100, M=50N=200, M=100
MesoscopicK-edge
0 1 2 3 4 5 6 7 8 9 10(ω − ω th) / d
0
2.5
5
7.5
10
⟨Α(ω
)⟩Bulk-likeL-edge
0 1 2 3 4 5 6 7 8 9 10(ω − ω th) / d
0
0.5
1
1.5
2
⟨Α(ω
)⟩N ↑: Anderson wins
N ↑: Mahan wins
|∆π−δ| 2
EF
λ0
Results:
2. Average Photoabsorption L-edge
è Coupling to the wave function: wjo~ ψj
-4 -2 0 2 4 6λ-50
-25
0
25
y
ε0 ε1 ε2 ε3 ε4 ε5
λ0 λ1 λ2 λ3 λ4 λ5 V big
bound state λ0:
• ψ0 (r0) piles up• screens core hole• s-like
-10 -5 0 5 10x
0
0.005
0.01
0.015
0.02
0.025
0.03
|ψi|2
-10 -5 0 5 10x
0
0.2
0.4
0.6
0.8
1
|ψi|2
i=0i=1i=2i=M=N/2 (EF)i=N
vc= - 0.1d vc= -10 d
100 random plane waves, N=100, M=50
0 1 2 3 4 5 6 7 8 9 10<ω−ωth>/d
0
10
20
30
40
<A(ω
)>
0 1 2 3 4 5 6 7 8 9 10<ω−ωth>/d
0
10
20
30
40
<A(ω
)>
N=100, M=50N=50, M=25
vc = -10 d, GOE, active spin
• small differences
mesoscopic vs. bulk-like,
and GOE vs. GUE
• edge peak ↑ with N ↑
Average Photoabsorption L-edge cont.
MesoscopicL-edge
Bulk-likeL-edge
0 1 2 3 4A(ω) / <A(ω)>
0
0.5
1
1.5
2
P(A
(ω)
/ <A
(ω)>
)
0 1 2 3 4A(ω) / <A(ω)>
0
0.5
1
1.5
2
P(A
(ω)
/ <A
(ω)>
)
Results:
3. Mesoscopic fluctuations in A(ω)
K-edge
à wjc ~ ψ’ large Porter-Thomaslike fluctuations
à overwhelm overlap correlations anddominate fluctuations of A(ω)
γ=Μ+1γ=Μ+2γ=Μ+3γ=Μ+8P.Th.
L-edge
à wjc ~ ψ narrowly distributedà Symmetry:
replacement through bound state acts like a ground state overlap with δF’ = π − δF ,results in highly peaked edge
= f ( |∆|2, ∆repl ,∆shup ; wjc)
γ=Μ+1γ=Μ+2γ=Μ+3γ=Μ+8P.Th.
K-e
dge
L-ed
ge
• x-ray photoabsorption with metallic nanoparticles: feasible in few years
• double quantum dots: constriction Abanin/Levitov, cond-mat/0405383
• photoabsorption via impurity states in semiconductor heterostructures
Experimental Realizations
“Fermi sea of electrons subject to a rank-1 perturbation”
GaAs
2DEG andimpurities
Quantum Dot Array (diam.~100 nm)etched from heterostructure
Control Experiment “bulk-like”: no dots, just 2DEG with impurities - already done?
Summary part III
Mesoscopic X-ray Edge Problem
bulk-like mesoscopic
s-like conduction electrons: δ0= π/2, δ1= 0
Dipole coupling changed because mesoscopic system is
- chaotic (loose l as quantum number)- coherent confinement- wave function and derivative independent
• rounded K-edge
• peaked L-edge
• (slightly) peaked K-edge
• peaked L-edgeAverage ⟨A(ω)⟩
Mesoscopic fluctuations• individual spectra can even zig-zag
IV. Conclusions• AOC in Mesoscopic Systems:
- broad distribution P(|∆|2)- scaling with |∆b|2, δF
• Mesoscopic Photoabsorption Spectra and X-Ray Edge Problem:
- K-edge: ⟨A(ω)⟩ from rounded to peaked as system becomes coherent,Porter-Thomas fluctuations
- L-edge: strongly peaked, same fluctuations as |∆|2
• Experimental realizations: - array of quantum dots, impurity level takes role of core electron
- nanoparticles, double dots M. Hentschel, D. Ullmo, H.U. Baranger,cond-mat/0402207
0 1 2 3 4 5 6 7 8(ω − ωth) / d
0.1
0.3
1
3
⟨Α(ω
)⟩
0 1 2 3|∆|2/ |∆ b|2
00.20.40.60.8
11.2
P(|∆
|2 /|∆b|2 )
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