Theory of Phonon Shakeup Effects on Photoluminescence from the Wigner Crystal in a Strong Magnetic Field

14 Pages·0.33 MB·English

by  D. Z. Liu

REVTEX 3.0 Das SarmaGroup Preprint, 1996
Theory of Phonon Shakeup E(cid:11)ects on Photoluminescence from the Wigner Crystal in
a Strong Magnetic Field
1(cid:3) 2 1
D. Z. Liu , H.A. Fertig , and S. Das Sarma
1
Center for SuperconductivityResearch,Departmentof Physics,
Universityof Maryland,CollegePark, Maryland20742
2
Departmentof PhysicsandAstronomy,Universityof Kentucky,Lexington,Kentucky40506-0055
(Submitted to Physical Review B on 5 January 1996)
We develop a method to compute shakeup e(cid:11)ects on photoluminescence from a strong magnetic
(cid:12)eld induced two-dimensional Wigner crystal. Only localized holes areconsidered. Our method treats
the lattice electrons and the tunneling electron on an equal footing, and uses a quantum-mechanical
calculation ofthecollective modesthatdoesnotdepend in anywayonaharmonic approximation. We
(cid:12)nd that shakeup produces a series of sidebands that may be identi(cid:12)ed with maxima in the collective
modedensityofstates,andde(cid:12)nitively distinguishes thecrystalstatefromaliquid stateintheabsence
of electron-hole interaction. In the presence of electron-hole interaction, sidebands also appear in the
liquid state coming from short-range density (cid:13)uctuations around the hole. However, the sidebands in
6 the liquid state and the crystal state have di(cid:11)erent qualitative behaviors. We also (cid:12)nd a shift in the
9 main luminescence peak, that is associated with lattice relaxation in the vicinity of a vacancy. The
9 relationship ofthe shakeup spectrum with previous mean-(cid:12)eld calculations is discussed.
1

n PACS numbers: 78.20.Ls,72.20.Jv, 73.20.Dx
a
J

5 7
I. INTRODUCTION spectrum. A mean-(cid:12)eld analysis of the latter type of

experiment showed that the PL spectrum has, in prin-

7 It was (cid:12)rst pointed out by Wigner1 more than sixty ciple, characteristic signatures of the WC:a \Hofstadter
1 8
years ago that an electron gas will undergo a zero- butter(cid:13)y" spectrum for the case of weak interactions
0
1 temperature,quantumphasetransitionintoacrystalline between the electrons and the hole, and a characteristic
0 phase as the density is lowered, since quantum (cid:13)uctua- shift in the PL spectrum upon meltingof the crystal.
6 tione(cid:11)ects diminishmorerapidlythanCoulombcorrela- In this paper, we go beyond the mean-(cid:12)eld approxi-
9 tion. Onlytwodecades ago,the(cid:12)rstconvincingevidence mation,to examineshakeup e(cid:11)ects on the PL spectrum;
t/ of an electron crystal was presented for a system ofelec- i.e.,wewillexaminehowthecollectivemodespectrumof
a trons on aHe surface2. The electron densities attainable the WC (which, at long wavelengths, corresponds to the
m
in this fashion are extremely low, however, making this classical phonon spectrum) , and the fact that some of
- an unattractive systemfor observing the quantumphase thesemodesmaybeexcitedintheelectron-holerecombi-
d
transition. Semiconductorsaremuchmoreattractivesys- nation process, modifythe results of the mean-(cid:12)eld the-
n
o temsinthissense, because onehasgreatcontroloverthe ory. Wewillconsiderindetailonlythecaseofalocalized
5;6
c electrondensities,throughdopantconcentrations. Apar- hole . Our method is purely quantum-mechanical,and
ticularlygoodcandidateforobservingtheWignercrystal treats both the tunneling electron and the other lattice
(WC)isthe two-dimensionalelectron gas(2DEG),asre- electrons on the same footing. Furthermore, we employ
alized in modulation doped semiconductors. Samples of a quantum treatment of the collective modes to realisti-
this type are now available with such high quality that callyaccountforcontributionsbothfromsmallandlarge
the electron groundstate is notnecessarily dominatedby wavevector collective excitations of the lattice. This ap-
disorder. The possibility of observing the WC is further proachhasthefurther advantageofdepending innoway
9
enhanced by the application of a strong perpendicular upon a harmonic approximation for the lattice . The
magnetic (cid:12)eld, which quenches the kinetic energy, and methodallowsinprincipleforshakeup ofarbitrarynum-
allowsthe formationofacrystalstate athigherdensities bers of these excitations. Since we are working in the
(forwhichdisordere(cid:11)ectsarelessimportant)thanwould strong magnetic (cid:12)eld limit,we consider only excitations
be possible without it. within the lowest Landau level (LLL). Our main results
Experimental evidence which may be associated with are: (1)Shakeupe(cid:11)ects shiftthemainPLpeaktohigher
7
the WC in 2DEG's has accumulated over the last sev- energies than found in a mean-(cid:12)eld treatment . (2) The
eral years3, including rf data, transport experiment, cy- HofstadterspectrumiseliminatedfromthePLspectrum,
6
clotron resonance, and photoluminescence (PL) experi- even in the case of weak electron-hole interactions (al-
ments. In the last of these probes, which has produced though we will argue below that it survives in the itin-
much intriguing data, either a valence band hole4 or a erant hole case7). The sudden shift of the PL spectrum
hole bound to an acceptor5;6 recombines with an elec- upon melting, by contrast, survives even when shakeup
tron in the 2DEG, producing a characteristic photon is included. (3) Phonon sidebands appear that corre-
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
spond tomaximainthe phonondensityofstates (DOS); To address the problem of photoluminescence from
some (but not all) of these sidebands are results of van a strong magnetic (cid:12)eld induced two-dimensional WC,
Hove singularities in the DOS, and so are characteristic we now consider a system with an interacting 2DEG
of an ordered WC state. For the case of weak electron- and a layer with a low density of holes (either local-
6
hole interactions , we do not see these sidebands in the ized or itinerant) separated from the electron plane by
liquid state, so that phonon satellites uniquely distin- a distance d, in the presence of a strong perpendicular
guish between a liquid and a solid state. Interestingly, magnetic (cid:12)eld. We assume the magnetic (cid:12)eld is very
2
for stronger electron-hole interactions, a shakeup satel- strong; i.e., (cid:22)h!c >> e =(cid:15)a, and the electronic (cid:12)lling
lite persists even above the melting temperature. We factor (cid:23) = nhc=eB << 1 in the WC regime, so that
expect this sideband to lose oscillator strength relative we can assume only the lowest Landau level is occupied
to the main peak, either with increasing temperature or by electrons. The general many-body Hamiltonian can
decreasing electron-hole interaction strength. The latter be expressed intermsofsingle-electron eigenstates (only
maybeaccomplishedbyexaminingPLfromseveralsam- N =0 eigenstates are included) as follows:
6
ples with di(cid:11)erent acceptor - 2DEG setback distances .
This paper is organized in the following way. In Sec H=Ho+Hee+Heh; (2)
II, we review the mean-(cid:12)eld theory for the photolumi-
nescence, as well as the time-dependent Hartree-Fock where
approximation for collective modes in strong magnetic X1 y X y
(cid:12)elds, and show how they maybe combined to calculate Ho = (cid:22)h!caXaX + Ehcici
2
the phonon shake-up e(cid:11)ect on photoluminescence. We X i
present anddiscuss ournumericalresults inSec. III,and 1 X X
Hee = Vc(q)<X1jexp(iq(cid:1)r)jX4 >
makesomeconcludingremarksinSec.IV.Abriefaccount 2S
10 q6=0X1X2X3X4
of some ofthese results has been published previously.
y y
(cid:2)<X2jexp((cid:0)iq(cid:1)r)jX3 >aX1aX2aX3aX4
1 X X X
II. THEORY Heh = Veh(q)<X1jexp(iq(cid:1)r)jX2 >
S
q X1X2 ij
y y
In this section, we (cid:12)rst set up the theoretical frame- (cid:2)<ijexp((cid:0)iq(cid:1)r)jj >aX1aX2cicj: (3)
work by reviewing the general expressions for the time-
dependent Hartree-Fock approximation(TDHFA) of the Here Ho is the single-particle zero-point energy which
2DEGinastrongmagnetic(cid:12)eldandoutliningthemean- is constant. Hee is the electron-electron interaction, in
2
(cid:12)eld theory for photoluminescence from Wigner crystal. which Vc(q) = 2(cid:25)e =(cid:15)q is the two-dimensional Fourier
Then we introduce the expressions for collective modes transform of the Coulomb interaction. Heh is the
in Wigner crystal. Finally we present the detailed for- electron-hole interaction, in which Veh(q) is the Fourier
malismtocalculatethe collective modesshake-up e(cid:11)ects transformoftheinteractionbetweenoneelectronandone
on the photoluminescence. hole. For a system with a small density of holes, it is a
goodapproximationtoalsoignorehole-holeinteractions,
which are extremely small in comparison with Heh. In
A. Single-ParticleProperties our current consideration, we also do not include the ef-
fectofimpuritiesandanyexternalpotential. Inpractice,
It is well known that non-interacting two-dimensional we may also drop Ho because it is just a constant. The
matrixelements in Eq.(3) are given by
electrons in a perpendicular magnetic (cid:12)eld (B = (cid:0)Bz^)
haveanenergyspectrumofdiscreteLandaulevels: EN =
1 (cid:3) <X1jexp(iq(cid:1)r)jX2 >
(N + 2)h(cid:22)!c;N = 0;1;2;:::, where !c = eB=m c is the 2 2
cyclotron resonance frequency. Working in the Landau i q lc
gauge(A = (cid:0)Bxy^), and with periodic boundary condi- =exp(2qx(X1+X2)(cid:0) 4 )(cid:14)X1;X2+qyl2c (4)
tionsinthey^-direction,thesingleparticleeigenstates are
Itiswellknownthatthegroundstatecon(cid:12)gurationfor
given by: 11
atwo-dimensionalWignercrystalisatriangularlattice .
1 2 In the presence of electron-hole interaction, the Wigner
<rjNX >= exp(iXy=lc)(cid:30)N(x(cid:0)X): (1) lattice shoulddeformaround the position ofthe hole. In
Ly
order to take into account this e(cid:11)ect, while still taking
1=2 advantageoftheperiodicityoftheWC,wedividetheWC
Here lc = (h(cid:22)c=eB) is the magnetic length and (cid:30)N is
system into hexagon unit cells. Each supercell contains
the one-dimensional harmonic oscillator eigenstate with
a (cid:12)nite numberof electrons, and one hole (in the center,
oscillationcenters X. The allowed values of X are sepa-
2 for localized hole case). The (cid:12)nite size of our unit cells
rated by 2(cid:25)lc=Ly. The degeneracy of each Landau level
2 willnot be signi(cid:12)cant for large enough supercells.
is given by g =S=2(cid:25)lc, with S the area of the 2DEG.
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
Denoting b as the distance between the centers of nearest-neighbor supercells, then the basis vectors for the super-
p
3 1
lattice are L1 = b(1;0;0) and L2 = b( 2 ;2;0). Then the basis vectors for the reciprocal lattice are given by
p (cid:16)p (cid:17)(cid:0)1
1 3 3
G1 =G0(1;0;0)and G2=G0((cid:0)2; 2 ;0),where G0 = 2 b .
In a lattice system, all the expectation values for single particle properties in momentumspace are non-zero only
for momentaequal to reciprocal lattice vectors. We (cid:12)rst de(cid:12)ne the density operator:
Z
2 2
2 G lc
n(G)= d rexp((cid:0)iG(cid:1)r)n(r)=g(cid:26)(G)exp((cid:0) ); (5)
4
where
(cid:26)(G)= g1 X e(cid:0)2iGx(X1+X2)(cid:14)X1;X2(cid:0)Gyl2cayX1aX2: (6)
X1X2
One mayeasily show that <(cid:26)(G=0)>=< N^e >=g =(cid:23),where N^e is the electron number operator.
In terms of many-bodyoperator decomposition,Hartree-Fock approximationcan be expressed as:
y y y y y y
aX1aX2aX3aX4 =<aX1aX4 >aX2aX3(cid:0)<aX1aX3 >aX2aX4 (7)
In this e(cid:11)ective mean-(cid:12)eld theory, a single electron moves as if in an average external potential provide by other
electrons through both the direct ((cid:12)rst term) and exchange (second term) interactions. For localized holes which
y
are very far away from each other, we can also apply the approximation that < cicj >= (cid:14)ij: After some algebra,
Hartree-Fock Hamiltonian for the 2DEG (with an electron-hole interaction appropriate for a localized hole) in the
lowest Landau level can be written as:
Xh 22 i
HHF =g W(G)<(cid:26)(G)>+nhVeh(G)e(cid:0)G lc=4 (cid:26)(G) (8)
G
where nh is the density of holes. Note that we ignore the negative sign in front of G because of apparent symmetry
in this problem. W(G) is the e(cid:11)ective Hartree-Fock interaction:
(cid:20) r (cid:21)
2 2 2
W(G)= (cid:15)elc2 G1lce(cid:0)G2l2c=2(1(cid:0)(cid:14)G;0)(cid:0) (cid:25)2e(cid:0)G2l2c=4I0((cid:0)G4 lc) (9)
12
where I0(x) is the modi(cid:12)edBessel function of the (cid:12)rst kind .
We de(cid:12)ne single electron Green's function:
y
G(X1;X2;(cid:28))=(cid:0)<T(cid:28)aX1((cid:28))aX2(0)> (10)
It is convenient to de(cid:12)ne the Fourier transform
G(G;(cid:28))= 1 X e(cid:0)2iGx(X1+X2)(cid:14)X1;X2+Gyl2cG(X1;X2;(cid:28)): (11)
g
X1X2
We willuse this formof the Fourier transformthroughout this work. The TDHFA is derived by writing the equation
of motionfor the Green's function,
@ @ y
G(X1;X2;(cid:28))=(cid:0) <T(cid:28)aX1((cid:28))aX2(0)> (12)
@(cid:28) @(cid:28)
y
=(cid:0)(cid:14)X1X2(cid:14)((cid:28))(cid:0)<T(cid:28)[H(cid:0)(cid:22)eNe;aX1]((cid:28))aX2(0)>:
One can compute the commutatorsexplicitly, and simplifythe result using a Hartree-Fock decomposition presented
12
in Eq.(7). After Fourier transforming with respect to time,the equation of motionfor G(G;!n) can be written as
X
0 0
(i!n+(cid:22)e)G(G;i!n)(cid:0) B(G;G)G(G;i!n)=(cid:14)G;0; (13)
G0
where
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
B(G1;G2)=[W(G1(cid:0)G2)<(cid:26)(G1(cid:0)G2)>
i
+nhVeh(G1(cid:0)G2)e(cid:0)(G1(cid:0)G2)2l2c=4 ei(G1(cid:2)G2)l2c=2: (14)
e
We can directly diagonalizematrix B and obtain its eigenvectors Vj(G) and eigenvalues !j, after which the Green's
function can be written as
XVj(G)Vj(cid:3)(G=0)
G(G;i!n)= e
j i!n+(cid:22)e(cid:0)!j
X We(G;j)
= : (15)
e
j i!n+(cid:22)e(cid:0)!j
The density of states for electrons is then given by:
1 g
D(E) =(cid:0) Im(G(G=0;E+i(cid:14)))( )
(cid:25) S
1 X We(G=0;j) 1
=(cid:0) Im( )( ) (16)
(cid:25) j E(cid:0)!je+i(cid:14) 2(cid:25)lc2
Finally,the density operator can be expressed as
(cid:0)
<(cid:26)(G)> =G(G;(cid:28) =0 )
X
(cid:3) e
= Vj(G)Vj (G=0)fFD(!j (cid:0)(cid:22)e): (17)
j
(cid:0)1
Here fFD(x) = [1+exp((cid:12)x)] is the Fermi-Dirac distribution. Since < (cid:26)(G = 0) >= (cid:23), we can self-consistently
calculate the chemical potential (cid:22)e, the density of states and the electron density in G space. By iteratively solving
Eqs.(13),(15), and (17), we can calculate the density con(cid:12)gurations for the Wigner crystal.
B. Mean-Field Theory for Localized Holes
We now present our theory for photoluminesence fromthe WC in a strong magnetic (cid:12)eld. The photoluminescence
intensity is given, for a single localized hole state, by
P(!)= I0 XXe(cid:0)En=kBTj<m;0jL^jn;h>j2(cid:14)(!(cid:0)En+Em); (18)
Z
n m
where Z =(cid:6)ne(cid:0)En=kBT, jn;h> is a many-bodyelectron state with energy En and N electrons when there is a core
hole present, jm;0 > is a many-body electron state with N (cid:0)1 electrons and energy Em, ! is the luminescence
R
frequency, and L^ = d2r (r) h(r) is the luminescence operator, with (x) the electron annihilation operator and
h(x) the hole annihilation operator. As written, the initial state is actually higher in energy than the (cid:12)nal state,
and we (cid:12)nd it convenient to rework the problemin terms of absorption rather than emission. To accomplishthis, we
0 y y
add a term H = (cid:0)E0c0c0 to the Hamiltonian,where c0 creates a localized hole, and take the limit E0 ! 1. It is
not di(cid:14)cult to show
0
P (!(cid:0)E0)
P(!)= lim (19)
E0!1 n0(E0)
0
where P is the absorption spectrum of the new Hamiltonian, and n0 is the average occupation of the hole state,
which just becomes one in the limitE0 !1. The absorption spectrum is identical to Eq.(18), except one needs to
13
add the energy E0 to allthe quantities En in the expression. After standard manipulations , one can show that
0 I0 1
P (!)= Im R(!+i(cid:14)) (20)
(cid:25) 1(cid:0)e!=kBT
The function R(!+i(cid:14)) is a response function, which continued to imaginaryfrequency has the form
4
REVTEX 3.0 Das SarmaGroup Preprint, 1996
Z (cid:12)
y i!n(cid:28)
R(i!n)=(cid:0) <T(cid:28)L((cid:28))L (0)>e d(cid:28) (21)
0
To compute this quantity, we consider (for the case of a localized hole state) instead of a single hole, a periodic
(hexagonal) lattice of them, with a unit cell that contains as many electrons as can be handled numerically. We
allow neither interactions between the holes nor tunneling between the hole sites, so that in the limit of large unit
(super)cells, one should expect the result to be the same as for the isolated hole case. Expanding the electron and
hole creation and annihilationoperators in L^ in Eq.(21) in terms of electronic states in the lowest Landau level and
localized hole states, one (cid:12)nds
R(!)= nhS XR(G;!)e(cid:0)G2l2c=4; (22)
2(cid:25)lc2 G
where R(G;!) is the Fourier transform of the followingquantity:
y y
Rij(X1;X2;(cid:28))=(cid:0)<T(cid:28)aX1((cid:28))ci((cid:28))cj(0)aX2(0)>; (23)
and
Rij(G;(cid:28))= g1 X e(cid:0)2iGx(X1+X2)(cid:14)X1;X2+Gyl2cRi;j(X1;X2;(cid:28)): (24)
X1X2
We write down the equation of motionfor Rij(X1;X2;(cid:28)) in terms of its commutatorwith the Hamiltonian:
@ @ y y
@(cid:28)Rij(X1;X2y;(cid:28)y)(cid:17)(cid:0)@(cid:28) <T(cid:28)aX1((cid:28))ci((cid:28))cj(0)aX2(0)> y y (25)
= (cid:0)<[aX1ci;cjaX2]>(cid:14)((cid:28))(cid:0)<T(cid:28)[Heff (cid:0)(cid:22)(Ne(cid:0)Nh);aX1ci]((cid:28))cjaX2 >;
P y
where Heff = H (cid:0) E0 icici and Ne and Nh are corresponding electron and hole number operator. Following
steps closely analogous to those used to (cid:12)nd the equation of motion for the Green's function in Sec. A above, the
7;12
Hartree-Fock approximationfor Eq.(25) becomes
1
(cid:0)(!+i(cid:14))Rij(G;!)=<(cid:26)(G)>(cid:14)ij +(E0(cid:0) (cid:22)h!c(cid:0)Eh)Rij(G;!) (26)
2
(cid:0)XW(G0)<(cid:26)(G0)>e(cid:0)i(G(cid:2)G0)l2c=2Rij(G(cid:0)G0;!)
G0
(cid:0) 2(cid:25)1lc2 XG0 Veh((cid:0)G0)<(cid:26)(G0)>e(cid:0)G02l2c=4Rij(G;!)
(cid:0)nhXVeh(G0)e(cid:0)G02l2c=4(cid:0)i(G(cid:2)G0)l2c=2Rij(G(cid:0)G0;!): (27)
G0
It is apparent that the solution to this satis(cid:12)es Rij(G;!)=R(G;!)(cid:14)ij. This result maybe expressed in the form
(cid:20) (cid:21)
X
0 0
(!+i(cid:14)+E0(cid:0)!0)(cid:14)G;G0 (cid:0)B(G;G) R(G;!)=(cid:0)<(cid:26)(G)>; (28)
G0
where
!0 = 21(cid:22)h!c+Eh+ 2(cid:25)1lc2 XG <(cid:26)(G)>Veh((cid:0)G)e(cid:0)G2l2c=4; (29)
and B is exactly given by Eq.(14).
We see that the form of R is essentially that of electron Green's function as in Eq.(13). By inverting Eq.(28), we
have
X(cid:20) (cid:21)(cid:0)1
0 0
R(G;!)=(cid:0) (!+i(cid:14)+E0(cid:0)!0)(cid:14)G;G0 (cid:0)B(G;G) <(cid:26)(G)>
G0
XVj(G)Vj(cid:0)1(G0)<(cid:26)(G0)>
=(cid:0)
e
jG0 !+i(cid:14)(cid:0)!0(cid:0)!j
XWe(G;j)fFD(!je(cid:0)(cid:22)e)
=(cid:0) : (30)
e
j !+i(cid:14)(cid:0)!0(cid:0)!j
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
Here we have already dropped E0 because it cancels out when we calculate the (cid:12)nal photoluminescence power using
Eqs.(19) and (20). We can see that R has poles at precisely the same energies as the poles in the electron Green's
14
functionforthesysteminthepresence oftheexternalinteractionVeh duetothehole ,uptotheconstantenergyshift
!0. However, the residues of the poles are not the same as for the Green's function. The residues are determined by
theoverlapsofthe mean-(cid:12)eldsingleparticleelectronwavefunctionswiththe holewavefunction. Itisthusappropriate
0
tothinkofRasaweightedGreen's function. SinceP (!)involvesthe imaginarypartofR(!),the photoluminescence
spectrum at this mean-(cid:12)eldlevel represents a weighted measure ofthe single particle density of states of the electron
system with a localized hole.
FIG.1. ElectrondensityofstatesandPLspectrumforperfectWCwithnoelectron-holeinteractionfor(cid:23) =2=7. (a)Electron
1
DOS below chemical potential at zero temperature (inset: DOS above chemical potential), where Ee= 2!c; (b) PL spectrum
at di(cid:11)erent temperatures: T = 0:0045Tmelt(solid), T = 0:45Tmelt(dotted), T = 0:9Tmelt(dashed) and T = 1:12Tmelt(inset:
dash-dotted).
7
The results of this mean-(cid:12)led approach have discussed previously , so we only brie(cid:13)y summarize them here. The
DOS for a perfect WC is a Hofstadter butter(cid:13)y. For fractional (cid:12)lling factor (cid:23) = p=q, this has q subbands, and at
zero temperature, the lowest p bands are (cid:12)lled. We should expect to observe this structure; i.e., p lines for (cid:23) = p=q,
in the photoluminescence spectrum of an ideal Wigner crystal if we turn o(cid:11) the electron-hole interaction. Presented
in Fig.1 are the density of states (a) and the photoluminescence spectrum (b) for an undisturbed Wigner crystal at
(cid:23) = 2=7 with no e-h interaction. We can clearly see that the DOS has 7 bands, with only the lowest 2 occupied at
low temperature. As expected, the photoluminescence has 2 peaks with a splitting identical to that of the DOS. An
observation of this behavior in experiments would directly con(cid:12)rm the presence of a Wigner crystal in the system.
Furthermore, as a function of temperature, the PL spectrum shifts very suddenly as the crystal melts to higher
frequencies. This energy shift represents the latent heat of melting of the crystal. Once again, observation of this
behavior would directly con(cid:12)rm the presence of a WC in the system.
While these e(cid:11)ects are relatively simple to understand within the mean-(cid:12)eld approach, one needs to question
whether they can survive (cid:13)uctuation e(cid:11)ects. In particular, the smallness of the Hofstadter gaps in Fig. 1 suggest
that they may not be robust enough to survive beyond the mean-(cid:12)eld level. We will see for localized holes that this
isindeed the case, and thatthis structure isreplaced bya spectrum that re(cid:13)ects the collectivemodedensity ofstates
rather than the mean-(cid:12)eldsingle particle spectrum. To show this, we proceed by reviewing how the collective modes
are computed in the TDHFA, and then show how they maybe incorporated into the PL spectrum.
C. Collective Modes in Wigner Crystal
We de(cid:12)ne the dielectric function as the density response of a two-dimensional electron system to an external
perturbation U(p;(cid:28)),i.e.
0
(cid:14) <(cid:26)((cid:0)q;(cid:28) )> 0
(cid:17)(cid:0)(cid:31)(p;q;(cid:28) (cid:0)(cid:28) ): (31)
(cid:14)U(p;(cid:28))
In the special circumstance ofWigner crystal, the density-density correlation functionhas the followingspecial form:
(cid:31)(G1+q;G2+q;(cid:28))=(cid:0)g <T(cid:26)~(G1+q;(cid:28))(cid:26)~((cid:0)G2(cid:0)q;0)>; (32)
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
where (cid:26)~= (cid:26)(cid:0) < (cid:26) >. From now on, we denote (cid:31)(G1+q;G2+q;(cid:28)) as (cid:31)G1G2(q;(cid:28)). The Fourier transform of this
response function contains poles at the collective mode frequencies of the system. This function can be numerically
12
computed using a generalized random phase approximation (GRPA) which does not assume small displacements
9
of the lattice electrons (as is necessary in classical approaches ), and thus gives a realistic dispersion relation for
the collective modes across the entire Brillouin zone. In this subsection, we present the formalism for this GRPA
(cid:16) (cid:17)1=2
calculation including the electron-hole interaction. (The magnetic length lc = (cid:22)hc=eB will be set to unity).
In the real space, the dielectric function is given by:
(cid:31)(X1;X2;X3;X4;(cid:28))=(cid:0)g <T(cid:28)(cid:26)~(X1;X2)((cid:28))(cid:26)~(X3;X4)(0)>; (33)
y y
where (cid:26)~(X1;X2)=aX1aX2(cid:0)<aX1aX2 >. The equation of motionfor (cid:31) can be written as:
@(cid:31)(X1;X2;X3;X4;(cid:28))
@(cid:28)
=(cid:0)g <[(cid:26)~(X1;X2);(cid:26)~(X3;X4)]>(cid:14)((cid:28))
(cid:0)g <T(cid:28)[H(cid:0)(cid:22)N^;(cid:26)~(X1;X2)]((cid:28))(cid:26)~(X3;X4)(0)>
=(cid:0)T1(cid:0)T2; (34)
where
T1 =g <[(cid:26)~(X1;X2);(cid:26)~(X3;X4)]>(cid:14)((cid:28))
y y
=g[<aX1aX4 >(cid:14)X2X3(cid:0)<aX3aX2 >(cid:14)X1X4](cid:14)((cid:28)): (35)
We de(cid:12)ne the Fourier transformof this quantity as
T1(q;q0;(cid:28))= 1 X e(cid:0)2iqx(X1+X2)(cid:14)X1;X2(cid:0)qy (36)
g
X1X2
(cid:2)1 X e(cid:0)2iqx0(X3+X4)(cid:14)X3;X4+qy0T1(X1;X2;X3;X4;(cid:28)):
g
X3X4
so that in frequency and momentumspace,
h
T~1(G+q;G0+q;!)=<(cid:26)(G(cid:0)G0)> e(cid:0)i(G+q)(cid:2)(G0+q)=2
i
i(G+q)(cid:2)(G0+q)=2
(cid:0)e (37)
The mostimportantterm T2 comes fromthe electron-electron interactionand electron-hole interaction inthe Hamil-
tonian. Followingthe method of Ref.12,a Hartree-Fock decompositionis applied to this term, to obtain
T~2(G+q;G0+q;(cid:28))
Xh 00 00 00 (cid:0)(G(cid:0)G00)2=4i
=2i W(G(cid:0)G )<(cid:26)(G(cid:0)G )>+nhVeh(G(cid:0)G )e
G00
00
(cid:2)sin[(G+q)(cid:2)(G +q)=2](cid:31)G00G0(q;(cid:28))
X
00 00
(cid:0)2i W(q+G )<(cid:26)(G(cid:0)G )> (38)
G00
00
(cid:2)sin[(G+q)(cid:2)(G +q)=2](cid:31)G00G0(q;(cid:28)):
Thus the equation of motionfor (cid:31)GG0(q) at speci(cid:12)c wavevector q in the Brillouinzone can be expressed in terms of
the followingmatrixequation:
h i
!+i(cid:14)(cid:0)A(q)(cid:0)D(q)W~(q) (cid:31)(q)=D(q); (39)
where
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
h i
0 2
0 0 0 (cid:0)(G(cid:0)G) =4
AGG0(q)=2i W(G(cid:0)G)<(cid:26)(G(cid:0)G)>+nhVeh(G(cid:0)G)e
0
(cid:2)sin[(G+q)(cid:2)(G +q)=2]
0 0
DGG0(q)=(cid:0)2i<(cid:26)(G(cid:0)G)>sin[(G+q)(cid:2)(G +q)=2] (40)
W~GG0(q)=W(q+G)(cid:14)GG0
The collective modes are given by the eigenvalues of the matrix A+DW~. Note that generally this matrix is not
Hermite. However, we can still diagonalize this matrixnumerically and it turns out that all the eigenvalues are real
(and symmetricaround zero) for this matrix.
D. Phonon Shake-up Theory
Wenowincorporate the e(cid:11)ect ofcollectivemodesintothe photoluminescencespectrum calculation. Recallthat we
need to calculate the quantity
Z (cid:12)
y y i!n(cid:28)
Rij(X1;X2;i!n)=(cid:0) <T(cid:28)aX1((cid:28))ci((cid:28))cjaX2 >e d(cid:28):
0
Working in the lowest Landau level, the equation of motion for Rij(X1;X2;(cid:28)) is given exactly by Eq.(25). In the
derivation of the terms for contribution from the electron-electron interaction and the electron-hole interaction, by
applying the many-bodyHamiltoniangiven by Eq.(3), we encounter the followingcorrelation function:
0 y 0 0 y y
Cij(X1X2;X3X4;(cid:28) ;(cid:28))(cid:17)(cid:0)g <T(cid:28)aX1((cid:28) )aX2((cid:28) )aX3((cid:28))ci((cid:28))cj(0)aX4(0)>; (41)
In the momentumspace, this correlation function can be written as:
Cij(q;q0;(cid:28)0;(cid:28))= 1 X e(cid:0)2iqx(X1+X2)(cid:14)X1;X2(cid:0)qy (42)
g
X1X2
(cid:2)1 X e(cid:0)2iqx0(X3+X4)(cid:14)X3;X4+qy0Cij(X1;X2;X3;X4;(cid:28)0;(cid:28)):
g
X3X4
Then the equation of motionfor Rij(G;!) maybe written as
@
Rij(G;(cid:28))=<(cid:26)(G;(cid:28))>(cid:14)ij(cid:14)((cid:28))(cid:0)(cid:15)0Rij(G;(cid:28)) (43)
@(cid:28)
X 0 iG0(cid:2)G=2(cid:0)G02=4 0
(cid:0)nh Veh(G)e Rij(G(cid:0)G;(cid:28))
G0
1 X iG(cid:2)q=2(cid:0)q2=2
(cid:0) Vc(q)Cij((cid:0)q;q+G)e
S
q6=0
1 X (cid:0)iq(cid:1)Ri(cid:0)q2=4
(cid:0) Veh(q)Cij((cid:0)q;G)e :
S
q
InEq.(43),Vc(q)andVeh(q)arethe Fouriertransformsoftheelectron-electron andelectron-holeinteractions,respec-
0
tively,(cid:15)0 istheenergy ofthe localizedhole,andthesumoverG isonlyoverreciprocal latticevectors, whilethe sums
(cid:0)G2=4
over q are over all wavevectors. Ri speci(cid:12)es the position of the hole in the i th unit cell, and < (cid:26)(G;(cid:28))> e is
the expectation value of a Fourier component of the electron density. While this quantity is independent of (cid:28) in the
groundstate, it will be convenient for later purposes to formallyleave it as an argument of the density. The method
7;12
for computingthese Fourier components has been described previously . Eq.(43) represents the (cid:12)rst in an in(cid:12)nite
15
series of equations relating an n particle Green's function to the n+1 particle Green's function . In the mean-(cid:12)eld
approximation,itwassimpli(cid:12)edbyemployingaHartree-Fock (HF)decompositionofEq.(41),whichconverts Eq.(43)
7
into a self-consistent equation for Rij .
Toincludeshakeupe(cid:11)ects,weinsteadextendthishierarchytoonemorelevel,writingdownaself-consistentformfor
Cij which explicitly contains the collective modeexcitations. To carry out this program,it is convenient to implicitly
de(cid:12)ne a self-energy by writing the last two terms in Eq.(43) as
XZ (cid:12)
0 0 0 0 0
(cid:0) d(cid:28) (cid:6)(G;G;(cid:28) (cid:0)(cid:28) )Rij(G;(cid:28) );
G0 0
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
which implicitlyde(cid:12)nes the self-energy forthis problem. The HF approximationforthe PL is equivalentto replacing
0 0 HF 0 0
(cid:6)(G;G;(cid:28) (cid:0)(cid:28) ) by (cid:6) (G;G)(cid:14)((cid:28) (cid:0)(cid:28) ), where
HF 0 0 0 iG(cid:2)G0=2
(cid:6) (G;G)=W(G(cid:0)G)<(cid:26)(G(cid:0)G;(cid:28))>e (44)
+ 1 XVeh(q)<(cid:26)((cid:0)q;(cid:28))>e(cid:0)q2=4(cid:0)iq(cid:1)Ri(cid:14)G;G0; (45)
2(cid:25)
q
7;12
and W is the sum of the direct and exchange Coulombpotentials . (Note here we keep the negative sign in front
of q because it is not necessarily a reciprocal lattice vector.) Thus we need to calculate the self-energy
HF 0
(cid:6)=(cid:6) (cid:14)((cid:28) (cid:0)(cid:28) )+(cid:14)(cid:6)
beyond mean-(cid:12)eld approximation.
15
Togenerateaself-consistentequationforCij,wetakeafunctionalderivative ofEq.(43)withrespect toanexternal
potential U which contributes an extra term in the Hamiltonian
X
y
Hext = U(X1;X2)aX1aX2:
X1X2
In doing this, it must be noted that a term directly coupling the Green's function Rij to the external potential U
(cid:0)XRij(p;(cid:28))U(G(cid:0)p;(cid:28))e2iG(cid:2)p
p
must be added to Eq.(43), and the sums over reciprocal lattice vectors must be extended to allwavevectors, because
U is not in general commensurate with the lattice.
We introduce a generalized Green's function F satisfying
X
0 0
Rij(p;(cid:28))(cid:17) F(p;p;(cid:28))<(cid:26)(p)>: (46)
p0
The equation of motionfor the generalized Green's function can be obtained followingEq.(43)
@
F(p;p1;(cid:28) (cid:0)(cid:28)1)=(cid:14)pp1(cid:14)((cid:28) (cid:0)(cid:28)1)(cid:0)(cid:15)0F(p;p1;(cid:28) (cid:0)(cid:28)1) (47)
@(cid:28)
X 2
iq(cid:2)p=2(cid:0)q =4
(cid:0)nh Veh(q)e F(p(cid:0)q;p1;(cid:28) (cid:0)(cid:28)1)
q
XZ (cid:12)
(cid:0) d(cid:28)~(cid:6)(p;q;(cid:28) (cid:0)(cid:28)~)F(q;p1;(cid:28)~(cid:0)(cid:28)1); (48)
q 0
Under Hartree-Fock approximation,the equation of motioncan be simpli(cid:12)edas
@ HF HF
F (p;p1;(cid:28) (cid:0)(cid:28)1)=(cid:14)pp1(cid:14)((cid:28) (cid:0)(cid:28)1)(cid:0)(cid:15)0F (p;p1;(cid:28) (cid:0)(cid:28)1) (49)
@(cid:28)
X 2
iq(cid:2)p=2(cid:0)q =4 HF
(cid:0)nh Veh(q)e F (p(cid:0)q;p1;(cid:28) (cid:0)(cid:28)1)
q
X
HF HF
(cid:0) (cid:6) (p;q)F (q;p1;(cid:28) (cid:0)(cid:28)1);
q
ComparingEq.(49) with the equation of motionfor the single particle Green's function (Eqs.(12) and (13)), one can
derive the followingrelation:
HF (E0(cid:0)!0(cid:0)(cid:22)e)((cid:28)1(cid:0)(cid:28)2)+ip1(cid:2)p2=2
F (p1;p2;(cid:28)1(cid:0)(cid:28)2)=(cid:0)e G(p1(cid:0)p2;(cid:28)1(cid:0)(cid:28)2): (50)
Here p1(cid:0)p2 has to be one of the reciprocal lattice vectors. Using these de(cid:12)nitions, we (cid:12)nd Eqs. (43) and (48) may
be rewritten as
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REVTEX 3.0 Das SarmaGroup Preprint, 1996
HF
F(p1;p2;(cid:28)1(cid:0)(cid:28)2)=F (p1;p2;(cid:28)1(cid:0)(cid:28)2) (51)
Z Z
X
HF
(cid:0) d(cid:28)~1d(cid:28)~2F (p1;q1;(cid:28)1(cid:0)(cid:28)~1)
q1q2
(cid:2)(cid:14)(cid:6)(q1;q2;(cid:28)~1(cid:0)(cid:28)~2)F(q2;p2;(cid:28)~2(cid:0)(cid:28)2)
and
Z Z
X
HF HF
Rij(G;(cid:28))=Rij (G;(cid:28))(cid:0) d(cid:28)~1d(cid:28)~2F (G;G1;(cid:28) (cid:0)(cid:28)~1) (52)
G1G2
(cid:2)(cid:14)(cid:6)(G1;G2;(cid:28)~1(cid:0)(cid:28)~2)Rij(G2;(cid:28)~2):
0 0
Finally,we de(cid:12)ne the zeroth order Green's functions Rij andF whichsatisfyEqs. (48)and(46)withthe self-energy
(cid:6) (and external potential) set to zero. It is not di(cid:14)cult to show that
Z Z
X
0 0
Rij(q;(cid:28))=Rij(q;(cid:28))(cid:0) d(cid:28)~1d(cid:28)~2F (q;q1;(cid:28) (cid:0)(cid:28)~1) (53)
q1q2
(cid:2)(cid:6)(q1;q2;(cid:28)~1(cid:0)(cid:28)~2)Rij(q2;(cid:28)~2):
Z
X
0
(cid:0) d(cid:28)~F (q;q1;(cid:28) (cid:0)(cid:28)~)
q1q2
iq1(cid:2)q2=2
(cid:2)U(q1(cid:0)q2;(cid:28)~)e Rij(q2;(cid:28)~): (54)
It is this form of the equation of motion for Rij that is most convenient for taking a functional derivative. Using
15
standard methods , we (cid:12)nd the Green's functions Rij and Cij satisfy the relation
(cid:14)Rij(p;(cid:28)) 0 0 0
(cid:14)U(p0;(cid:28)0) =Cij(p;p;(cid:28) ;(cid:28))(cid:0)gRij(p;(cid:28))<(cid:26)(p)>(cid:28)0
0 0
=(cid:14)Cij(p;p;(cid:28) ;(cid:28)): (55)
HF
Taking the functional derivative of Eq. (54), and de(cid:12)ning Cij as the Hartree-Fock decomposition of Cij that turns
Eq. (43) into Eq. (27), we arrive after much algebra at the relation
HF PH
(cid:14)Cij =(cid:14)Cij +Cij ;
where
Z
X
PH 0 0 HF
Cij (p;p;(cid:28) ;(cid:28))=(cid:0) d(cid:28)~1d(cid:28)~2F (p;q1;(cid:28) (cid:0)(cid:28)~1)
q1q2
(cid:14)(cid:6)(q1;q2)(cid:28)~1(cid:0)(cid:28)~2
(cid:2) (cid:14)U(p0;(cid:28)0) Rij(q2;(cid:28)~2): (56)
PH
Cij is the correction to Cij beyond mean-(cid:12)eld theory, and it will contain the phonon (i.e., collective mode) contri-
bution to Cij.
PH
To this point we have made no approximations. To obtain a closed expression for Cij , we need an approximate
formfor(cid:6). ThebestchoiceavailableistheHartree-Fockform,Eq. (45),andsowemakethissubstitution. Combining
Eqs.(31) and (44), we get:
HF
(cid:14)(cid:6) (q1;q2)(cid:28)~
(cid:14)U(p0;(cid:28)0)
iq1(cid:2)q2=2 0 0
=(cid:0)W(q1(cid:0)q2)e (cid:31)(p;q2(cid:0)q1;(cid:28) (cid:0)(cid:28)~)
1 X (cid:0)q2=4(cid:0)iq(cid:1)Ri 0 0
(cid:0) Veh(q)e (cid:14)q1q2(cid:31)(p;q;(cid:28) (cid:0)(cid:28)~): (57)
2(cid:25)
q
PH
Upon substituting the resulting Cij into the followingequation (the phonon mode contribution to Eq.(43))
10