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Magnetoexcitons in coupled quantum wells
17 March 1997
PHYSICS
ELSEXIER
LETTERS
A
Physics Letters A 227 ( 1997) 271-284
Magnetoexcitons in coupled quantum wells Yu.E. Lozovik a,‘, A.M. Ruvinsky b a Institute of Spectroscopy, Academy of Sciences of Russia, 142092 Troitsk, Moscow Region. Russian Federation h Department of Theoretical Physics, Moscow Institute for Steel and Alloys, Leninsky Prospekt 4. 117934 Moscow, Russian Federation
Received 10 December 1996; accepted for publication 20 december 1996 Communicated by V.M. Agranovich
Abstract Excitons in coupled quantum wells with spatially separated electrons (e) and holes (h) in a high transverse magnetic field (H) are considered. The exciton energies are calculated for various Landau levels and arbitrary interwell separations (D). The change of parameter D/l (where 1 = ,,/m) leads to a reconstruction of the exciton dispersion laws E( P) (P is the conserved “magnetic” momentum of the magnetoexciton, being proportional to the mean distance between e and h along the wells). Noncentral (“roton”) extrema exist only for interlayer separations D/l smaller than the critical one (the latter depends on exciton quantum numbers). Impurity states in CQWs are also calculated. PAC.% 7 I .3S.+z: 73.20.D~
1. Introduction
Systems of excitons (or pairs) with spatially separated electrons (e) and holes (h) recently have attracted interest long after the prediction of superfluidity in these systems [ 11, manifesting itself in a sense as a superconductivity along the electron and hole layers. An interesting quasi-Josephson phenomenon in this system was also studied [2J. The two-dimensional exciton system in a strong transverse magnetic field considered in Refs. 13-51 (see also Refs. [6,7]) revealed very curious properties. Particularly it was shown that the ground state of a one-layer exciton system in a strong magnetic field is an ideai gas of excitons at any density [ 3,5]. This is supported experimentally in fine experiments of Chemla and others (see Ref. [ 81 and references therein). Moreover, exact solutions for the ground state (corresponding to a Bose condensate of non-Bose particles) and for some excited states were obtained [5] (see also Ref. [6]), being the consequence of the supersymmetry in the problem [ 5,7]. The properties of a system of spatially separated electrons and holes was investigated recently in a set of interesting experiments [ 9- 141. ’ E-mail: [email protected] 0375-9601/97/$17.00 Copyright @ 1997 Published by Elsevier Science B.V. All rights reserved PI! 50375-960 I (97)00039-X
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Recent experimental investigations [ 1 I] of the photoluminescence of spatially separated electron-hole systems in strong transverse magnetic fields in coupled quantum wells (CQW) discovered interesting collective properties of the system. Fairs of spatially separated e and h (indirect excitons) may condense in the liquid phase and give rise also to a variety of different phases in the system of indirect excitons [ 151. The condition for the appearance of these phases can be satisfied only if the exciton lifetime is much longer than the thermalization time. The electrons and holes localized mainly in opposite quantum wells may satisfy this criterion because the probability of annihilation of electrons and holes is suppressed by the small overlap of electron and hole wave functions. The application of an electric field normal to the layers [ 9,111 diminishes this overlapping and also suppresses the recombination rate. In such a regime a magnetic field was found to result in a strong change of both the photoluminescence, decay time and diffusion coefficient [ II]. As claimed in Ref. [ 111 superfluidity of indirect excitons (predicted in Ref. [ 1] taking into account magnetic field effects [ 31) can be responsible for the observed effects. All discussed above and recent detailed experimental investigations of magnetoexciton spectra in CQWs [ 12,141 stimulate, as a first step, performing a detailed analysis of the properties of an isolated magnetoexciton with spatially separated electrons and holes. It is just this problem our paper is devoted to (collective properties are discussed elsewhere [ 151). The organization of this paper is as follows. In the present paper the exciton dispersion laws E(P) as a function of conserved “magnetic” momentum for various Landau levels and arbitrary interwell separation (D) are calculated for nondegenerate states. It turns out that noncentral (“roton”) minima (at P # 0)in the exciton spectrum arise only at an interwell separation smaller than some critical one. The equation for the critical value (D/r),, (I = dw is the magnetic length) is obtained for arbitrary exciton states. Degenerate exciton states and splitting of the exciton dispersion laws as a function of D are considered. The variation of D/Z leads to the reconstruction of the dispersion laws in an analogous manner. The problem of the impurity states in CQWs in a strong magnetic field (for an electron or hole bounded to the spatially separated impurity) is analyzed. The effect of the width of CQWs on magnetoexciton dispersion is discussed. The possibility of an experimental study of magnetoexciton dispersion laws, particularly an exciton in roton minima, is shortly discussed.
2. An Exciton with a spatially separated case
electron and hole in a strong magnetic field: nondegenerate
We consider a magnetoexciton in coupled quantum wells in high magnetic fields and thin quantized wells. In order to use the perturbation method we suppose that the separation eH/,uc between adjacent energy levels connected with Landau quantization (,u = memh/(m, + mh) is the reduced mass) and also with the size quantization N 7r2fi2/2m,,,d~,, greatly exceeds the characteristic Coulomb energy of the e-h interaction (i.e. magnetoexciton energy). The magnetoexciton energy is of order e*/k at D < 1 and of order e212/D3e at D >> 1 (see below). These conditions lead to the inequalities 1<
d* << &,hl,
%,hr
for interwell
separations
l4 << a,+hD3,
D
(1)
or to
d*l* << a,,hD3,
l
(2)
for D>l, where &J, = fi*/m,,he* are the effective Bohr radii of e and h, m,Q are the effective masses of the e and h (for H = O), E = (~1 + ~2) /2, ~1.2are the dielectric susceptibilities of the media surrounding e and h layers, de,h are the widths of the wells, 1 = &$% is the magnetic length. The Hamiltonian of the electron and hole located in different 2D layers (in the symmetrical gauge for the vector potential A = ;H x r) has the form
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Letters A 227 (1997) 271-284
213
(3) where re.h are the radius vectors of e and h along the layers (the role of the interlayer tunneling will be considered in another paper). The Schrodinger equation for the exciton in a magnetic field is invariant on the translation of the electron and hole by the same vector and a simultaneous gauge transformation of A(r) (more general cases are considered in Ref. [ 151). This invariance leads to the conservation of the “magnetic momentum” of the exciton P, being equal at H = 0 to the ordinary momentum of the center of mass. The existence of this conserved quantity essentially simplifies the calculations for 3D [ 16,171 and 2D one-layer magnetoexcitons [4]. We use this conservation law below in our calculations of the 2D indirect exciton. The operator of the magnetic momentum may be obtained through the operator of infinitesimal translation of e and h and a simultaneous infinitesimal gauge transformation and has the form
(4) Contrary to the case H = 0 the quantity P in a strong magnetic field is connected to the mean displacement of electron relative to the hole along the layers: pa = Z2(H x P)/TiH. Using [A, $‘I = 0 one obtains the orthonot-malized exciton wave functions as eigenfunctions of the magnetic momentum P, P+;Hxr
>I
exp( $iyr . P)@(r
- PO),
where P is an eigenvalue of P, y = (mh - m&/(mh + m,), R = (m,r, + mhrh)/(me eigenfunction of the relative motion @(r - po) is a solution of the equation efi -$A+--yH.(rxV)+2&c
e2 8pcLc2
H2r2 -
e* EJD*
+ (r + &,I2
Q(r)
+ mh), r = r, - rh. The
= E@(r).
(6)
In accordance with inequalities ( 1 ), (2) exciton energies may be calculated with the use of perturbation theory for the Coulomb interaction. In the first order of the small parameter I/a,.h at D K 1 or f4/a,,hD3 at D B I one can neglect transitions on other Landau levels. So one may use for Q(r) the eigenfunctions
(7) here L; is the Laguerre
polynomial.
The unperturbed
~~,=~,[n+~(lml-ym+l)l (WC= eH/,uc is the cyclotron energy, II = min(
(by Coulomb
interaction)
spectrum
is discrete,
(8)
nt , n2), m = ]nt - Q(, nt.2 are the e and h quantum numbers). If y = 1 this energy is degenerate in the positive angular-momentum projection m. Real masses of quasiparticles have finite values and hence y < 1 and the degeneration in the momentum of the relative motion is absent. But the exciton energy &$,, is degenerate in the magnitude of P. At some values of the ratio m,/mh magnetoexciton levels (8) become closely spaced. For example, if m, = mh then levels with the quantum numbers n = 0, m = 1 and n = 0, m = - 1 coincide; the energy level n=l,m=Ocoincideswiththelevelsn=O,m~2aty=((m(-2)/ m ( we consider this case in more detail in Section 3).
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Letters A 227 (1997)
271-284
The operator of e-h interaction
(9) commutes with p and may be diagonalized on P. Thus in the nondegenerate case the exciton energies E,,,(P) in the first order perturbation are equal to E,,(P) = E$ + E,,,(P), where E,,,(P) = -(nmPle*/~dD* + (re - rh)*lnmP) and ,I$“, is the unperturbed spectrum (8). As a result one can obtain the exciton dispersion laws in the first order perturbation theory,
e2(n+ 'm')!j-n",(v,P),
&n,,(P*~D)= -2 f,,,(P,D)
=
gp*k(g)(1$(;;;)! 2 k=O
x
(10)
fin!
s,s2=o
cnslc~(-l)s’+s* (Iml+~l)!(lml+R?)!
[(v2:p2)k+“2 Y(lrnlfSl
+ (D2;2P2)k
+S*+k+l,;(D*+P*))
T(lml+s,+S2--+$,~(02+P2))
1 ,
(11)
~(a, X) is the incomplete gammawhere g = [ 1 + (P/‘D)21- ‘i2, T(a x) is the complete gamma-function, function, pk (x) is the Legendre polynomial [ 181, C,f is the number of combinations of n elements taking s at a time. The indirect magnetoexciton dispersion law is defined by the two dimensionless parameters P = H/h, D = D/Z. The analysis of the dispersion laws of a two-dimensional magnetoexciton at D = 0 (i.e. for a direct exciton) was performed in Ref. [4]. Below it will be shown that the exciton spectrum consists of zones with width monotonically decreasing as a function of D (for a direct exciton this width is of order e*/1). At small 2, (see below) the dispersion laws are found to be nonmonotonic for all states except the ground state (n = m = 0). In this case the exciton energy in the state (n, m) has n + 1 minima, and for m = 0 the deepest minimum corresponds to zero momentum. The dispersion laws with nonzero m turn out to have a maximum for P = 0. Analogously to Ref. [4] we rewrite Eq. (9) in the form
n!
e2 Ll(P~~)
= -const
= -Tr12
x
(n +
J
Nnn,(r)
Iml)!
J d*r
&r+~o)~+D*’
(12)
This expression one can interpret as the interaction between the 2D charged distribution -eN,,, (r) and a charged circle spot with radius D and the center at the end of the vector -pO. The function N”,,,(r) possesses axial symmetry and has Nmax = n + 1 maxima with width N 1 and Nen = n + 1 - 80,~ minima. The first maximum is the highest one for arbitrary quantum numbers rz, m. The center of symmetry (0,O) corresponds to the maximum of N,,,(r) for m = 0 and to the minimum for m # 0. The main contribution to the integral in Eq. ( 12) above gives the region of the crossing of spot and rings corresponding to the maxima of Nnm (r) . Using the electrostatic analogy described above we give a qualitative analysis of the magnetoexciton dispersion laws in the state with quantum numbers n = 0, m # 0. The function NOm(r) has one maximum with radius r,,, = ml and a minimum at rti,, = 0. By changing p,, one moves the charged spot and simultaneously changes its crossing with the ring corresponding to the maximum of No,(r). For P < 1 and D < I the
Yu.E. Lozovik.
A.M. Ruvinsky/Physics
4
2
0
275
Letters A 227 (1997) 271-284
6
Ll
Fig. I. The energies of magnetoexcitons at P = 0 with quantum numbers (n, tn) = in units !!? = (e2/el)
fi
of the magnetoexciton energy Em(O)
(0, 0), (0,
at D = 0, P in units l/1;
I), ( I, 0) as functions of 23. Energy is given
1= m
is the magnetic length.
crossing region is small and because of the axial symmetry of the charge configuration the value of the integral reaches its minimum at P = 0. This means that the momentum P = 0 for D < Icorresponds to the maximum of the dispersion law. For P x rmaxh/l* and D < 1 the crossing region is maximal. Therefore the momentum P = rm,,li/Z2 corresponds to the minimum of the dispersion law (“roton” minimum). On further increasing the momentum the crossing region decreases and hence for P >> 1 the magnetoexciton energy asymptotically goes to zero. Now we consider the influence of increasing D on the integral for E,,,,(P, D). For P = 0 the value of the crossing region changes with increasing D from zero for D = 0 to SC, = 2hmax for D > rmax+ l/2. So the type of the extremum changes from maximum to minimum at D,, N r,,. Simultaneously the “roton” minimum disappears. The charge density on the spot decreases as e/z-D*. Hence the depth of the minimum diminishes. We summarise our consideration for the state with arbitrary quantum numbers n, m. The dispersion curve always has an extremum at P = 0. For m = 0 and arbitrary D it is a minimum but for m # 0 it is a maximum at D < I and a minimum at D >> 1. For D < 1 the dispersion function has n + 1 minima but for D >> 1 (more strictly for D >> r$J; r$ is the ring with the largest radius) only a single minimum at P = 0 is observed. As D increases one observes the hierarchy of the “roton” extrema disappear. The order of disappearance of extrema depends on the corresponding value at the extrema depth and on the distance between adjacent rings. For an indirect exciton (2, # O), as our calculation shows, the width of the exciton zones decreases versus D (see Figs. 1, 2). The type of the dispersion extrema in the case D < 1 does not change in comparison with extrema for the direct exciton (Figs. 2a-2c). One can obtain expressions for the energies of excitons on any level at P = 0, for example (Fig. 1) &m(D)
Eto(Dj=-EQ
= -l?ev2/*erfc
l7 ( Jz’ )
[(~+~+$)ePi”erfc(~)
EOl CD)
z-9
[(
,/2-z
--&-($)‘--&I,
2)e”2/ie~c(~)+-$====7
( 13)
where E? = ( e2/eI) m. Dispersion curves have an extremum at P = 0. So we can derive the effective magnetoexciton mass as c?*&~,,( P)/c?P*Ip+ For the dependence of the effective magnetoexciton mass on 2, we obtain 1/Mm, =
276
Yu.E. bzovik.
A.M.
RuvinskyIPhysics
Letters
A 227 (1997)
271-284
1
(W
oo-
,,” ,
3
,, ” I’
-02
-
-03
/
-
,,.”
/
4
,’
/’ c
-04
6
b
.
““..,
/
2
L_ 4’
.04
,/
,,,
-
41”
6
,.,’
3
,,--),
P Fig. 2. The dispersion n = 0, m =
I
of the magnetoexciton (curves
21 (n+ ItIll)!
1 M,,(D)
laws &(P,D)
for ‘D = 0,0.25,0.5,1
= 25/2
@rz!E
I-4);
n c s,s2=o
(c)
MO1CD) =
in the state n = m = 0 for ‘D = 0, 1.5 (curves
Q:,‘tJlml+ 31 +
(T (k, (Y,x) is the degenerate hypergeometrical to (0,0),(1,0),(0,fl) wehave (seeFig.3)
MdD/O =
(a)
in the state n = I,m
function
= 0 for D = 0,0.1,0.3,0.5,1
[ 181). For levels with quantum
- IT@’ MO
(3 + ZY)D/G
l-3);
(b)
in the state
l-5).
sz)!
MO ( 1 + D2)ev2/* erfc(D/Ji)
(curves
- ( f + 2V2 + P/2)eD2/*
erfc(V/JZ)
’
numbers
(n, m)
equal
Yu.E. Lmovik,
A.M. Ruvinsky/Physics
Letters A 227 (1997) 271-284
Fig. 3. Effective masses M of the magnetoexciton at the levels (n. m) = (00). (01). ( 10) as a function given in units of the mass MO = 2 3/2fi2/e21fi of the magnetcexciton at the (0,O) level, D in units 1.
MlO(Vo)=
211
of 23. The effective
masses are
MO
(3 + yV2 + +P +
$V6>ev’/2erfc(V/JZ>
- (7
+ 5V2 + i’o4)V
where MO = 23/2efi2/e21& is the mass of the direct magnetoexciton at n = 112= 0. Below we consider dispersion laws &,,(P, V) and effective masses of the magnetoexciton functions of V. For V < 1 expressions ( 13) transform to E@J(O,V) =-i? (
1 - 20 fi+:
&,0(O,V)=-$?
EOl(07V)
1 ’
(15)
x l/&G’
M,,,,,(V)
V2 =-+O
( ‘-z
1 ’
I-=+?
3J21;
(
For small magnetic
momenta
‘P < 1 and V < 1 the dispersion
+
2M nm)l (V)
n
v2 (n+Iml)! c
n!
Q:,XlmI
+
z
SI +
s2 -
l/2),
(17)
S,SZ=o
where Qy,“j, =C~‘C~(-l>S’+S2/(~m~+s~)!(~m~
(r-7 nm.P=o =
laws are
P2 W?l~l,P=O
4J?;
(16)
>
G”,>l (P, V) = -;
+Eo--
and (T-‘),,~~,P=o = (nm,P =O\r-‘Inm,P
n
1 (n+lml>! n!
+sz)!
1
Q;;~W4
+
SI +
s2 +
=O) or
l/2).
s, S2=0 It is interesting
as
that for m = 0 all dispersion
+f?
laws change when D grows as first order of the parameter
Q;;J2W~ +
P2 ~2 -
l/2)
mass of the indirect exciton increases
+
(18) 2M!Io(V3).
t!O,P=O
The effective magnetoexciton
V,
as V grows,
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Yu.E. Lozovik, A.M. Ruvinsky/Physics
1
Mm(~)
=--_-1 Mno
1)Z3j2 + 9D2
D (2n +
J;;MO
(n+lml>! n!
+ iD2fiMo
~2 -
3/Z),
t7%= 0,
2
Q,:‘Jh
+
$2 -
l/2),
WI= 1,
S]S&l
”
C
e~,~~~clml +sl +s2 - 3/2h
m 3 2,
SIS2=0
where M,,,is the mass of the direct exciton 1 1 -=------2fiMn MIl,
+
51SF0
fiMo
1
nn1~2
cn Q$%,
J;jMo
&MO
1
= -M
1
1)23/Z +g02(n+‘)
(n + =$+v Ill
8
Letters A 227 (1997) 271-284
(n+lml>! n!
(e and h are in the same layer),
’ Qf,yzUlml + SI + ~2 - l/2). c s,sz=o
(19)
For m = 0 and arbitrary n, direct and indirect excitons have positive effective masses. For n = 0 and m # 0 the masses of direct and indirect excitons are negative and for m >> 1 the effective exciton mass of the indirect exciton is
MO,,, = -2fiMd/ml-
l/2j3j2
1
1 - $D2/( Irnl - l/2)
At small interlayer separations D all dispersion curves (excluding only the state n = m = 0) are and have IZ+ 1 minima. As our calculations show, at increasing 2) the noncentral “roton” minima gradually disappear and for V > DCr only a single minimum with zero momentum is observed. --i0 After some algebra the equation for the the extremum type at P = 0 occurs if MO,,,(V)-' DCr depending on the quantum number m may be written as 31y(lmj+ 1, Irnl - l/2,JjDzr> = !P(jml+
nonmonotonic of E,, (P, D) The change of critical values
1, Jml + l/2, $Dzr>.
It is clear from the electrostatic analogy that DC, N rmax. For m = 1 and m = 2 we obtain V;; = 1.233. For D >> 1 and P < 1 the dispersion laws are
(r2>,,P=O P2 2D; > + 2Mnm(D)’
Dz’ = 0.756,
(20)
where (r2)nnt,p,o = 212 (n
+nP’) 12
Qf,Y2(Iml+s~+s2+1)!. s,SF0
For the lowest levels ELq. (21) gives (at P = 0) (r2)~= 212, (r2)et = 412, (r2)to = 612. The effective exciton mass is increased with 2) at D >> 1 as
For n = 0, m >> 1 the last relation Man,(V)
= V3
J;r SMol-
gives for the exciton mass 1
(3/V2)((ml+1)’
(21)
Yu.E. Lazovik. A.M. Ruvinsky/Physics
For P >
1 and P >> D dispersion
Gm,(P~~D)=
e2 --$j
laws have the following
(
2p2
asymptotics,
(~*)n”wl
v2
1-
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Letters A 227 (1997) 271-284
+
4P212
(22)
>
With the magnetic momentum increasing the mean e-h separation along the layers in the magnetoexciton increases as (r) = I*( H x P)/hH and &,,(P,D) decreases. So at sufficiently large D the magnetoexciton effects may be suppressed in real (imperfect) systems due to more essential interactions with impurities and the roughness of the layers in comparison with the e-h interaction (if the magnetoexciton energy is greater than the interactions with impurities the magnetoexciton still exists but the exciton as a whole may be localized in random fields of impurities [ 191).
3. Indirect magnetoexcitons
on excited Landau
levels: role of degeneration
In some cases the unperturbed spectrum is quasi-degenerate i.e. the distance between unperturbed levels becomes smaller than the typical e-h interaction energy I,,,,(P) [4]. For example, if y = 0 then levels with the quantum numbers n = 0, m = 1 and n = 0, m = - 1 are degenerate. The energy level n = 1, m = 0 coincides with the levels II = 0, m 2 2 at y = ( Irnl - 2) /m. Hence, due to the possible quasidegeneration the applications of the expressions given above are restricted by the following conditions,
Y>-$ e.
Ym -
2
14 fl
k&,h
’
for D/l << 1 and
y> -
l4
w-1%
&,hD3’
+1
>>
2
l4 ae,t,D3 ’
for D/l >> 1. If m e = mh the energy (8) depends on the quantum number N = 2n + 1rn) and all levels (8) are ( N + 1 )-fold degenerate (the ground state N = 0 is nondegenerate). In the first order of the perturbation theory the dispersion laws for the lowest level N = 1 gives EfJ(P,V)
= LiJc +l5o,(P9q
The last expression
f
may be represented
e*
(0 - 1pt
1OlP) . + (r + PO)=
q/D*
in the form @eWo
(O-
1PI e+’
.= IOlP) = fiW/P)2 + (r + po)*
X [(D2:pl)k+“2++k,~)+
(23)
(-l)k+‘(2k + 11 @t)(g)2”(k+
(~2;P2)k+2-k,~2;P2)]
l)!(k
- 2)! _ l)!
(24)
Here P::‘(g) is the second derivation of the Legendre polynomial P2k(g); g is defined above in the text following below Eq. ( lo), Eat (P, 27) is defined by Eq. (lo), the upper index I in Eq. (23) corresponds to plus and 2 to minus, +a is the polar angle of the vector po. As (0 - 101e*/~~~(OlO) = 0 the degeneration with zero momentum stays but the exciton dispersion curve splits into two branches with effective masses (Figs. 4a, 4b):
Yu.E. Lozovik, A.M. Ruvinsky/Physics Letters A 227 (1997) 271-284
280
1
04 -0 14
,” /
-0 24
I
t
4
0
L 1
2
3
4
P
P
Fig. 4. Two magnetoexciton branches corresponding to the degenerate state with the quantum number N = 1; N = 2 min( nl, IQ) + (nl - nz 1, nl,nz are Landau quantum numbers of e and h (see Eq. (23)). (a) ‘D = 0. Branch 2 has a “roton” minimum. (b) ‘D = 3.
1
-=
_
M1,2G9
(25)
f a(V),
MO1 CD>
MO + $V2 + $ ev2/2erfc(V/JZ) -(5+V2)(0/2)Ji75+(~ >
1
a(V) -= For V <
1
1 and P <
(26)
1 we obtain
Ef’2(P,v) = Tcwc- iE*(
1 - $V,‘) -
P2
(27)
2M1,2CD9 40) ’
where MI(D) Increase
= 4Mo( 1 + 2V2) 2 ’
M2(V)
of 2) leads to the disappearance
Efp2(P,V)
= b,
-
=
-;Mo( 1 + YmV
of the “rofon” minimum
- ;VO’). in E:. For V >> 1 we get
$
(28)
and the effective exciton masses are
For P >> 1 we obtain E,‘B2(P,V) N -@‘m/P. Analogously the level N = 2 splits into three dispersion n = 1, m = 0 and n = 0, m = 352,respectively,
Eis2(P, V) = $tkoc + ;[E,o(P,2)) + Eo2(P, f
&E,o(P,V)
E;(P,V) = &02(P.
- Eo2(P,V) 2)) - A2(P,
- A2U’,W12 27,
branches
(Figs.
5a, 5b) with quantum
numbers
27 + A2(P, DD) +
8A1(7’,W21~
(29) (30)
LE.
Lozovik. A.M. Ruvinsky/Physics
281
Letrers A 227 (1997) 271-284
/(a)
P
P
Fig. 5. Three magnetoexciton branches corresponding to the degenerate state with the quantum number N = 2; N = 2 min( n, , n2) +
nl, n2 are (b)
D =
Landau quantum numbers of e and h (see Eqs. (29).
I
(30));
(n, -n2 1,
(a) V = 0. “Roton” minimuma are present on all exciton branches.
“Roton” minima have disappeared (compare with (a) ).
where
e2
(IO1 E (O-21
]Of 2) = e**‘~Ar(P,D),
D2+(r+p0)2 q/D*
102) = e4’&A2(P, D). eL + (r + PO)*
(31)
For P = 0 we have Ar(O,D) = A2(0, D) = 0 and hence the degeneration remains for levels Ez and Ez at P = 0. When D grows the extrema at P # 0 gradually disappear and for 27 > I only a single minimum at P = 0 exists.
4. Impurity
states in CQW
The problem of the interaction between an electron (or hole) and a spatially separated charge impurity can be treated analogously to the consideration above (compare Ref. [ 4]), but the results follow directly from those given above if one assumes y = 1 (mh -+ 00). In this case the unperturbed spectrum is degenerate for positive momentum. The operator of the electron-impurity interaction -( Ze2/1) ( 1/dm) does not depend on the angle variables, and so it may be diagonalized on quantum number m (Ze is the impurity charge). Therefore, we may calculate Coulomb corrections by the usual perturbation method
(32) For m = 0 and D > m
> 1 the electron spectrum
is written as (33)
((r2),c
is
defined by Eq. (21)).
In the case D
one gets
Yu.E. Lozovik, A.M. RuvinskyIPhysics Letters A 227 (1997) 271-284
282
(34) where
(rl)n~
is defined below Eq. (17). For n = 0 and m >
1 >> 2) one easily finds
(35)
(36) So the interaction of the electron and Coulomb impurity removes tirely. For m >> 1,n = 0 the impurity levels bunched - -Z&/dm - -Z&( [ml + l)/D3 if 2) >> &Gi.
the degeneration if V <
in momentum m en2J;ti and repelled as
5. Role of layer width Now we consider a more realistic layered system with finite widths of the electron and hole layers (dt, = d, f 0). As before, we assume that particles e and h do not penetrate through the interlayer barrier (tunneling effects will be discussed separately). We consider the case of ultrathin e and h layers (see inequalities ( 1 ), (2) ) when the widths of layers d are the smallest spatial scale of the system. Until now the results given correspond to the zeroth order of d. Taking into account inequalities (l), (2) we may use the adiabatic approximation. So we suggest the transverse motion in the layer is the quickest system and so we may average the electron-hole potential on the wave functions @ sin[ (&t/d) ~1 ] , flsin [ (rkz/d) (~2 - d) ] of the transverse motion in e- and h-layers. As a result the dispersion laws gain new terms which do not lead to qualitative changes of the exciton spectrum. For D < 1, P = 0 we obtain
&“I,
I
(0,V)
x (n+Iml)! n! (kt , k2 are quantum
=
I?1
4J;;
2 Q~,“i$Ylml + $1 + ~2 - l/2) SlSZ=o numbers
of the e and h transverse
(37) motion)
and
for m = 0. In the case D >> 1 we obtain S&,(O,D) - .??(d1/D2)( l/v!%>. As an illustration we calculated dispersion laws for the exciton located in two adjoining layers with thicknesses equal to d = 0.51 for each layer. In Fig. 6 the dispersion curves &~,Eot,Eto for this case are presented in comparison with the dispersion laws for the magnetoexciton in two planes with a distance D = 0.5 between them. We can see some quantitative change of dispersion laws, however, qualitatively they change insignificantly.
Yu.E. Lozovik.
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Letters A 227 (1997) 271-284
50
____- --
_I :
,A’
-a 4
,
/’
,’ _’
EO, ,I”
-a 5
I’
:
‘---
Fig. 6. The dispersion laws of the magnetoexciton in double quantum wells with widths de = dh = 0.51 (solid lines); e- and h-layers have a common interface in comparison with the case of two spatially separated layers D = 0.51 with negligible widths of layers (dotted lines).
The difference between the dispersion curves in these two cases reduces with increasing of momentum P. This is quite natural because the magnetic momentum P is proportional to the longitudinal distance between e and h.
6. Conclusions We have investigated the 2D magnetoexciton spectrum in CQWs in strong magnetic fields. We found that the exciton spectrum consists of separate zones. The width of the zones and the character of the dispersion laws I( P, DD) (where P is the conserved “magnetic” momentum proportional to e-h separation along layers) essentially depends on the interlayer separation 23. For 23 < 1 the dispersion laws are found to be nonmonotonic in all nondegenerate states except the ground state (n = m = 0). The dispersion laws in the state (n, m) have n + 1 “roton” minima. For m = 0 and any n zero momentum corresponds to the minimum of E,,( P, D) and, on the contrary, to a maximum of &,,,( P, ‘0) for n = 0 and m Z 0. The extrema at P # 0 gradually disappear with increasing 2, and for 23 > V,, only a single extremum at P = 0 is observed. The dispersion laws of the degenerate states were also calculated. For y = 0 (m, = mh) the energy depends on the quantum number N = 2n + [ml,and every level is (N + I)-fold degenerate (the state N = 0 is nondegenerate). On increasing 23 the extrema at P # 0 also gradually disappear and for D B 1 the states with N = 1 and N = 2 have only a single minimum at P = 0. Note that “roton” magnetoexcitons with P f 0 may be created using a layer with a high Miller’s index in the plane or an artificial superlattice constructed by nanotechnological methods on the surface of quantum wells. “Roton” magnetoexcitons can arise also after relaxation from a maximum with P = 0 into a noncentral minimum of the dispersion law.
Acknowledgement This work was supported by the Russian Foundation of Fundamental of Solid State Nanostructures” and “Surface Atomic Structures”.
Investigations,
by the Programs
“Physics
284
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Letters A 227 (1997)
271-284
References [ 1] YuE. Lozovik and V.I. Yudson, Pis’ma Zh. Eksp. Tear. Fiz. 22 ( 1975) 556 [JETP Lett. 22 ( 1975) 2741; Zh. Eksp. Teor. Fiz. 78 (1976) 1167 [Sov. Phys. JETP 44 (1976) 3891; Solid State Commun. 19 (1976) 391; 21 (1977) 211; YuE. Lozovik, report at All-Union Conf. on Dielectr. Electronics, Tashkent, 1973. [2] Yu.E. Lozovik and V.I. Yudson, Zh. Eksp. Tear. Fiz. Pis’ma 25 ( 1977) 18 [Sov. Phys. JETP 25 ( 1977) 141; Solid State Commun. 22 (1977) 117; 1.0. Kulik and S.I. Shevchenko, Solid State Commun. 21 (1977) 409; Yu.E. Lozovik and A.V. Klyuchnik, J. Phys. C 11 ( 1978) L483; J. Low. Temp. Phys. 38 (1980) 761; A.V. Klyuchnik and Yu.E. Lozovik. Zh. Eksp. Tear. Fiz. 76 (1979) 670 [Sov. Phys. JETP 49 ( 1979) 3351; Yu.E. Lozovik and A.V. Poushnov, to be published. 131 I.V. Lemer and Yu.E. Lozovik, Zh. Eksp. Tear. Fiz. 80 (1981) 1488 [Sov. Phys. JETP 53 (1981) 7631; 82 (1982) 1188 155 (1982) 6911. 141 IV. Lemer and Yu.E. Lozovik, Zh. Eksp. Tear. Fiz. 78 ( 1980) 2167 [Sov. Phys. JETP 51 (1980) 5881. [5] A.B. Dzuybenko, YuE. Lozovik, Fiz. Tverd. Tela 25 ( 1983) 1519 [Solid State Phys. 25 ( 1983) 8741; 26 ( 1984) 1540 [Solid State Phys. 26 (1984) 9381; J. Phys. A 24 (1991) 415; Yu.E. Lozovik, report at Conf. on Nonhomogeneous Electronic Systems, Novosibirsk, 1982. [6] D. Paquet, T.M. Rice and K. Ueda, Phys. Rev. B 32 ( 1985) 5208. [ 7 ] A.H. MacDonald and E.H. Rezayi. Phys. Rev. B 42 ( 1990) 3224. [S] D.S. Chemla, J.B. Stark and W.H. Knox, in: Ultrafast phenomena VIII, eds. J.-L. Martin et al. (Springer, Berlin, 1993) p. 21; G. Finkelstein, H. Strikman and I. Bar-Joseph, to be published. [9] T. Fukuzawa, E.E. Mendez and J.M. Hong, Phys. Rev. L&t. 64 ( 1990) 3066; J.A. Kash et al., Phys. Rev. Lett. 66 ( 1991) 2247; J.E. Golub et al., Phys. Rev. B 45 ( 1992) 9477. [ IO] U. Sivan, PM. Solomon and H. Strikman, Phys. Rev. Let. 74 ( 1995) 450. [ 111 L.V. Butov, A. Zrenner, Cl. Abstreiter, G. Biihm and G. Weigmann, Phys. Rev. L&t. 73 (1994) 304. [ 121 A. Zrenner, L.V. Butov, M. Hang, G. Abstreiter, G. Bijhm and G. Weigmann, Phys. Rev. Lett. 72 (1994) 3383; L.V. Butov, A. Zrenner, G. Abstreiter, A.V. Petinova and K. Eberl, Phys. Rev. B 52 (1995) 12153. [ 131 J.P Cheng, 1. Kono, B.D. McCombe, I. Lo, W.C. Mitchell and C.E. Stuts, Phys. Rev. Lett. 74 ( 1995) 450. [ 141 M. Bayer, V.B. Timofeev, F. FalIer, T. Gutbrod and A. Forchel, Phys. Rev. B (August 1996). [ 151 Yu.E. Lozovik, reports at Adriatic0 Conf. on Low Dimensional Electronic Systems, Trieste, 1996; Yu.E. Lozovik and O.L. Berman, Pisma Zh. Eksp. Tear. Fiz. 64 ( 1996) 526; Phys. Ser., to be published. [ 161 R.J. Elliott and R. Loudon, J. Phys. Chem. Solids 15 ( 1960) 196; H. Hasegawa and R.E. Howard, J. Phys. Chem. Solids 21 (1961) 179. [ 171 L.P. GorKov and I.E. Dzyaloshinskii, Zh. Eksp. Tear. Fiz. 53 ( 1967) 717. [ 181 A.P Prudnikov, Yu.A. Brychkov and 0.1. Marichev, Integrals and series: special functions (Nauka, Moscow, 1983) [in Russian], [ 191 Zh.S. Gevorkyan and Yu.E. Lozovik, FIZ. Tverd. Tela 27 (1985) 1800; Yu.E. Lozovik and A.M. Ruvinsky, to be published.
PHYSICS
ELSEXIER
LETTERS
A
Physics Letters A 227 ( 1997) 271-284
Magnetoexcitons in coupled quantum wells Yu.E. Lozovik a,‘, A.M. Ruvinsky b a Institute of Spectroscopy, Academy of Sciences of Russia, 142092 Troitsk, Moscow Region. Russian Federation h Department of Theoretical Physics, Moscow Institute for Steel and Alloys, Leninsky Prospekt 4. 117934 Moscow, Russian Federation
Received 10 December 1996; accepted for publication 20 december 1996 Communicated by V.M. Agranovich
Abstract Excitons in coupled quantum wells with spatially separated electrons (e) and holes (h) in a high transverse magnetic field (H) are considered. The exciton energies are calculated for various Landau levels and arbitrary interwell separations (D). The change of parameter D/l (where 1 = ,,/m) leads to a reconstruction of the exciton dispersion laws E( P) (P is the conserved “magnetic” momentum of the magnetoexciton, being proportional to the mean distance between e and h along the wells). Noncentral (“roton”) extrema exist only for interlayer separations D/l smaller than the critical one (the latter depends on exciton quantum numbers). Impurity states in CQWs are also calculated. PAC.% 7 I .3S.+z: 73.20.D~
1. Introduction
Systems of excitons (or pairs) with spatially separated electrons (e) and holes (h) recently have attracted interest long after the prediction of superfluidity in these systems [ 11, manifesting itself in a sense as a superconductivity along the electron and hole layers. An interesting quasi-Josephson phenomenon in this system was also studied [2J. The two-dimensional exciton system in a strong transverse magnetic field considered in Refs. 13-51 (see also Refs. [6,7]) revealed very curious properties. Particularly it was shown that the ground state of a one-layer exciton system in a strong magnetic field is an ideai gas of excitons at any density [ 3,5]. This is supported experimentally in fine experiments of Chemla and others (see Ref. [ 81 and references therein). Moreover, exact solutions for the ground state (corresponding to a Bose condensate of non-Bose particles) and for some excited states were obtained [5] (see also Ref. [6]), being the consequence of the supersymmetry in the problem [ 5,7]. The properties of a system of spatially separated electrons and holes was investigated recently in a set of interesting experiments [ 9- 141. ’ E-mail: [email protected] 0375-9601/97/$17.00 Copyright @ 1997 Published by Elsevier Science B.V. All rights reserved PI! 50375-960 I (97)00039-X
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Iatters A 227 (1997) 271-284
Recent experimental investigations [ 1 I] of the photoluminescence of spatially separated electron-hole systems in strong transverse magnetic fields in coupled quantum wells (CQW) discovered interesting collective properties of the system. Fairs of spatially separated e and h (indirect excitons) may condense in the liquid phase and give rise also to a variety of different phases in the system of indirect excitons [ 151. The condition for the appearance of these phases can be satisfied only if the exciton lifetime is much longer than the thermalization time. The electrons and holes localized mainly in opposite quantum wells may satisfy this criterion because the probability of annihilation of electrons and holes is suppressed by the small overlap of electron and hole wave functions. The application of an electric field normal to the layers [ 9,111 diminishes this overlapping and also suppresses the recombination rate. In such a regime a magnetic field was found to result in a strong change of both the photoluminescence, decay time and diffusion coefficient [ II]. As claimed in Ref. [ 111 superfluidity of indirect excitons (predicted in Ref. [ 1] taking into account magnetic field effects [ 31) can be responsible for the observed effects. All discussed above and recent detailed experimental investigations of magnetoexciton spectra in CQWs [ 12,141 stimulate, as a first step, performing a detailed analysis of the properties of an isolated magnetoexciton with spatially separated electrons and holes. It is just this problem our paper is devoted to (collective properties are discussed elsewhere [ 151). The organization of this paper is as follows. In the present paper the exciton dispersion laws E(P) as a function of conserved “magnetic” momentum for various Landau levels and arbitrary interwell separation (D) are calculated for nondegenerate states. It turns out that noncentral (“roton”) minima (at P # 0)in the exciton spectrum arise only at an interwell separation smaller than some critical one. The equation for the critical value (D/r),, (I = dw is the magnetic length) is obtained for arbitrary exciton states. Degenerate exciton states and splitting of the exciton dispersion laws as a function of D are considered. The variation of D/Z leads to the reconstruction of the dispersion laws in an analogous manner. The problem of the impurity states in CQWs in a strong magnetic field (for an electron or hole bounded to the spatially separated impurity) is analyzed. The effect of the width of CQWs on magnetoexciton dispersion is discussed. The possibility of an experimental study of magnetoexciton dispersion laws, particularly an exciton in roton minima, is shortly discussed.
2. An Exciton with a spatially separated case
electron and hole in a strong magnetic field: nondegenerate
We consider a magnetoexciton in coupled quantum wells in high magnetic fields and thin quantized wells. In order to use the perturbation method we suppose that the separation eH/,uc between adjacent energy levels connected with Landau quantization (,u = memh/(m, + mh) is the reduced mass) and also with the size quantization N 7r2fi2/2m,,,d~,, greatly exceeds the characteristic Coulomb energy of the e-h interaction (i.e. magnetoexciton energy). The magnetoexciton energy is of order e*/k at D < 1 and of order e212/D3e at D >> 1 (see below). These conditions lead to the inequalities 1<
d* << &,hl,
%,hr
for interwell
separations
l4 << a,+hD3,
D
(1)
or to
d*l* << a,,hD3,
l
(2)
for D>l, where &J, = fi*/m,,he* are the effective Bohr radii of e and h, m,Q are the effective masses of the e and h (for H = O), E = (~1 + ~2) /2, ~1.2are the dielectric susceptibilities of the media surrounding e and h layers, de,h are the widths of the wells, 1 = &$% is the magnetic length. The Hamiltonian of the electron and hole located in different 2D layers (in the symmetrical gauge for the vector potential A = ;H x r) has the form
Yu.E. Lozovik, A.M. Ruvinsky/Physics
Letters A 227 (1997) 271-284
213
(3) where re.h are the radius vectors of e and h along the layers (the role of the interlayer tunneling will be considered in another paper). The Schrodinger equation for the exciton in a magnetic field is invariant on the translation of the electron and hole by the same vector and a simultaneous gauge transformation of A(r) (more general cases are considered in Ref. [ 151). This invariance leads to the conservation of the “magnetic momentum” of the exciton P, being equal at H = 0 to the ordinary momentum of the center of mass. The existence of this conserved quantity essentially simplifies the calculations for 3D [ 16,171 and 2D one-layer magnetoexcitons [4]. We use this conservation law below in our calculations of the 2D indirect exciton. The operator of the magnetic momentum may be obtained through the operator of infinitesimal translation of e and h and a simultaneous infinitesimal gauge transformation and has the form
(4) Contrary to the case H = 0 the quantity P in a strong magnetic field is connected to the mean displacement of electron relative to the hole along the layers: pa = Z2(H x P)/TiH. Using [A, $‘I = 0 one obtains the orthonot-malized exciton wave functions as eigenfunctions of the magnetic momentum P, P+;Hxr
>I
exp( $iyr . P)@(r
- PO),
where P is an eigenvalue of P, y = (mh - m&/(mh + m,), R = (m,r, + mhrh)/(me eigenfunction of the relative motion @(r - po) is a solution of the equation efi -$A+--yH.(rxV)+2&c
e2 8pcLc2
H2r2 -
e* EJD*
+ (r + &,I2
Q(r)
+ mh), r = r, - rh. The
= E@(r).
(6)
In accordance with inequalities ( 1 ), (2) exciton energies may be calculated with the use of perturbation theory for the Coulomb interaction. In the first order of the small parameter I/a,.h at D K 1 or f4/a,,hD3 at D B I one can neglect transitions on other Landau levels. So one may use for Q(r) the eigenfunctions
(7) here L; is the Laguerre
polynomial.
The unperturbed
~~,=~,[n+~(lml-ym+l)l (WC= eH/,uc is the cyclotron energy, II = min(
(by Coulomb
interaction)
spectrum
is discrete,
(8)
nt , n2), m = ]nt - Q(, nt.2 are the e and h quantum numbers). If y = 1 this energy is degenerate in the positive angular-momentum projection m. Real masses of quasiparticles have finite values and hence y < 1 and the degeneration in the momentum of the relative motion is absent. But the exciton energy &$,, is degenerate in the magnitude of P. At some values of the ratio m,/mh magnetoexciton levels (8) become closely spaced. For example, if m, = mh then levels with the quantum numbers n = 0, m = 1 and n = 0, m = - 1 coincide; the energy level n=l,m=Ocoincideswiththelevelsn=O,m~2aty=((m(-2)/ m ( we consider this case in more detail in Section 3).
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Letters A 227 (1997)
271-284
The operator of e-h interaction
(9) commutes with p and may be diagonalized on P. Thus in the nondegenerate case the exciton energies E,,,(P) in the first order perturbation are equal to E,,(P) = E$ + E,,,(P), where E,,,(P) = -(nmPle*/~dD* + (re - rh)*lnmP) and ,I$“, is the unperturbed spectrum (8). As a result one can obtain the exciton dispersion laws in the first order perturbation theory,
e2(n+ 'm')!j-n",(v,P),
&n,,(P*~D)= -2 f,,,(P,D)
=
gp*k(g)(1$(;;;)! 2 k=O
x
(10)
fin!
s,s2=o
cnslc~(-l)s’+s* (Iml+~l)!(lml+R?)!
[(v2:p2)k+“2 Y(lrnlfSl
+ (D2;2P2)k
+S*+k+l,;(D*+P*))
T(lml+s,+S2--+$,~(02+P2))
1 ,
(11)
~(a, X) is the incomplete gammawhere g = [ 1 + (P/‘D)21- ‘i2, T(a x) is the complete gamma-function, function, pk (x) is the Legendre polynomial [ 181, C,f is the number of combinations of n elements taking s at a time. The indirect magnetoexciton dispersion law is defined by the two dimensionless parameters P = H/h, D = D/Z. The analysis of the dispersion laws of a two-dimensional magnetoexciton at D = 0 (i.e. for a direct exciton) was performed in Ref. [4]. Below it will be shown that the exciton spectrum consists of zones with width monotonically decreasing as a function of D (for a direct exciton this width is of order e*/1). At small 2, (see below) the dispersion laws are found to be nonmonotonic for all states except the ground state (n = m = 0). In this case the exciton energy in the state (n, m) has n + 1 minima, and for m = 0 the deepest minimum corresponds to zero momentum. The dispersion laws with nonzero m turn out to have a maximum for P = 0. Analogously to Ref. [4] we rewrite Eq. (9) in the form
n!
e2 Ll(P~~)
= -const
= -Tr12
x
(n +
J
Nnn,(r)
Iml)!
J d*r
&r+~o)~+D*’
(12)
This expression one can interpret as the interaction between the 2D charged distribution -eN,,, (r) and a charged circle spot with radius D and the center at the end of the vector -pO. The function N”,,,(r) possesses axial symmetry and has Nmax = n + 1 maxima with width N 1 and Nen = n + 1 - 80,~ minima. The first maximum is the highest one for arbitrary quantum numbers rz, m. The center of symmetry (0,O) corresponds to the maximum of N,,,(r) for m = 0 and to the minimum for m # 0. The main contribution to the integral in Eq. ( 12) above gives the region of the crossing of spot and rings corresponding to the maxima of Nnm (r) . Using the electrostatic analogy described above we give a qualitative analysis of the magnetoexciton dispersion laws in the state with quantum numbers n = 0, m # 0. The function NOm(r) has one maximum with radius r,,, = ml and a minimum at rti,, = 0. By changing p,, one moves the charged spot and simultaneously changes its crossing with the ring corresponding to the maximum of No,(r). For P < 1 and D < I the
Yu.E. Lozovik.
A.M. Ruvinsky/Physics
4
2
0
275
Letters A 227 (1997) 271-284
6
Ll
Fig. I. The energies of magnetoexcitons at P = 0 with quantum numbers (n, tn) = in units !!? = (e2/el)
fi
of the magnetoexciton energy Em(O)
(0, 0), (0,
at D = 0, P in units l/1;
I), ( I, 0) as functions of 23. Energy is given
1= m
is the magnetic length.
crossing region is small and because of the axial symmetry of the charge configuration the value of the integral reaches its minimum at P = 0. This means that the momentum P = 0 for D < Icorresponds to the maximum of the dispersion law. For P x rmaxh/l* and D < 1 the crossing region is maximal. Therefore the momentum P = rm,,li/Z2 corresponds to the minimum of the dispersion law (“roton” minimum). On further increasing the momentum the crossing region decreases and hence for P >> 1 the magnetoexciton energy asymptotically goes to zero. Now we consider the influence of increasing D on the integral for E,,,,(P, D). For P = 0 the value of the crossing region changes with increasing D from zero for D = 0 to SC, = 2hmax for D > rmax+ l/2. So the type of the extremum changes from maximum to minimum at D,, N r,,. Simultaneously the “roton” minimum disappears. The charge density on the spot decreases as e/z-D*. Hence the depth of the minimum diminishes. We summarise our consideration for the state with arbitrary quantum numbers n, m. The dispersion curve always has an extremum at P = 0. For m = 0 and arbitrary D it is a minimum but for m # 0 it is a maximum at D < I and a minimum at D >> 1. For D < 1 the dispersion function has n + 1 minima but for D >> 1 (more strictly for D >> r$J; r$ is the ring with the largest radius) only a single minimum at P = 0 is observed. As D increases one observes the hierarchy of the “roton” extrema disappear. The order of disappearance of extrema depends on the corresponding value at the extrema depth and on the distance between adjacent rings. For an indirect exciton (2, # O), as our calculation shows, the width of the exciton zones decreases versus D (see Figs. 1, 2). The type of the dispersion extrema in the case D < 1 does not change in comparison with extrema for the direct exciton (Figs. 2a-2c). One can obtain expressions for the energies of excitons on any level at P = 0, for example (Fig. 1) &m(D)
Eto(Dj=-EQ
= -l?ev2/*erfc
l7 ( Jz’ )
[(~+~+$)ePi”erfc(~)
EOl CD)
z-9
[(
,/2-z
--&-($)‘--&I,
2)e”2/ie~c(~)+-$====7
( 13)
where E? = ( e2/eI) m. Dispersion curves have an extremum at P = 0. So we can derive the effective magnetoexciton mass as c?*&~,,( P)/c?P*Ip+ For the dependence of the effective magnetoexciton mass on 2, we obtain 1/Mm, =
276
Yu.E. bzovik.
A.M.
RuvinskyIPhysics
Letters
A 227 (1997)
271-284
1
(W
oo-
,,” ,
3
,, ” I’
-02
-
-03
/
-
,,.”
/
4
,’
/’ c
-04
6
b
.
““..,
/
2
L_ 4’
.04
,/
,,,
-
41”
6
,.,’
3
,,--),
P Fig. 2. The dispersion n = 0, m =
I
of the magnetoexciton (curves
21 (n+ ItIll)!
1 M,,(D)
laws &(P,D)
for ‘D = 0,0.25,0.5,1
= 25/2
@rz!E
I-4);
n c s,s2=o
(c)
MO1CD) =
in the state n = m = 0 for ‘D = 0, 1.5 (curves
Q:,‘tJlml+ 31 +
(T (k, (Y,x) is the degenerate hypergeometrical to (0,0),(1,0),(0,fl) wehave (seeFig.3)
MdD/O =
(a)
in the state n = I,m
function
= 0 for D = 0,0.1,0.3,0.5,1
[ 181). For levels with quantum
- IT@’ MO
(3 + ZY)D/G
l-3);
(b)
in the state
l-5).
sz)!
MO ( 1 + D2)ev2/* erfc(D/Ji)
(curves
- ( f + 2V2 + P/2)eD2/*
erfc(V/JZ)
’
numbers
(n, m)
equal
Yu.E. Lmovik,
A.M. Ruvinsky/Physics
Letters A 227 (1997) 271-284
Fig. 3. Effective masses M of the magnetoexciton at the levels (n. m) = (00). (01). ( 10) as a function given in units of the mass MO = 2 3/2fi2/e21fi of the magnetcexciton at the (0,O) level, D in units 1.
MlO(Vo)=
211
of 23. The effective
masses are
MO
(3 + yV2 + +P +
$V6>ev’/2erfc(V/JZ>
- (7
+ 5V2 + i’o4)V
where MO = 23/2efi2/e21& is the mass of the direct magnetoexciton at n = 112= 0. Below we consider dispersion laws &,,(P, V) and effective masses of the magnetoexciton functions of V. For V < 1 expressions ( 13) transform to E@J(O,V) =-i? (
1 - 20 fi+:
&,0(O,V)=-$?
EOl(07V)
1 ’
(15)
x l/&G’
M,,,,,(V)
V2 =-+O
( ‘-z
1 ’
I-=+?
3J21;
(
For small magnetic
momenta
‘P < 1 and V < 1 the dispersion
+
2M nm)l (V)
n
v2 (n+Iml)! c
n!
Q:,XlmI
+
z
SI +
s2 -
l/2),
(17)
S,SZ=o
where Qy,“j, =C~‘C~(-l>S’+S2/(~m~+s~)!(~m~
(r-7 nm.P=o =
laws are
P2 W?l~l,P=O
4J?;
(16)
>
G”,>l (P, V) = -;
+Eo--
and (T-‘),,~~,P=o = (nm,P =O\r-‘Inm,P
n
1 (n+lml>! n!
+sz)!
1
Q;;~W4
+
SI +
s2 +
=O) or
l/2).
s, S2=0 It is interesting
as
that for m = 0 all dispersion
+f?
laws change when D grows as first order of the parameter
Q;;J2W~ +
P2 ~2 -
l/2)
mass of the indirect exciton increases
+
(18) 2M!Io(V3).
t!O,P=O
The effective magnetoexciton
V,
as V grows,
278
Yu.E. Lozovik, A.M. Ruvinsky/Physics
1
Mm(~)
=--_-1 Mno
1)Z3j2 + 9D2
D (2n +
J;;MO
(n+lml>! n!
+ iD2fiMo
~2 -
3/Z),
t7%= 0,
2
Q,:‘Jh
+
$2 -
l/2),
WI= 1,
S]S&l
”
C
e~,~~~clml +sl +s2 - 3/2h
m 3 2,
SIS2=0
where M,,,is the mass of the direct exciton 1 1 -=------2fiMn MIl,
+
51SF0
fiMo
1
nn1~2
cn Q$%,
J;jMo
&MO
1
= -M
1
1)23/Z +g02(n+‘)
(n + =$+v Ill
8
Letters A 227 (1997) 271-284
(n+lml>! n!
(e and h are in the same layer),
’ Qf,yzUlml + SI + ~2 - l/2). c s,sz=o
(19)
For m = 0 and arbitrary n, direct and indirect excitons have positive effective masses. For n = 0 and m # 0 the masses of direct and indirect excitons are negative and for m >> 1 the effective exciton mass of the indirect exciton is
MO,,, = -2fiMd/ml-
l/2j3j2
1
1 - $D2/( Irnl - l/2)
At small interlayer separations D all dispersion curves (excluding only the state n = m = 0) are and have IZ+ 1 minima. As our calculations show, at increasing 2) the noncentral “roton” minima gradually disappear and for V > DCr only a single minimum with zero momentum is observed. --i0 After some algebra the equation for the the extremum type at P = 0 occurs if MO,,,(V)-' DCr depending on the quantum number m may be written as 31y(lmj+ 1, Irnl - l/2,JjDzr> = !P(jml+
nonmonotonic of E,, (P, D) The change of critical values
1, Jml + l/2, $Dzr>.
It is clear from the electrostatic analogy that DC, N rmax. For m = 1 and m = 2 we obtain V;; = 1.233. For D >> 1 and P < 1 the dispersion laws are
(r2>,,P=O P2 2D; > + 2Mnm(D)’
Dz’ = 0.756,
(20)
where (r2)nnt,p,o = 212 (n
+nP’) 12
Qf,Y2(Iml+s~+s2+1)!. s,SF0
For the lowest levels ELq. (21) gives (at P = 0) (r2)~= 212, (r2)et = 412, (r2)to = 612. The effective exciton mass is increased with 2) at D >> 1 as
For n = 0, m >> 1 the last relation Man,(V)
= V3
J;r SMol-
gives for the exciton mass 1
(3/V2)((ml+1)’
(21)
Yu.E. Lazovik. A.M. Ruvinsky/Physics
For P >
1 and P >> D dispersion
Gm,(P~~D)=
e2 --$j
laws have the following
(
2p2
asymptotics,
(~*)n”wl
v2
1-
279
Letters A 227 (1997) 271-284
+
4P212
(22)
>
With the magnetic momentum increasing the mean e-h separation along the layers in the magnetoexciton increases as (r) = I*( H x P)/hH and &,,(P,D) decreases. So at sufficiently large D the magnetoexciton effects may be suppressed in real (imperfect) systems due to more essential interactions with impurities and the roughness of the layers in comparison with the e-h interaction (if the magnetoexciton energy is greater than the interactions with impurities the magnetoexciton still exists but the exciton as a whole may be localized in random fields of impurities [ 191).
3. Indirect magnetoexcitons
on excited Landau
levels: role of degeneration
In some cases the unperturbed spectrum is quasi-degenerate i.e. the distance between unperturbed levels becomes smaller than the typical e-h interaction energy I,,,,(P) [4]. For example, if y = 0 then levels with the quantum numbers n = 0, m = 1 and n = 0, m = - 1 are degenerate. The energy level n = 1, m = 0 coincides with the levels II = 0, m 2 2 at y = ( Irnl - 2) /m. Hence, due to the possible quasidegeneration the applications of the expressions given above are restricted by the following conditions,
Y>-$ e.
Ym -
2
14 fl
k&,h
’
for D/l << 1 and
y> -
l4
w-1%
&,hD3’
+1
>>
2
l4 ae,t,D3 ’
for D/l >> 1. If m e = mh the energy (8) depends on the quantum number N = 2n + 1rn) and all levels (8) are ( N + 1 )-fold degenerate (the ground state N = 0 is nondegenerate). In the first order of the perturbation theory the dispersion laws for the lowest level N = 1 gives EfJ(P,V)
= LiJc +l5o,(P9q
The last expression
f
may be represented
e*
(0 - 1pt
1OlP) . + (r + PO)=
q/D*
in the form @eWo
(O-
1PI e+’
.= IOlP) = fiW/P)2 + (r + po)*
X [(D2:pl)k+“2++k,~)+
(23)
(-l)k+‘(2k + 11 @t)(g)2”(k+
(~2;P2)k+2-k,~2;P2)]
l)!(k
- 2)! _ l)!
(24)
Here P::‘(g) is the second derivation of the Legendre polynomial P2k(g); g is defined above in the text following below Eq. ( lo), Eat (P, 27) is defined by Eq. (lo), the upper index I in Eq. (23) corresponds to plus and 2 to minus, +a is the polar angle of the vector po. As (0 - 101e*/~~~(OlO) = 0 the degeneration with zero momentum stays but the exciton dispersion curve splits into two branches with effective masses (Figs. 4a, 4b):
Yu.E. Lozovik, A.M. Ruvinsky/Physics Letters A 227 (1997) 271-284
280
1
04 -0 14
,” /
-0 24
I
t
4
0
L 1
2
3
4
P
P
Fig. 4. Two magnetoexciton branches corresponding to the degenerate state with the quantum number N = 1; N = 2 min( nl, IQ) + (nl - nz 1, nl,nz are Landau quantum numbers of e and h (see Eq. (23)). (a) ‘D = 0. Branch 2 has a “roton” minimum. (b) ‘D = 3.
1
-=
_
M1,2G9
(25)
f a(V),
MO1 CD>
MO + $V2 + $ ev2/2erfc(V/JZ) -(5+V2)(0/2)Ji75+(~ >
1
a(V) -= For V <
1
1 and P <
(26)
1 we obtain
Ef’2(P,v) = Tcwc- iE*(
1 - $V,‘) -
P2
(27)
2M1,2CD9 40) ’
where MI(D) Increase
= 4Mo( 1 + 2V2) 2 ’
M2(V)
of 2) leads to the disappearance
Efp2(P,V)
= b,
-
=
-;Mo( 1 + YmV
of the “rofon” minimum
- ;VO’). in E:. For V >> 1 we get
$
(28)
and the effective exciton masses are
For P >> 1 we obtain E,‘B2(P,V) N -@‘m/P. Analogously the level N = 2 splits into three dispersion n = 1, m = 0 and n = 0, m = 352,respectively,
Eis2(P, V) = $tkoc + ;[E,o(P,2)) + Eo2(P, f
&E,o(P,V)
E;(P,V) = &02(P.
- Eo2(P,V) 2)) - A2(P,
- A2U’,W12 27,
branches
(Figs.
5a, 5b) with quantum
numbers
27 + A2(P, DD) +
8A1(7’,W21~
(29) (30)
LE.
Lozovik. A.M. Ruvinsky/Physics
281
Letrers A 227 (1997) 271-284
/(a)
P
P
Fig. 5. Three magnetoexciton branches corresponding to the degenerate state with the quantum number N = 2; N = 2 min( n, , n2) +
nl, n2 are (b)
D =
Landau quantum numbers of e and h (see Eqs. (29).
I
(30));
(n, -n2 1,
(a) V = 0. “Roton” minimuma are present on all exciton branches.
“Roton” minima have disappeared (compare with (a) ).
where
e2
(IO1 E (O-21
]Of 2) = e**‘~Ar(P,D),
D2+(r+p0)2 q/D*
102) = e4’&A2(P, D). eL + (r + PO)*
(31)
For P = 0 we have Ar(O,D) = A2(0, D) = 0 and hence the degeneration remains for levels Ez and Ez at P = 0. When D grows the extrema at P # 0 gradually disappear and for 27 > I only a single minimum at P = 0 exists.
4. Impurity
states in CQW
The problem of the interaction between an electron (or hole) and a spatially separated charge impurity can be treated analogously to the consideration above (compare Ref. [ 4]), but the results follow directly from those given above if one assumes y = 1 (mh -+ 00). In this case the unperturbed spectrum is degenerate for positive momentum. The operator of the electron-impurity interaction -( Ze2/1) ( 1/dm) does not depend on the angle variables, and so it may be diagonalized on quantum number m (Ze is the impurity charge). Therefore, we may calculate Coulomb corrections by the usual perturbation method
(32) For m = 0 and D > m
> 1 the electron spectrum
is written as (33)
((r2),c
is
defined by Eq. (21)).
In the case D
one gets
Yu.E. Lozovik, A.M. RuvinskyIPhysics Letters A 227 (1997) 271-284
282
(34) where
(rl)n~
is defined below Eq. (17). For n = 0 and m >
1 >> 2) one easily finds
(35)
(36) So the interaction of the electron and Coulomb impurity removes tirely. For m >> 1,n = 0 the impurity levels bunched - -Z&/dm - -Z&( [ml + l)/D3 if 2) >> &Gi.
the degeneration if V <
in momentum m en2J;ti and repelled as
5. Role of layer width Now we consider a more realistic layered system with finite widths of the electron and hole layers (dt, = d, f 0). As before, we assume that particles e and h do not penetrate through the interlayer barrier (tunneling effects will be discussed separately). We consider the case of ultrathin e and h layers (see inequalities ( 1 ), (2) ) when the widths of layers d are the smallest spatial scale of the system. Until now the results given correspond to the zeroth order of d. Taking into account inequalities (l), (2) we may use the adiabatic approximation. So we suggest the transverse motion in the layer is the quickest system and so we may average the electron-hole potential on the wave functions @ sin[ (&t/d) ~1 ] , flsin [ (rkz/d) (~2 - d) ] of the transverse motion in e- and h-layers. As a result the dispersion laws gain new terms which do not lead to qualitative changes of the exciton spectrum. For D < 1, P = 0 we obtain
&“I,
I
(0,V)
x (n+Iml)! n! (kt , k2 are quantum
=
I?1
4J;;
2 Q~,“i$Ylml + $1 + ~2 - l/2) SlSZ=o numbers
of the e and h transverse
(37) motion)
and
for m = 0. In the case D >> 1 we obtain S&,(O,D) - .??(d1/D2)( l/v!%>. As an illustration we calculated dispersion laws for the exciton located in two adjoining layers with thicknesses equal to d = 0.51 for each layer. In Fig. 6 the dispersion curves &~,Eot,Eto for this case are presented in comparison with the dispersion laws for the magnetoexciton in two planes with a distance D = 0.5 between them. We can see some quantitative change of dispersion laws, however, qualitatively they change insignificantly.
Yu.E. Lozovik.
A.M. Ruvinsky/Physics
383
Letters A 227 (1997) 271-284
50
____- --
_I :
,A’
-a 4
,
/’
,’ _’
EO, ,I”
-a 5
I’
:
‘---
Fig. 6. The dispersion laws of the magnetoexciton in double quantum wells with widths de = dh = 0.51 (solid lines); e- and h-layers have a common interface in comparison with the case of two spatially separated layers D = 0.51 with negligible widths of layers (dotted lines).
The difference between the dispersion curves in these two cases reduces with increasing of momentum P. This is quite natural because the magnetic momentum P is proportional to the longitudinal distance between e and h.
6. Conclusions We have investigated the 2D magnetoexciton spectrum in CQWs in strong magnetic fields. We found that the exciton spectrum consists of separate zones. The width of the zones and the character of the dispersion laws I( P, DD) (where P is the conserved “magnetic” momentum proportional to e-h separation along layers) essentially depends on the interlayer separation 23. For 23 < 1 the dispersion laws are found to be nonmonotonic in all nondegenerate states except the ground state (n = m = 0). The dispersion laws in the state (n, m) have n + 1 “roton” minima. For m = 0 and any n zero momentum corresponds to the minimum of E,,( P, D) and, on the contrary, to a maximum of &,,,( P, ‘0) for n = 0 and m Z 0. The extrema at P # 0 gradually disappear with increasing 2, and for 23 > V,, only a single extremum at P = 0 is observed. The dispersion laws of the degenerate states were also calculated. For y = 0 (m, = mh) the energy depends on the quantum number N = 2n + [ml,and every level is (N + I)-fold degenerate (the state N = 0 is nondegenerate). On increasing 23 the extrema at P # 0 also gradually disappear and for D B 1 the states with N = 1 and N = 2 have only a single minimum at P = 0. Note that “roton” magnetoexcitons with P f 0 may be created using a layer with a high Miller’s index in the plane or an artificial superlattice constructed by nanotechnological methods on the surface of quantum wells. “Roton” magnetoexcitons can arise also after relaxation from a maximum with P = 0 into a noncentral minimum of the dispersion law.
Acknowledgement This work was supported by the Russian Foundation of Fundamental of Solid State Nanostructures” and “Surface Atomic Structures”.
Investigations,
by the Programs
“Physics
284
Yu.E. Lozovik. A.M. Ruvinsky/Physics
Letters A 227 (1997)
271-284
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