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Metals Mesoscopic SystemsExpected:A(
th
A(
th
Observed:
core level
{}
bare
EF
EC
Absorptionh
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
{}
NIRT program
A(
th
?
e.g. Peaked Edge L2,3-edge simple metals like Al, Mg, (Na)
from K. Othaka, Y. Tanabe, RMP 62 2929 (1990):GaAs-AlxGa1-xAs Quantum well, Lee et al. (1987)
A(I(
Martina Hentschel:
Yu und Cardona
Fund.of SC, p.477
Martina Hentschel:
Yu und Cardona
Fund.of SC, p.477
The ‘classical’ X-Ray Edge ProblemSingularities at the Fermi edge threshold in
X-Ray Emission or Absorption Spectra of, e.g., metals
although iias N
+
1 of 1023 electrons
Many-body ground state made of single particle wf. |i
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~~
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 state
initially
ground state
under perturbation V
Orthogonality Block
Or any state entirely described in terms of plane waves
Or any state entirely described in terms of plane waves
Perturbation can be small there is NO ADIABATICITY in those systems!
Perturbation can be small there is NO ADIABATICITY in those systems!
CHECK zero-bias anomaly (in dosordered systems)
CHECK zero-bias anomaly (in dosordered systems)
screeningdipole selection rules
Orthogonality block due 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
)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 l )
counteracting(Mahan) many-body process (lo 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)
...},{ˆˆ :perturbed
...},{ˆ :dunperturbe
VH
H kk
M
i
N
Mjijij
ijij
filled0
empty
1
22
))((
))((
= f (eigenvalues only)
00 rrVN
e.g. core hole left behind at r0
overlap between perturbed and unperturbed ground states:
• unperturbed level k: equidistant (“picket fence”, “bulk-like”)
• perturbed level : Schrödinger equation
Example for a rank-1 perturbation
Vk
k
11:
V big
V small
Martina Hentschel:
Check this – der ist gar nicht constant!!!
Martina Hentschel:
Check this – der ist gar nicht constant!!!
d
V
d kkb arctan)(:) (Nshift phase
-4 -2 0 2 4 6 8-50
-25
0
25
50
y
6 level, attractive pertubation V
d
• Assumptions:
{k} GOE / GUE distribution
{|k(r0)|2} Porter-Thomas distribution
Motivation: Random matrix theory
chaotic systems: quantum dots, nanoparticles
• Joint probability distribution
N i = const. = V-1 for GOE (GUE)
(Aleiner/Matveev, PRL 1998)
iii
ji ji
ji jiji
ii iP )(
2exp
))((}){},({
,
2/1
Rank-1 perturbation in the mesoscopic case
VN
r
kk
ok 1|)(| 2
Fluctuations: k k(ro) :
Boundary effects
• run-away level
• “pressure” from far away level level-dependent potential and phase shift
1/0
Vde
dN
i
iN
dViln
111
-4 -2 0 2 4 6 8-50
-25
0
25
50
y
V big
d
Workhorse: Metropolis algorithm on the circle
• Start: picket fence
(N+1 level , N+1 level , mean level spacing dshift b)
• Random number in (0, 2N+1) level i or i shifted within interval given by neighboring levels
• Every third step: move pair (i, i)
• Memory lost after ~ N steps
• Metropolis step: accept / reject change with PM=min(1, P({i,{i
i
i+1
i+2
i-1i
i+2
i+1
i-1
N0N
0
M of N level filled
generate many ensembles [kk distribution of overlaps
Circle:constant DOS
d
Results:
1. Ground state overlap distribution P()
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 1
overlap 2
0
1
2
3
4
5
P(
2 )
N=1000N=500N=250N=100N=50N=10
bulk values
22
2
1
2
2||
bb
eN
b
|Vc|
N
P(|determined by phase shift F at Fermi energy (as in metallic x-ray edge problem)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
P(2 )
GOE: N, M, v c /d250, 78, -1100, 50, -5100, 31, -1100, 14, -0.5 50, 16, -1
b2
0 1 2 32/ b2
0
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 F at Fermi energy
F
1. P() cont.
Results:
2. Origin of Fluctuations in P()
M
i
N
Mjijij
ijij
filled0
empty 1
2
))((
))((
:Reminder
Reference |b - evaluate |
starting at the Fermi edge EF :
{} {}
EF
M+1
M
bulk case
bulk case
M
range 2
EF
M+1
M
bulk case
bulk case
M+1
M
M+2
M-1
range 1
emptyj
filledi
2
1
12
)1r(
)(
)(b
bMM
MMd
d
2. Fluctuations in P() 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
deviation of range-n result from P():
vc /d-0.25-10
2. Fluctuations in P() cont.
analytically understanding
of overlap fluctuations:
• consider two level i = 0,1 around EF
in the mean field of other level
• s = ) Wigner surmise
• |u0|2 , |u1|2 Porter-Thomas
• i 2: i , |ui|2, i 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
• {} equidistant {},{} fix
• single value b
• bulk: N b
• {},{} fluctuating (GOE/GUE)
• RMT treatment justified
• broad distribution P()
• 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___
AbsorptionA(
EmissionI(
A(I(
A(I(
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
{}
bare
Model: Fermi golden rule approach
CF
EEWA
F
cF 2
ˆ2)(
] h.c.[ˆ :operator dipole
0~...~~ ),...,1( ~ : perturbed
0 :electron core
0... ),...,1( : dunperturbe
1
010
010
N
k
ckkc
MM
cccc
MMkk
ccwW
cccNc
cc
cccNkc
cF W ˆelement matrix Dipole
Tanabe and Othaka, RMP 1990
Model: Fermi golden rule approach
CF
EEWA
F
cF 2
ˆ2)(
direct process replacement shake-updirectrepl.
EF
core o
j
M
i
0
{}
F
h___
{}
~ |wjc|2 || ~|wc|2 ||
Dipole matrix element wjc
jE
cjcrayxr
w
uφuφw ccjc
)( rujknnonlocjro
s-likebulk-like mesoscopic
il
l
lrki
erkJe j )(
||'||,'
''
jjj
j
j
kkk
rkik ea
l orbital channel, partial wave decomp. l not conserved in chaotic systems
c = s-like: ~J0’(ro) = 0 = 0 wjc =0 at K-edge rounded
c = p-like: = 0 ~J0(ro) wjc 0 at L-edge peaked
bulk-like
V= (r-r0) l=0: s-like cond. el.
mesosc. c = s-like: wjo ~ ’(r0) ~ j` peaked or rounded K-edge (j`, j indep.)
c = p-like: wjo ~ (r0) ~ j stronger correlations at L-edge
Results:
1. Average Photoabsorption K-edge
a) Contributions from the various processes
V = 0.25
0 5 10 15 200
100
200
300
A(
)
directdirect+repl.shake up
0 5 10 15 200
100
200
300
400
A(
)
V = 1
0 5 10 15 20T
0
100
200
A(
)
V = 10
N=40, GOE (, V varied
M/N=1/2, full spectra
0 5 100
50
100
150
A(
)
0 5 100
100
200
A(
)
0 5 10T
0
50
100
150
A(
)
M/N=1/4, egde region
V = 0.25
V = 1
V = 10
total (direct+repl.+shake up)naive bare (norm.)
direct process replacement
j
M
i
0
{} {}
direct + replacement
shake-up
direct process replacement
j
M
i
0
{} {}
direct + replacement
0 1 2 3 4 5 6 7 8(th) / d
0
0.5
1
1.5
2
{}
bare
~|wjc|2
~|wjc|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
E F
width of | in basis of perturbed final states |F
active spin
E F
active
spectator
full spin
0 1 2 3 4 5 6 7 8(th) / d
0
0.5
1
1.5
2
0 1 2 3 4 5 6 7 8(th) / d
0.1
0.3
1
3
Comparison with bulk-like case
vc = -10 d, GOE
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
Mesoscopic K-edge
0 1 2 3 4 5 6 7 8 9 10(th) / d
0
2.5
5
7.5
10
Bulk-like
L-edge
0 1 2 3 4 5 6 7 8 9 10(th) / d
0
0.5
1
1.5
2
N Anderson wins
N Mahan wins
E F
0
Results:
2. Average Photoabsorption L-edge
Coupling to the wave function: wjo~ j
-4 -2 0 2 4 6 8-50
-25
0
25
50
y
V big
bound state
• (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
|i|2
-10 -5 0 5 10x
0
0.2
0.4
0.6
0.8
1
|i|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< th>/d
0
10
20
30
40
<A(
)>
0 1 2 3 4 5 6 7 8 9 10< th>/d
0
10
20
30
40
<A(
)>
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 withN
Average Photoabsorption L-edge cont.
Mesoscopic L-edge
Bulk-like L-edge
0 1 2 3 4A() / <A()>
0
0.5
1
1.5
2
P(A
()
/ <A
()>
)
0 1 2 3 4A() / <A()>
0
0.5
1
1.5
2
P(A
()
/ <A
()>
)
Results:
3. Mesoscopic fluctuations in A()
K-edge
wjc ~ ’ large Porter-Thomas like fluctuations overwhelm overlap correlations and dominate fluctuations of A()
GOE, N=40, M=20, Vc=-10
P.Th.
GOE, N=40, M=20, Vc=-10
P.Th.
L-edge
wjc ~ narrowly distributed Symmetry: replacement through bound state acts like a ground state overlap with F’ = F ,
results in highly peaked edge
= f ( ||2, repl shup
; wjc)GOE, N=40, M=20, Vc=-10
P.Th.
GOE, N=40, M=20, Vc=-10
P.Th.
K-e
dge
L-e
dg
e
• 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: 0= 1= 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(
Mesoscopic fluctuations
• individual spectra can even zig-zag
IV. Conclusions• AOC in Mesoscopic Systems:
- broad distribution P(- scaling with bF
• Mesoscopic Photoabsorption Spectra and X-Ray Edge Problem:
- K-edge: A( from rounded to peaked as system becomes coherent, Porter-Thomas fluctuations
- L-edge: strongly peaked, same fluctuations as
• 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(th) / d
0.1
0.3
1
3
0 1 2 32/ b
2
00.20.40.60.8
11.2
P(2 /
b2 )