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

II. Anderson Orthogonality Catastrophe

in Mesoscopic Systems

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

III. Mesoscopic X-ray Edge Problem

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 )