Two-dimensional magneto-polarons in GaAsGaAlAs heterostructures

Surface Science 170 (1986) 549-555 North-Holland, Amsterdam

549

TWO-DIMENSIONAL MAGNETO-POLARONS IN GaAs-GaAIAs HETEROSTRUCTURES

Rudolf LASSNIG lnstitut ]'fir Experimentalphysik, Universiti~t Innsbruck, A-6020 lnnsbruck, Austria

Received 23 July 1985; accepted for publication 13 September 1985

A consistent, parameter-free analysis of 2D magneto-polaron effects is presented and applied to the GaAs-GaAIAs system. Experimental results of several groups are well described within lowest order perturbation theory. With the dynamical polarisability of the electron gas included, polaron pinning to the TO phonon energy is expected.

I, Introduction

Polaron effects have been treated theoretically [1-6] and detected experimentally [7-11] in several quasi-2D systems. The results appeared partially contradictory, which was attributed to the variable experimental conditions. Especially at high magnetic fields, where resonant polarons are usually detected, the effective strength of the interaction is strongly dependent on the system parameters, such as tilling factors. The present work concentrates on the consistent description of polaron effects in quasi-2D systems. The presentation is divided into four steps: (1) Comparison of 2D and 3D polarons for a parabolic band, within single particle approximation (SPA). (2) Exact inclusion of nonparabolicity (which is mostly stronger than polaron band bending). (3) Influence of occupation number effects. (4) Inclusion of the dynamic polarizability of the electron gas and phonon-Coulomb mixing, which changes the quasiparticle structure strongly. Throughout the calculations I do not use any adaptable parameters (except the band edge values), since fitting does not improve the physical understanding. A bare gaAs mass m 0 = 0.065 is used, which is determined from the analysis of 3D polarons [12]. 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation

R. Lassnig / 2D magneto-polaronsin GaAs-GaAIAs heterostructures

550

2. 2D versus 3D polarons

In an ideal 2D system the electron is described by a delta function in z-direction (the interface is taken in the x-y plane and q denotes the parallel wave vector). Due to the strong concentration of the wave function, the ideal 2D Fr6hlich interaction is considerably stronger than in 3D [2-4]. However, in a realistic system, the electron wave function is spread out over several 100 ,~, which reduces the effective interaction. Formally, this is expressed by the appearance of a form factor F(q) in the quasi-2D interaction:

e2h~L( 1 4~

V~fr(q)=

1 ) F(q). co

c~:

(1)

F(q)

includes the influence of the z-dependent part of the electron wave function and is equal to 1 for the ideal case. For the S t e r n - H o w a r d wave function (with variational parameter b), F(q) is given by [5]:

F(q)

9q/8b + 3q2/8b2)/(1 + q/b) 3. interaction is suppressed by 3b/8q for

= (1 +

(2)

Thus the high q-values. Starting the perturbation series from the bare energy levels E °, the polaron shift in SPA is given by: ( ~ E n ~---

~ M)m (q)

Vef f

( q ) / ( E ° + 8E,, - E,,° - hWL -- 3Eo ).

(3)

r¢/, q

Mn.,(q) are

1,2

matrix elements between Landau levels (n, m) and a shift

....... ideal 2D real 2D 3D

6Eo has

,"

,," / II

.';" ///iI

o1" E

,°

E 1,1

."

°,*°°

1

/

//

j/

L

l

0.5

1

wc / wLO Fig. l. Comparison of the ideal 2D, real 2D and 3D polaron mass shift for the 0-1 CR transition in single particle approximation.

R. Lassnig / 2D magneto-polarons in GaAs- GaA 1.4s heterostructures

5 51

been included in order to guarantee a consistent perturbation series [12]. It is essential to point out that the nonresonant terms give large contributions even in the resonant regime. Therefore the interaction with at least ten higher Landau levels must be included in order to obtain reasonable results. A quantitative comparison of the strength of the polaron effect is given in fig. 1 for a parabolic band and GaAs material parameters. The mass shift of the 0 - 1 cyclotron resonance (CR) transition is plotted as a function of the bare C R frequency ~0¢ = eH/moC. The dashed line indicates the 3D polaron effect for k s --- 0. The dotted line is the ideal 2D result and the full line is the realistic quasi-2d polaron mass shift for an electron density of N~l = 4 × 1 0 n / c m 2. It is seen that due to the spreading of the wave function normal to the interface the effective interaction is reduced close to the 3D value.

3. Nonparabolicity Already in 3D polaron experiments, it has been demonstrated [12] that nonparabolicity is usually as strong as polaron effects. Therefore, any polaron analysis must be based on a reliable k.p theory, which yields the bare energy levels. A simple analytical two-level approximation for polarons in the nonresonant regime has recently been derived by Das Sarma and Mason [6]. However, for a quantitative analysis it is necessary to insert the correct nonparabolic energy levels E°(nonp.) into eq. (3). For heterostructures, I have

1.2 .....

nonparabolicity ,, degenerate levels p°lar°n effect', spin split levels

....

't

o

~x

J]

1

~0

E

E 1.1

× O0 G

0 0 0

•

• •

•

•

•

w

x ..... exp., Horst et el. o ..... exp., Sigg 1

i

I

1

100

140

180

220 H[kG] Fig. 2. CR mass shift due to nonparabolicity and polaron interaction compared with the experimental data of refs. [9,10].

552

R. Lassnig / 2D magneto-polarons in GaAs-GaAIAs heterostructures

recently developed a self-consistent k .p theory [13] (a 5-level scheme including the penetration into the barrier material and many-body effects, but neglecting warping). The essential point is the derivation of an effective Schr~Sdinger equation of the form:

+/' 2m(,, z) where Uc(z ) is the potential of the conduction band edge, /3 is the generalized momentum operator and re(z) is a local effective mass obtained by matrix diagonalization. For further details see ref. [13]. The dotted line in fig. 2 shows the calculated effective mass increase due to nonparabolicity for a heterostructure with charge densities of Ne~ = 4 × 1011/cm 2 and Ndepl = 5 >( 101°/cm 2, plotted against the magnetic field. A comparison with the experimental results of Horst et al. [9] and Sigg [10] shows that the nonparabolicity contribution to the mass shift is stronger than the polaron effect. Thus it is emphasized that an exact k . p model is required for any polaron analysis.

4. Occupation number effects In lowest order many-body approximation, the effective polaron interaction with a partially occupied Landau level m (with filling factor um) is reduced by a factor of 1 - v,,, which must be added in eq. (3). This is simply Pauli's exclusion principle, which forbids the interaction with an occupied state [15]. For GaAs-GaA1As, especially for low-mobility samples, it is not clear at which magnetic field the spin degeneracy is lifted. Thus in fig. 2 the experimental data are compared with the theoretical calculation for a degenerate lowest Landau level (full line) and for a spin-split Landau level (dashed line). The overall agreement with the experiment is satisfactory. With respect to the scattering of the experimental data, the best theoretical explanation is given for spin split levels. An even more conclusive experiment has been recently performed by Seidenbusch et al. [11], who studied the energy spliting of the 0-1 from the 1-2 CR transition (a so-called three-level experiment). The occupation of the Landau levels is varied by heating the carriers with an electric field. A temperature dependent line splitting is found, increasing from 0.7 kG at low fields to 1.59 kG at high electric fields (for the 118 /.tm laser line). The theoretical single particle value for this configuration is 1.58 kG, which means that the reduction of the polaron effect due to Landau level filling is lifted by redistributing the carriers.

R. Lassnig / 2D magneto-polarons in GaAs-GaAIAs heterostructures

553

5. Dynamic polarizability and phonon-Coulomb mixing The approximation discussed in the previous section can be interpreted as a zeroth order many-body approach, since the electron gas in the occupied Landau levels is incompressible and not influenced by the additional Coulomb interaction. If a less restrictive scheme is used, including Coulomb interaction and allowing the electron system to be polarizable, the calculation becomes complex. In particular, the traditional notion of "screening" (which is a concept of multiplicative renormalization) cannot be transferred to the dynamical (and nonmultiplicative) case. The consequences are discussed in two steps:

5.1. Quasiparticle spectrum / magnetoplasmon-phonon mixture First of all, the inclusion of Coulomb interaction leads to a drastic change of the 2D boson spectrum. The combined interaction is shown in fig. 3a as a perturbation series. This series can be summed up exactly to give the full electron-electron interaction [14]: Vtot(q, ¢0)=

[Vq AcVph D0(~0)]/[1 --Uq H(q, ~o)-vp, Do(~O) Fl(q,

~0)].

(5)

Here Vq is the Coulomb potential, Vph is the phonon vertex, D O is the bare phonon propagator and H(q, ~) is the full dynamic polarizability. The poles of Vto~(q, ~0) correspond to the physical magnetoplasmon-phonon hybrids. To illustrate the consequences for a two-level system (for a partially occupied zeroth Landau level with H - go/(~2 _ ~02) and the same parameters as in fig. 2), fig. 4 shows the dispersion relation of the bosons for ~oc = 31 meV and for ¢¢ = 35 meV. The essential difference is seen from the lower branch: If

ca} Vto t .....

.

.v....~, ----~

....

b}

Fig. 3. (a) Feynman graphs for the full Coulomb-phonon mixing. Dashed lines are bare photons, wiggled lines are bare LO phonons, and the circles denote the full dynamic polarizability of the electron gas. (b) Example for partially cancelling graphs for the electron self-energy in fourth order.

R. Lassnig / 2D magneto-polarons in GaAs-GaAl,4s heterostructures

554 /.5 m

>o

.....

Wc=31meV

......

wc =35meV

E LLI

40

•

>

°

°

°

°

,

w--

~

LO

35

~

~"

"--" - - -

35meV °

'

•

•

,

.

o

•

•

,

,

•

•

•

°

*

•

,

•

,

•

,

,

i

•

•

"

°

'

°

"

~

"

•

°

wTO

f

/

30 0

--

31meV

I 0.5

.

r 1

.

.

.

.

s 1.5 q.I/fT

Fig. 4. Boson resonances in the coupled p h o n o n - p h o t o n system for two bare CR energies in a two-level system, plotted against the parallel wave vector. / is the Landau length.

w~ < ~TO, the resonance is more plasmon-like and above the bare CR energy. For ~c > wTO, the resonance is more phonon-like and the energy is smaller than ~c. This indicates that pinning to the TO phonon level can be expected: The polaron pinning energy is generally determined by the possibility of emitting a real boson. In the 2D system, if the bare CR frequency exceeds the TO phonon energy, an electron in the first Landau level should be shifted below the TO phonon energy, due to the modified boson structure. Such a behaviour has been recently reported by Nicholas et al. [15] in the I n P - G a I n A s system. 5.2. Electron self-energies cannot be calculated within R P A

Due to the complexity of the system, it is hard to find a "physical" and rapidly converging perturbation series. It is not at all clear if a perturbation procedure should be introduced using the bare particles (such as 3D phonons) or the physical particles (the poles of the full boson propagator). In the magneto-polaron problem, it can be shown that the RPA (which uses the physical particles) violates the exclusion principle and cannot be used as a reasonable approximation for electron self energies. As an example, the graphs shown in fig. 3b cancel partially. This means that the traditional RPA summation is not applicable, indicating that an

R. Lassnig / 2D magneto-polarons in GaAs-GaAIAs heterostructures

555

order-by-order p e r t u r b a t i o n procedure should be applied. T r a d i t i o n a l "screening" calculations, especially static screening [2,16], lead to u n p h y s i c a l results.

6. Conclusion I have d e m o n s t r a t e d that a consistent p o l a r o n analysis ( i n c l u d i n g wave f u n c t i o n effects, n o n p a r a b o l i c i t y a n d n o n r e s o n a n t c o n t r i b u t i o n s ) can well describe available experimental data within lowest order m a n y - b o d y app r o x i m a t i o n . The inclusion of a d y n a m i c polarizability a n d c o m b i n e d C o u l o m b - p h o n o n interaction modifies the b o s o n spectrum a n d m a y lead to a p i n n i n g to the T O p h o n o n energy.

Acknowledgements I acknowledge fruitful discussion with E. G o r n i k , K. Svozil, W. Zawadzki a n d S. Das Sarma. I t h a n k H. Sigg for sending his thesis.

References [1] [2] [3] [4] [5] [6] [7] [8]

S. Das Sarma and A. Madhukar, Phys. Rev. B22 (1980) 2823. S. Das Sarma, Phys. Rev. B27 (1983) 2590. D.M. Larsen, Phys. Rev. B30 (1984) 4595. F.M. Peeters and J.T. Devreese, Phys. Rev. B31 (1985) 3689. R. Lassnig and W. Zawadzki, Surface Sci. 142 (1984) 388. S. Das Sarma and B.A. Mason, Phys. Rev. B31 (1985) 1177. M. Horst, U. Merkt and J.P. Kotthaus, Phys. Rev. Letters 50 (1983) 754. E. Gornik, R. Lassnig, H.I. St~rmer, W. Seidenbusch, A.C. Gossard, W. Wiegmann and M. von Ortenberg, in: Proc. 17th Intern. Conf. on Physics of Semiconductors, San Francisco, 1984, Eds. D.J. Chadi and W.A. Harrison (Springer, Berlin, 1985) p. 303. [9] M. Horst, U. Merkt, W. Zawadzki, J.C. Maan and K. Ploog, Solid State Commun. 53 (1985) 403. [10] H. Sigg, Thesis, University of Nijmegen (1985). [11] W. Seidenbusch, E. Gornik and G. Weimann, Physica 134B (1985) 314. [12] G. Lindemann, R. Lassnig, W. Seidenbusch and E. Gomik, Phys. Rev. B28 (1983) 4693. [13] R. Lassnig, Phys. Rev. B31 (1985) 8076. [14] G.D. Mahan, Many Particle Physics (Plenum, New York, 1981). [15] R.J. Nicholas, L.C. Brunel, S. Huant and M.A. Brumell, preprint. [16] S. Das Sarma and B.A. Mason, Physica 134B (1985) 301.