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EISEVIER

18 December 1995

Physics Letters A 209 (1995) 201-205

PHYSICS LETTERS A

Quasi-one-dimensional exciton crystals in a high magnetic field

Andrei V. Korolev, Michael A. Liberman Condensed Matter Theory Group, Department of Physics, Uppsala University, Box 530, S-75121 Uppsala, Sweden

Received 25 August 1995; accepted for publication 17 October 1995 Communicated by J. Flouquet

Abstract

Formation of quasi-one-dimensional exciton crystals is possible in a high magnetic field. It is shown that different types of exciton complexes, from crystals to molecular complexes, can be obtained both by varying the direction and intensity of the magnetic field and by changing the exciton density. Possible physical effects are discussed from the point of view of the low-lying excitation spectrum, the stability criterion and the calculated crystallization temperature of the system.

PACS: 7 1.35.+2

Keywords: Excitons; High magnetic field; Quantum crystals

The study of possible properties of quantum crystals has become a starting point for verifying a number of fundamental concepts in modem solid-state physics. For instance, the concept of a Wigner electron crystal (WEC) attracts considerable attention due to its con- nection to the problem of the metal-insulator transi- tion [ 1,2]. The difficulty is, however, that in order to obtain the WEC as a ground state of the system one needs a rather dilute 2D electron system with rs > 10 (r, is the ratio of the interparticle distance and the elec- tron Bohr radius calculated with an effective electron mass and a background dielectric constant E) which is difficult to deal with from the experimental point of view [3]. At intermediate exciton densities a 1D system of excitons in a semiconductor quantum wire has recently been proposed as an alternative candidate for a possible observation of exciton quantum crystals in GaAs/Gal _,Al,As quantum-well wires (QWW) [ 41. Ivanov and Haug have shown that, due to the re- pulsive pair interaction of excitons created by a circu- larly polarized light pulse along a QWW, formation of

an exciton crystal may become possible at rs 2 $UO (here r, is the interparticle distance and a0 is the bulk exciton Bohr radius). This situation is in many re- spects similar to an atomic chain in a closed ring trap [ 51. However, experimental realisation of the exciton crystal is rather difficult due to a number of reasons. Below we discuss these difficulties and present some qualitative and quantitative arguments for a possible way of avoiding them.

Let us consider a quasi-one-dimensional exciton system in a semiconductor quantum wire in a high magnetic field and show that excitons under these conditions are capable of forming a quantum crystal at relatively low exciton densities, rs N a~, and at higher temperature. Unlike the situation considered in Ref. [ 41 where the exchange repulsion of excitons is conditioned by a special way of their excitation which may turn out to be difficult to realize, in our case the re- pulsive exchange part of the pair interaction necessary for creating a quantum crystal is guaranteed by a high magnetic field. Due to the strong angular dependence

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202 A.V. Koroleu, M.A. Liberman / Physics Letters A 209 (1995) 201-205

Fig. 1. Geometry of the exciton system in a semiconductor quantum

wire in a high magnetic field. Here xg is the effective size of the

exciton along the wire, rs = l/n is the equilibrium separation of two excitons (lattice parameter), and RW is the transverse size of

the wire.

of the pair interaction, it is also possible to change the sign of the interaction by changing the direction of the field and thus to obtain different exciton phases: from crystals to molecular complexes. Another advantage is that there is no such rigid restrictions on the radius of the wire as in the case without magnetic field [ 41 where the transverse size of the wire RW (see Fig. 1) is assumed to be P 0.2~0 or less and the system under consideration is thus almost purely one-dimensional. The main shortcomings of the earlier proposal are thus removed and a quasi-1D exciton crystal will be much easier to realize experimentally.

Excitons in a high magnetic field such as B* = B/BC >> 1, where B, = m2e3c/e2h3 - 5 T and m = m,mh/(m, + mh) is the reduced effective mass of the electron and the hole, have been considered in Refs. [ 6,7]. In this case, the exciton wave function of the ground state can approximately be written in terms of the Whittaker function,

Cc &p, ev-p2/4m2) exp(-IZ I/uoT>,

(1)

where po = so/v% is the radius of the first Landau orbital and 7 is related to the exciton binding energy EO = RexC~-2. Here ‘PC&, is the bulk exciton Rydberg and 7-l ---f In B” as In B* goes to infinity. The exact values of T for the region of fields 1 K B* 6 100 to which we confine ourselves in this Letter may be taken from Ref. [ 81. For instance, r = 0.5349 at B* = 10 and r = 0.4062 at B* = 50. In ( 1) we have chosen the z axis in the direction of the magnetic field. The mo- tion of the particles is more reduced by the field in the perpendicular plane, with the excitonic size along the

field being also greatly decreased. Thus the exciton in very strong fields resembles a needle stretched out along the field. It is clear that in equilibrium all exci- tons will have the same singlet spin structure, i.e., all electron spins will be opposite to the field and all hole spins will be fixed in the direction of the field. The exchange interaction of these excitons will be there- fore repulsive. At distances R between the excitons much greater than their effective size in the direction of the interaction two such excitons interact as two quadrupoles with the potential energy proportional to the Legendre polynomial Pa( cos 0))

U,,(R,8,B*) = ;+‘4(cost’),

where the quadrupole moment can be written as Q = 2(z2) = iao2T2.

Thus, if we take a$nite quasi-one-dimensional, e.g., GaAs/Gai _,Al,As medium in a high magnetic field and excite the excitons along this system with a light pulse, we shall see that the interaction of the excitons at distances greater than their effective size x0 along the wire is determined by the exchange interaction plus the quadrupole interaction and depends a great deal upon the angle B between the direction of the field and the direction of the wire (Fig. 1) . Here, by quasi- one-dimensionality of the wire, we imply the situa- tion when the transverse size of the wire RW is much larger than the transverse size of the exciton, which is Tao < ao, that is, say RW N a0 at B* = 10. This re- quirement is easily achieved experimentally and one can believe that at such RW the pair interaction of ex- citons along such a wire will not be seriously distorted by the dimensional effects due to the finite transverse size of the system. It is important to emphasize that the excitons in this semiconductor system remain suf- ficiently three-dimensional because RW is greater than the effective size of the exciton across the wire. On the other hand, RW remains sufficiently small not to allow the excitons to pass each other along the wire.

Because of the strong angular dependence of the quadrupole interaction, there are regions of angles 0 < 0 < 0, and d2 < 8 6 90” (0, = 30.5556”, 02 = 70.1243” ) where we have repulsion between excitons at large R. At such angles we should apparently obtain a linear chain of 3D excitons with equal masses M = m, + mh, interacting through the repulsive potential energy U( R, 8, B*) which will drive the whole system

A.V. Koroleu, MA. Liberman/Physics Letters A 209 (1995) 201-205 203

into the state with equal separation R = I, s l/rz (where YE is the linear density of the system).

In the region 01 < 8 < B2 there is weak attraction at large R and strong repulsion due to the exchange interaction at R N x0. Therefore, there should be either weakly bonded molecules as those of hydrogen-like type in a strong magnetic field [ 91 or, close to them, a system of liquid crystal type.

Now the question arises whether the zero-point mo- tion of light excitons will significantly change this in-

teraction picture. As is well known, harmonic oscilla- tions of a chain of this kind have the following exci- tation spectrum,

where lJ”(rS, 8, B*) is the second spatial derivative of the interaction energy of two excitons at a distance rS from each other along the line. Below we write all final expressions in terms of the exciton Bohr radius aa and the exciton Rydberg R,,,. The integration over the first Brillouin zone yields the energy of zero-point motion of the chain,

where L is the length of the wire, rs = l/nao is the dimensionless equilibrium separation (lattice param- eter), and the energy is written in units of 27X?,,,. According to the Mott criterion, at T # 0 we have to compare the zero-point energy per one elementary cell plus the thermal energy of the low-lying collec- tive excitations (3) with the corresponding expression for U( rs, 8, B*). Thus the condition of stability of the crystal against melting yields

In a high magnetic field the pair interaction between the Wannier excitons in semiconductors such as GaAs with relatively heavy holes is similar, in the effective mass approximation, to that of hydrogen-like atoms. The potential energy at distances between excitons rs greater than no and with due regard for excitons with a heavy hole can be obtained in a similar way as in Ref. [9],

2- 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3

rr = l/(na,)

Fig. 2. Calculated zero-point energy together with the thermal

energy versus the dimensionless lattice parameter rS = 1 jnno at

different temperatures (dashed lines) compared to the potential

energy (solid line) for B * = IO and 0 = 0’. The binding energy

of the exciton is Ec = 17 meV and its effective size along the

wire xg 2 0.54.

U(rs,8,B*) = iAE(r,,6,B*) +Ulq(rs,9,B*),

(6)

where the exchange part of the energy for 0 6 0 < &r in dimensionless notation is

A&r,, 0, B*) 2r,C( 8, B*) 2 cos 19 airSsin = T2cos28 -+

T 2P;

( 2r, COST a2r2sin28 xexp -____- 0 S

7 > 4P; . (7)

Here C(0, B*) N 1 is a slowly varying function of the angle and the field. In particular, at B* = 10 C(O”) = 1.7871, C(l0”) = 1.7288, C(20”) = 1.6035, and C(30”) = 1.4842, and C(B, B*) -+ 1 at higher fields.

The pictures of interaction in accordance with (5)) (6) are calculated for a GaAs/Gai _,AI,As medium at B* = 10 (0 = 0’) and are shown in Fig. 2. The crystallization temperature calculated for two values of the field (B* = 10 and B* = 50) and for the angles 8 varying from 0” to 30” is plotted in Fig. 3. As can be seen from these pictures, there are always regions of rs (r, > x0) where the crystallization is possible in the interval 0 < 19 < 81. Apparently, the most prefer- able lattice parameter at B* = 10 is rs cy 1.0, whereas for B* = 50 the crystal should exist at rs 2 0.7 (in units of a0 II 1.1 x IO+ cm). Comparing these val- ues of equilibrium rs with the corresponding values

204 A.V. Koroleu, MA. Liberman/Physics Letters A 209 (1995) 201-205

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

rs = l/(na,)

Fig. 3. Crystallization temperature of excitons in a quantum wire

in the magnetic field B* = IO (solid lines) and B* = 50 (dashed lines) and at different angles 0 between the direction of the field

and the wire. The binding energy of the exciton at B* = 50 is

E(, = 29.3 meV and its effective size along the wire changes from

xg Y 0.41 to xg F” 0.40 with H changing from O” to 30”.

in Ref. [ 41, we see that, e.g., rs N 1 .O in this paper corresponds to r, N 4.0 in Ref. [ 41. At these values of equilibrium separation the crystallization tempera- ture of the exciton system in a semiconductor wire in a high magnetic field is much higher than the one of Ref. [ 41 for the system without magnetic field. Thus the exciton crystal may exist already at not so high densities, when the lattice parameter x0 < rs < xi (for example, xl = 1.22 at B* = 10 and XI = 0.88 at B* = 50), in a high magnetic field and therefore be easier to realize. Changing the intensity of the field, one can also shift the region of densities where the crystal exists in accordance with a specific experimen- tal situation. As follows from Fig. 2, the energy of the zero-point oscillations begins to give a main contri- bution at large separations between excitons rs > XI.

This destroys the lD-diagonal order and the system turns into a quasi-l D exciton gas. Another interest- ing feature of crystallization of this kind is the depen- dence of the crystal formation on the angle 0 between the direction of the field and the wire. For instance, at 8 = 0” (B* = 10) and at rs = 1.18 there is a crys- tal in the temperature interval 0 < T < 20 K. Having changed B to 30”, we see that the crystal disappears. This is quite natural to expect due to the strong de- crease of the quadrupole interaction at this angle. As follows from considering the momentum distribution

in the crystalline phase, the probability to find an ex- citon in a certain momentum state at T < 40 K is N 2 orders of magnitude smaller than in the gaseous phase. Thus changing the angle should result in radi- cally changing the optical properties of the semicon- ductor wire: e.g., from incoherent luminescence in the gaseous phase to its disappearance in the crystalline phase. Another experimental confirmation of the exis- tence of the exciton lattice might be of course a direct observation of light difraction by the exciton lattice. An investigation of a phase diagram and optical prop- erties of this system having both spatial and magnetic- field-induced anisotropy is an interesting problem in itself which for sure will provide some surprises. The behavior of the system in the interval 92 < 8 6 90” is not qualitatively changed. That is, in the intermedi- ate region of the densities (r, N 1 .O) there exists an exciton crystal, whereas at lower densities we obtain an exciton gas. However, the situation rapidly changes with increasing field. At fields B* P 30 and higher apparently there is no crystal formation because of the large values of the zero-point energy.

In the interval of the angles 0, < 0 < t9z, we have a quite different situation. The energy of the zero-point oscillations dominates at intermediate rs N 1.0 in a wide region of fields, which means the existence of a gaseous phase of excitons. Below a certain density of excitons n, = 1 /r,,,a0 (e.g., at 8 = 40” r,,, varies from 2.8 to 1.3 by changing the field from B* = 10 to B* = 50) an instability of this phase occurs, which comes from the quadrupole attraction of excitons at large distances in this angular interval. At these den- sities the second derivative of the potential energy W” becomes negative which implies a phase transition to a system of very weakly bonded molecular complexes of two or several excitons.

So far we have considered a situation when the dis- tance between excitons is much greater than their size along the wire ~0. It is clear (see, e.g., the discussion in Ref. [ 41) that at very high densities ( rs < XO) due to hybridization of the neighboring excitons separate excitons cannot be identified. Instead of this, there are e bands filled up with electrons, whose spins are fixed against the magnetic field, and h bands filled up with holes of opposite spin configuration. In this case, e-h correlations should resemble a collective crystal-like behavior with properties also depending on the direc- tion of the field.

A.V. Koroleu, M.A. Liberman/ Physics Letters A 209 (I 995) 201-205 205

It may be interesting to apply arguments similar to those which have been discussed in this Letter to an exciton system in afinite quasi-two-dimensional semi- conductor layer in a high magnetic field. Obviously, under the same conditions the variation of the direc- tion and intensity of the magnetic filed will lead to changes of the ground state of the system. Suppose the magnetic field is perpendicular to the layer. Then the pair interaction between excitons is purely repul- sive. This should make the formation of a quasi-2D exciton crystal possible unless the zero-point energy is greater than the potential energy. On the other hand, if the direction of the field lies in the same plane as the layer, the excitonic configuration is rebuilt so that the pair interaction becomes repulsive at intermediate densities and weakly attractive at low densities. To- gether with the large zero-point energy, this should ap- parently lead to quasi-2D superfluidity very much like in a 3D exciton system in a high magnetic field 1 lo] taking into account the corrections due to the quasi- 2D character of this kind of superfluidity [ 111. The corresponding calculations for the quasi-2D exciton system as well as a more specific investigation of the connection of the optical properties with the spatial and magnetic-field-induced anisotropy for the quasi- 1D exciton system along with their phase diagrams as

a function of the magnetic field, angle, temperature, and density will be published elsewhere.

The authors would like to express their gratitude to B. Johansson and P. Omling for stimulating discus- sions.

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