Shakeup effects on photoluminescence from the Wigner crystal

Solid State Communications, Vol. 95, No. 7, pp. 435-441, 1995 Elsevier Science Ltd Printed in Great Britain 0038-1098/95 $9.50+.00 00%1098(95)00297-9

Pergamon

SHAKEUP

EFFECTS FROM

ON PHOTOLUMINESCENCE

THE’ WIGNER

CRYSTAL

D.Z. Liu’ Center

for Superconductivity

h!esearch,

H.A. of Physics

of Physics,

University

of

College Park, MD t07&’

Maryland,

Department

Department

Fertig

and Astronomy, Kentucky

University

of Kentucky,

Lezington,

40506-0055

S. Das Sarma Department

of Physics,

University

(Received

We develop a method

to compute

crystal from localized neling electron the collective shakeup

7 April

holes.

1995 by A. Pinczuk)

Our ,nethod

treats

a series of sidebands

the lattice electrons

throughout

the Brillouin

that may be identified

collective mode density of states, and definitively

Keywords:

distinguishes

We also find a shift ‘n the main luminescence

with lattice relaxation

in a Wigner and the tun-

and uses a quantum-mechanical

modes that is realistic

a liquid state.

College Park, MD 20742

shakeup effects on photoluminescence

on an equal footing,

produces

of Maryland,

calculation

zone.

of

We find that

with maxima

in the

the crystal state from peak, that is associated

in the vicinity of a vacancy.

A. heterojunctions;

D. anharmonicity,

optical properties,

phonons;

E.

luminescence.

Sixty years ago, Wigner[l]

pointed

tron gas will undergo a zero-temperature, transition

into a crystalline

quantum

phase as the density

ered. Forty-five years later, the first convincing of an electron

crystal was presented

trons on a He surface[2].

this an unattractive phase transition. tive systems

Present

atta:.n-

low, however, making the quantum

densities,

address:

sity of Chicago,

James Chicago,

dopant

Franck Institute,

good candidate

for observing

(2DEG),

semiconduc-

as realized

Samples

in modulation

doped

of this type are now available

high quality that the electron

groundstate

sarily dominated

The possibility

by disorder.

is not neces-

by the application

a strong

magnetic

field, which quenches

perpendicular

accumulated 435

(for which disorder

than would be possible

Experimental

IL 60637.

of observ-

enhanced

less important)

Univ:r-

with such

ing the WC is further

state at higher densities

concentca-

the

electron gas

the kinetic energy, and allows the formation

are much more attrac-

through

A particularly

Wigner crystal (WC) is the two-dimensional

tors.

evidence

densities

for observing

Semiconductors

is low-

in this sense, because one has great cant rol

over the electron [*]

system

phase

for a system of elec-

The electron

able in this fashion are extremely

tions.

out that an elec-

evidence

of a crystal effects are

without

it.

for the WC in 2DEG’s

over the last several years[3].

of

has

One probe

436

SHAKEUP

that has produced nescence

much intriguing

(PL), in which either

or a hole bound to an acceptor[5, electron

in the ZDEG, producing

spectrum.

A mean-field

experiment

data is photolumi-

a valence

ple, characteristic butterfly”[8]

band hole[4]

see these sidebands

a characteristic

photon

satellites

type of

solid state.

spectrum

shift in the PL spectrum

a shakeup

increasing

and the hole, and a characteristic

Interestingly,

strength

for stronger

we do not

a liquid and a electron-hole

persists

We expect

For

so that phonon

between

satellite

in-

even above the

this sideband

to lose

relative to the main peak, either with

temperature

action strength.

upon melting of the crystal.

interactions[6],

distinguish

temperature.

oscillator

for the case of weak interactions

between the electrons

melting

WC state.

in the liquid state,

uniquely

teractions,

has, in princi-

of the WC: a “Hofstadter

of an ordered

the case of weak electron-hole

with an

analysis[7] of the latter

signatures

and so are characteristic

61 recombines

showed that the PL spectrum

Vol. 95, No. 7

EFFECTS ON PHOTOLUMINESCENCE

or decreasing

electron-hole

The latter may be accomplished

interby ex-

amining PL from several samples with different acceptor In this Letter, mation,

we go beyond the mean-field

how the collective

mode spectrum

of the WC (which, at long wavelengths,

of these modes may be excited combination

process,

field theory.

We will consider

a localized

modify

and treats

the results

in detail only the case of is purely quantum-

treatment

to account

for contributions

wavevector

collective

numbers

of these

working in the strong only excitations

Further-

[6].

of the lattice

excitations.

survives in the itinerant

field limit, we consider level (LLL).

spectrum

teraction

plane.

and v = 2/7, for

1010n-2.

Our hole is as-

localized,

and located

For the case of no electron-hole

[Fig. l(a)], at low temperature,

shakeup

250A from in-

a well-defined

peak may be seen approximately

2 meV be-

low the main PL peak; a second very weak satellite observed

approximately

is

3.5meV below the main peak.

The origins of these peaks may be understood

in terms of

Photoluminescence 1600

Power

500

? f 7

2 250

1200800 -

c 4

0 -12

2

-8

-4

2. 4

is eliminated

hole case[‘l]). The sudden shift

upon melting,

even when shakeup

is included.

sity of states

the electron

x

v = l/5

-13.5

-15.5

E-E,-E,(meV)

we will argue below that it

of the PL spectrum

that correspond

sumed to be strongly

even in the case of weak electron-

(although

N, = 6

are shown in

tha.n found in a mean-field

(2) The Hofstadter

hole interactions[6]

density

PL spectra

of

Since we are

within the lowest Landau

from the PL spectrum,

1 for filling fractions

electron

using a

Our main results are: (1) Shakeup effects shift the main

treatment[7].

Fig.

of our calculated

approximation,

allows for the shakeup

magnetic

PL peak to higher energies

Examples

both from small and large

excitations

The method

and

of the collective

modes that does not involve a harmonic

single method.

electron

on the same footing.

more, we employ a quantum

re-

of the mean-

both the tunneling

the other lattice electrons

to

in the electron-hole

hole[5, 61. Our method

mechanical,

corresponds

, and the fact that some

the classical phonon spectrum)

appear

- 2DEG setback distances

to examine shakeup effects on the PL spectrum;

i.e., we will examine

arbitrary

approxi-

by contrast, (3) Phonon

to maxima

survives sidebands

in the phonon

den-

(DOS); some (but not all) of these side-

bands are results of van Hove singularities

in the DOS,

Figure 1: PL spectra for (a)v = l/5, T = 0, no electron(b)v = l/5, T = 0, with electron-hole

hole interaction;

= 2/7, T = 0, no electron-hole

interaction;(c)v action.

Inset:

PL spectra

above the melting different

spectra,

inter-

for (a) and (b), with T just

temperature.

In order to distinguish

they were seperated

by 250 units.

Vol. 95, No. 7 the phonon

SHAKEUP

DOS,

which is illustrated

in Fig.

Hove singularity,

arising

pears as a strong

double peak near 0.4meV.

peaks

from

may be seen near

are weak sidebands

associated

in the PL spectrum. peaks

is unclear;

these

represent

point

out that

chanical lattice

unusual

phonons,

peaks

interpretation

speculated

these higher

of the phonons[9,

excitations

order satellites;

We meof the

classical

111 do not produce

these

excitations.

than

the energy

the phonon

DOS

is a strong

at which appears.

is actually

reason

for this

renormalization

PL peak.

Physically,

the final state of the crystal

which is lowered in energy

between

the

larger

the van Hove singularity The

self-energy

in the main

contains

by a distortion

_ i.e., by allowing the electrons to relax inward.

leading

The self-energy

The

inset

the melted

to Fig. state

sideband melted

present.

local

is tl.at

modes in the LLL [lo, 12). By contrast,

when

renormalized

density modes

the oscillator

strength

is present,

further

between

allowing

even above

increase

some

the melt-

in temperature,

the hole and the 2DEG,

of this mode will significantly

Fig.

l(c)

except

arises

a vacancy,

the vacancy for this

are excited rel,zx-

of States

illustrates

our results

of electron-hole

de-

for a change

the magnetic

in energy

results

electron-hole

are difficult

in practice

characteristic

splittings

is related

latter

function

However,

they

shakeup

ef-

We note

hole, we expect

these

shakeup

effects.

excitonic

effects),

the PL spec-

of Green’s

For localized

The

functions

for

holes, only the spectrum

level, which are wiped out by shakeup for itinerant

by shakeup.

ciple be able to identify and gaps that

field in itinerant

While

that

evidently

holes,

the hole Green’s

also has poles of the Hofstadter

are unaffected

mode DOS for Y = l/5, T = 0.

situation

in the form of a Hofstadter

at the mean-field effects.

hole.

even in principle.

to the product

has poles

p/q gen-

to survive

the hole and the electron.

bands

in that

for the case of an itinerant

trum

3.0

a filling v =

to resolve,

fects wipe out this structure

with the [7], where

lines for a localized

are so small

identical

sharply

approximation

interactions,

yields p distinct

by changing

is essentially

This contrasts

found in the mean-field

without

As can be seen,

scale caused

field, the lineshape

to the case of Y = l/5.

for Y = 2/7 in the

interactions.

reason is that (neglecting

2: Collective

is a non-

crease.

that

Figure

there

near the hole,

to persist

With

setback

this

by lattice

is uniform,

interaction

electron

the splittings

are not nearly

case, there is no phonon

are no

the density

or increased

absence

Our calculation

in which phonons

an electron-

and there

to an upward shift of the main PL peak. shows that the final states

and with

in

in the

collective

due to the

shift accounts

the PL spectrum

in

of the lattice

surrounding

Density

between

so, because

erally

Phonon

splitting

This is necessarily

an electron-hole uniform

without

In the former

phase,

collective

1 illustrates

both

hole interaction.

lowering in energy of the final state of the WC, and leads

so strongly

to the increased

the main PL peak and the sidebands.

ing temperature.

PL peak and the first sideband

because

that

to use a fully quantum

main

phonons

of these

excitations[lO].

of the collective

ation effects,

437

Two other

It should also be noted that the splitting

there

ap-

with each of these

vacancy-interstitial

to observe

2. A van

There

it haa been

it is crucial

ON PHOTOLUMINESCENCE

and 1.9 meV.

The precise

however,

treatment

treatments

zone-edge

1.2 meV

EFFECTS

Thus,

one should

this characteristic

is unique

form,

which in prin-

spectrum

of

to a WC in a magnetic

hole experiments[l3].

SHAKEUP

438

Vol. 95, No.

EFFECTS ON PHOTOLUMINESCENCE In Eq.(l),

,u(q) and V(q) are the Fourier transforms

the electron-electron spectively,

and electron-hole

~0 is the energy

interactions,

of the localized

the sum over G’ is only over reciprocal

lattice

of the electron

is independent R(w) =

$ c

and we have approximated

nient for later purposes

the core-hole

ment of the density.

wavefunctions

as delta-functions,

nh is the density of holes, R the vol-

Fourier components

ume of the system,

and the vectors G are the reciprocal

The correlation

lattice vectors of the superlattice. IO = (3)“’

paper.

The magnetic

length

wd be set to unity in the remainder

of this

R(G,w)

density.

The method

function C,,(p,,

it will be conve-

leave it as an argufor computing

has been described

these

elsewhere[‘l,

lo].

pz) in the last two terms

is just Fourier transformation

of the following correla-

tion function:

Eq.(l)

is defined as

represents

the first in an infinite series of equaGreen’s function

to the n + 1

iG=X+iG=%DR;.(X, X _ G,; W)r

= ;&-

particle

where g is the Landau level degeneracy, center quantum

While this quantity

to formally

tions relating an n particle R(G,w)

R; spec-

value of a Fourier

of ‘f in the groundstate,

R(G,W)e-@y

OG

vectors,

of the hole in the i th unit cell, and

< p( G, T) > e-G2/4 is the expectation component

re-

hole, and

while the sums over q are over all wavevectors. ifies the position

of

number,

Green’s function[l5].

imation,

X is the guiding

and

it is simplified

In the mean-field

by employing

(HF) decomposition

of Eq.(2),

into a self-consistent

equation

approx-

a Hartree-Fock

which converts

Eq.(l)

for Rij [7]. To include

shakeup effects, we instead extend

this hierarchy

to one

&j(X17XZ;iWa)

=

-

P J0

more level, writing < TTaxl (~)c;(~)c,ta!,

down a self-consistent

which explicitly

contains the collective mode excitations.

To carry out this program, with axt creating

an electron

in state X and c,t creating

define a self-energy

a hole in the unit cell i, and /3 is the inverse tempera-

Eq.(l)

ture.

HF approximation

Working in the lowest Landau level, the equation

of motion for Ri,(G,w)

zffF(G,

may be written

G’)

= W(G

-

by substituting

by -CG,J:~T’C(G,G’;~

the last two terms in - T’)RU(G’,T’).

for the PL is equivalent

< p(G

- G’; T) > eiGxG”’

+

&c

The

to taking

v(g) < p(-q, T) > e-92'4-iqR'6G,Gfr (3)

=< p(G, T) > Sij6(~) - coR,j(G, T) - ~~ C V(G')e'G'XG'2-GR'4Ril(G - G’, T) G' -~q~oO(q)Cij(-q,q+

to implicitly

C(G, G’; T - 7’) by CHF(G, G’)S(7 - r’), where

as

G’)

it is convenient

9

iRAG, T)

form for C,,

> eiwnTdT,

G)~~GX~~2-~2~2 - ~~l/(q)l:,(-q,G)e-‘q’Rl-Q2/4. q

(1)

Vol. 95, No. 7

SHAKEUP

439

EFFECTS ON PHOTOLUMINESCENCE

where W is the sum of the direct and exchange

Coulomb

Eq.(5)

potentials[7,

a self-consistent

equation

ously

derivative[I5]

of Eq.(l)

able to compute

potential

CHFG(r - r’) + SC into Eq.(l),

for c;,,

lo]. To generate

we take a functional

with respect

periodic

external

In doing this, it must be noted

U(P,T').

directly

to a spatially

coupling

nal potential

wavevectors,

because

Tfi(GI +

vectors

must be extended

+ q,Gz +

x(Gi

q, r)fi(-Gs

of this response

-

q;7) =

q; 0)

function

tive mode frequencies x that shakeup

contains

essary in classical approaches[9]), relation

the entire Brillouin

computed

one obtains

- ~2)

in this computation,

lattice sums and an apwavevector

q.

We

results.

Finally,

version

per unit cell, once we have

to substitute

of Eq.(l),

obtain

this

R(G,w),

the PL spectrum.

modes across In summary,

the result-

the self-energy

we have developed

shakeup effects in the PL spectrum

C =

ized holes may be computed,

cqx(G1-G::)/2-iGlxG:/2+iG2xG;l2w(G: = kGc,

The computa-

this by using 469 q points in the first

and from there compute

CHF6(r - r’) + 6X, with

~C(GI,'&,TI

methods[l5].

SC, it is straightforward

into the frequency

(as is nec-

zone. Upon substituting

ing Cij into Eq.(l),

sums may be then

sum over the continuous

which give very similar

ran-

To

iw - iwn).

x as a sum over its col-

Brillouin zone, for one and three electrons

and thus gives a real-

for the collective

F(iw,)x(q;

two reciprocal

have accomplished

We have com-

electrons

this, we represent

proximate

which does not assume

of the lattice

accomplish

since it requires

it is thus through

using a generalized

Ciwn

tion of 6C is clearly the bottleneck

<

poles at the collec-

of the system;

approximation[lO]

istic dispersion

-g

>. The Fourier transform

numerically

small displacements

=

lead-

of the form (suppressing

arguments)

using standard

that

will be necessary,

summations

wavevector

computed

C =

This means

lective mode poles[lO]; the frequency

correlation

XG,G2(q;T)

effects are introduced.

puted this function

to all

we are now

We substitute

time.

of Eq.(4)

previ-

and Fourier transform

to imaginary

transform

discussed

this expression,

the PL intensity.

ing to frequency

the result

Cij in terms of the density-density

dom phase

a Fourier

U is not in general commensurate

After an arduous calculation,

[7, 10, 161. With

this with respect

Ri, to the exter-

the Green’s function

lattice

with the lattice.

function

a term

U must be added to Eq.( l), and the sums

over reciprocal

expresses

that

may be solved using methods

a method

of a WC from local-

that treats

+q)W(GI,

by which

the tunneling

+q)

Jszd2qe : ;

xF(GI

-G;

-

q,G2

- G; -- q;rl - 72)XG;G;(q;?

d2qe-(G;+9:“/4-(G;+s)2/4v(G:

+

-

72)

q)V(G;

+

q)

+&GyBz

xF(G,

In Eq.(4),

JBZd2q represents

G2;

71

an integral

-

r2)XG;G;(q;

over wavevecand F

tors in the first Brillouin zone of the superlattice, is a generalized

Green’s function

satisfying

the equation

of motion

&F(G, GI,T

71 -

T22)

electron

(4

and the lattice

and uses a fully quantum modes

that

Our method

- 71)

electrons

is realistic

treatment

over the entire

is quite general,

G’

- q) - ~C”‘(G,G’)F(G’,GI,T G’

- n),

of the collective Brillouin

zone.

and should be applicable

= SGG16(7- TV) - n,,C V(G’)e’G’XG~2-GR~4F(G - G’, GI, T - 71) -c,,F(G,G~,T

on an equal footing,

(5)

440

SHAKEUP

to other shakeup

problems

are important.

We find that

found in a mean-field (although

where quantum

fluctuations

the Hofstadter

analysis

we expect

EPPECTS

it to survive in itinerant

iments),

and is replaced

creation

of phonons

the WC.

These

sidebands

are a unique

WC, and can in principle a liquid and crystal

collective

hole experdue to

excitations signature

be used to distinguish

state of the electrons.

upon melting

of the crystal.

is lost

by a series of sidebands

and other

Vol. 95, No. 7

there is a sudden shift in the PL spectrum

spectrum

of this experiment

ON PHOTOLUMINESCENCE

of

of the between

We find that

Acknowledgments.

The authors

for helpful discussions. NSF,

through

91-23577, support search

Dr.

Ren6

This work was supported

Grant

Nos.

DMR

and by the US-ONR.

through

92-02255

C&d by the

and DMR

HAF acknowledges

of the Alfred P. Sloan Corporation,

thank

Foundation

a Cottrell

the

and the Re-

Scholar

Award.

References

[l] E.P. Wigner,

Phys.

[2] C.C.

and G. Adams,

Grimes

Rev. 46,

[9] P. Johansson

1002 (1934). Phys.

Rev.

Lett.

795 (1979). [3] For

[lo]

a review

Combe

of recent

experiments,

and A. Nurmikko,

ties of Two-Dimensional New York,

eds., Electronic Systems,

Mc-

et al., Surface

Goldys

et al., Phys.

R.G.

Clark,

Physica

(North-Holland,

Science

305,

Rev. B 46, 7957 (1992);

Scripta

T39,

45 (1991)

and

R. C&k and A.H. MacDonald, 2662 (1990);

[ll]

L. Bonsall

Rev. Lett.

et al., Phys.

D.Z. 70,

Liu,

Rev.

Rev. B 15,

lead to damped

Lett.

D.Z.

Science

67 (1994).

collective

could, in principle,

modes in the liquid state

in the LLL. However, so long as these modes are not they would only lead to broad structure

(1992).

sharp peaks such as those seen in the crystal

72,

3594

Phys.

11184 (1993);

Liu, and S. Das Sarma,

Phys.

calculations

[13] This conclusion

to introduce

may be modified

to the electron

layer that

relatively state.

if the hole is close excitonic

effects

Liu, and S. Das Sarma,

unpub-

are important.

D.Z. Liu, H.A. Fertig,

Phys. Rev. B 48,

H.A. Fertig,

[8] D. Hofstadter,

Phys.

in the PL, and are unlikely

and S. Das Sarma,

1545 (1993);

and S. Dss Sarma,

305,

65,

Rev. B 44, 8759 (1991).

and A.A. Maradudin,

(121 More sophisticated

enough Fertig,

Phys.

Phys. Rev. Lett.

(1991);

(1994). [7] H.A.

Lett.

1435 (1993).

long-lived,

et al, Phys. Rev. B 45,4532

Kukushkin

Rev.

42 (1994);

et al., Phys. Rev. Lett. 66,926

I.V. Kukushkin

Phys.

Proper-

therein.

[5] H. Buhmann

Kinaret,

1959 (1977).

E.M.

[6] I.V.

see B.

1994).

[4] S.A. Brown

references

71,

42,

and J.M.

Surface

Rev. B 14, 2239 (1976).

[14] H.A. Fertig,

D.Z.

lished.

[15] L.P.

Kadanoff

Mechanics,

and G. Baym,

(Benjamin,

Reading,

Quantum 1981).

Statistical

SHAKEUP

vol. 95, No. 7 [16] An improved the phonon

approximation self-energy

to a self-consistent energy.

processes

present

approach,

further

shakeup

scheme could include

in Eq.(5), which would lead

equation

Such an approach

phonon

EFFECTS ON PHOTOLUMINBSCENCE

beyond

for the phonon would include those

included

which in principle satellites

and further

self-

multiple in the

could lead t.o broadening

441

small, and the main PL peak is only very slightly broadened

by the phonon

self-energy,

it seems un-

likely that such multiple phonon processes a pronounced

effect on our main result, at least at

low temperatures. processes ature

will have

We believe, however,

are important

of the lattice,

of the the main PL peak. Because the single collec-

an accurate

tive mode emission shakeup peaks are already quite

passes through

that such

near the melting

and must be included

picture of the PL spectrum the melting transition.

temperto get

as the WC