Electronic g factor in biased quantum wells

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376

ELECTRONIC

g FACTOR

IN BIASED QUANTUM

B. The first contribution gi, is axially symmetric while Yf,’ arises from the lack of an inversion center in the compositional materials. The Zeeman effect is described by the g factor tensor components g, = gyy = g,, g, = gll and gY = g,,x = CF. The electric-field-induced in-plane anisotropy of the g factor, due to the nonzero off-diagonal components g,, g,, has been predicted by Kalevich and Korenev [9] and recently observed by Hallstein et al. [8]. Here we concentrate on the calculation of the diagonal components gl and gll.

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Vol. 102, No. 5

functions complicates the problem. In this geometry it is convenient to apply perturbation theory and find g, from the matrix element (el, 1/2]6?/s,s]el, -l/2). Here (el, +1/2) are the electron states at the subband bottom (/& = k,, = 0) for B = 0, &t8x8 = (e/c)C,&,, C,, = W8x8/d(li$) is the Kane velocity operator, its components are constant and we choose the vector potential in the form A(r) = (0, -B,z, 0). The transverse g-factor is related to the above matrix element by $ggl/.+Bx = $ggopBBx+ (el, 1/2]&flel,

3. LONGITUDINAL

g FACTOR

We use the Kane model with a 8 X 8 matrix Hamiltonian, Ytsx8, characterized by an axial symmetry [lo]. Thus in this model the coefficients /3, X and {in equation (2) are equal to zero but it can be used to calculate the parameters of equation (1). The longitudinal magnetic field B 11z as well as the electric field F 11t retain the axial symmetry which allows to seek solutions of the Schrodinger equation in the form

- 3el,1/2]t$,lel,

AE=E,,2-E_1,2

= (z+

?)p~Bz

where m. is the free electron mass, - e is the electron charge and we assume that ma/M > g11/2.At low magfield the netic fields, AE CCIBzI’“. With increasing square-root law changes to a linear dependence AE 0~ B,. Thus, values of both n and gll can be found from calculations of the spin splitting as a function of the magnetic field. 4. TRANSVERSE

g FACTOR

The transverse magnetic field, B IIx, reduces the axial symmetry and the application of harmonic oscillator

- l/2)@,),

(5)

where go is the free electron Land& factor (go = 2) and we have excluded the diamagnetic term - (l/c)]drj(r)A(r), where j(r) is the current density, because according to equation (1) the electron with the wave vector k = 0 and the spin polarized along the n axis moves with the group velocity vY = vF/li = (el, 1/2]i$,lel, - l/2). It follows then that g, is given by g, = go - 4T(el,

where ]j) are the eight Bloch amplitudes ]I’,, m), ]F7, m) (m = + l/2), II’s, m) (m = 2 3/2, t1/2), 4n,(p) are the harmonic oscillator functions and the integers nj are connected with the conduction-band Landau-level number, n, by the relation nj = n - m + l/2. The coupled differential equations for the envelope functions h(z) were solved for a fixed IZ by means of the scattering matrix approach [ 111. The boundary conditions imposed on the envelopes at the interfaces were taken to be the same as in [l]. According to equation (l), in the longitudinal magnetic field, the spin splitting of the lowest Landau level with (k2) = e]B,]/lic is given by

-l/2)

1/2]$Jz - Q)]el,

- l/2),

(6)

where (z) = (el, s(z]el, s) with s = l/2 or s = -l/2.

5. RESULTS

AND DISCUSSION

The electric field effect on the longitudinal and transverse g factor components in the lattice-matched Gao,4,1no,53As/InP QW structures is demonstrated in Fig. 1. The band parameters used in the calculation are as follows: The band gap Eg = 0.813 eV, the spin-orbit splitting of the valence band A = 0.356 eV, 2pfv/mo = 25.5 eV for bulk Gao,&o,5&, Eg = 1.423 eV, A = 0.108 eV, 2&,/m. = 20.4 eV for InP, the valence band offset AE, = 0.356 eV (pcy is the interband matrix element i(S&,lZ)). In order to take into account the contribution to the g factor from remote bands, we added a constant of Ag = - 0.13 to the bulk values of g respectively for the well and barrier materials in which case g(Gao,4,1no,&s) = -4.5 and g(InP) = 1.2. The dotted curve in Fig. l(b) presents the electric-field dependence of g, for a single heterointerface. In this case the transverse g factor increases from the bulk value of - 4.5 at zero field to values gl > -3 at F > 1.25 mVA_‘. Comparing Fig. l(a) and (b) one concludes that the quantum confinement in unbiased QWs has a stronger influence on g, than on gll, in agreement with the results of Kowalski et al. [7]. In QWs, at low electric fields, both g factor components can be approximated by a sum of field-independent and quadratic-in-F terms. At sufficiently high electric fields the influence of the second interface upon the electron confinement

Vol. 102, No. 5

ELECTRONIC

g FACTOR

IN BIASED QUANTUM

-2.0

377

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-2.0 (a) *

InGaAdlnP

-2.5 -

- 200 -4.5 0.0

- 100

@I

A

-2.5 -

A I 0.5

I

I 1.0

I

-4.5 -

I, 1.5

2.0

Electric field F (mv/A)

0.0

0.5

1.0

1.5

2.0

Electric field F (mv/A)

Fig. 1. The longitudinal (a) and transverse (b) electron g fcctor aa a functiop of the electric field applied to GA0,4&r0,53As/InP QW structures with the well width of 100 A, 150 A and 200 A. The dotted curve in Lb) presents the transverse g factor for a biased single heterointerface. The dashed curve in (a) is calculated for a 100 A QW in the single-band approximation, other curves are calculated in the Kane model. Inset: The schematic representation of the electron and heavy-hole envelop functions in a biased QW. becomes negligible, which fxplains wh,y in Fig. l(b) the curves gl(F) for the 150 A and 200 A QWs approach asymptotically the dotted curve. For the 100 A QW, similar asymptotic behavior takes place at higher fields. The developed theory allows one to estimate an influence of the built-in electric field on the measured values of g, and gll in GaInAs/InP structures. The quantitative comparison with the experiment [7] could be performed only taking into account a particular field variation within the structure. The dotted curve in Fig. l(a) is calculated for the 100 A QW in the one-band approximation by averaging the bulk g factor values g(k,) = g(0) + hkz in the well and barrier (see for details [l]). We remind that in this approximation the electron g factor remains isotropic. In Fig. 2 we show the electric-field dependence of the transverse g factor for a GaAs/A10,35Ga0,&s QW structure. The parameters used are Eg = 1.52 eV, A = 0.34 eV, 2&r0 = 28.9 eV for bulk GaAs and Eg = 1.94 eV, A = 0.32 eV, 2p~&ne = 26.7 eV (dashed and solid curves) and 24.7 eV (dotted curves) for the barrier material, the band offset AE, : AE, = 2 : 3. The contribution from remote bands is taken into account by adding a constant Ag = -0.12 to the values of gll and g, calculated in the Kane model. The difference between dashed and dotted curves in Fig. 2 illustrates the sensitivity of gll,, to the barrier

.I

1 Well width

(A)

Fig. 2. The electron g factor as a function of the GaAs/ Alo,35Gao,6+4s QW width. Dashed (dotted) curves represent the transverse and longitudinal g factor components in an unbiased QW calculated assuming that the bulk value of the g factor in the barrier material equals to 0.57 (0.67). The transverse g factor calculated for a,biased QW with the electric fieldF = 0.56 and 1.0 meV A-’ and for g(A10,35Ga0,65As) = 0.57 is shown by solid curves.

378

ELECTRONIC

g FACTOR

IN BIASED QUANTUM

interband matrix element. Note that, for the above two values ofp,, the g factor in the bulk barrier material is equal, respectively, to 0.57 and 0.67 and, for a 100 A QW GaAs/Alo,jsGao,&s, gl(F = 0) = - 0.17 and -0.13. The calculated field-induced contribution gl(F = 0.56 meV A- ‘) - g, (0) = 0.007 is in a reasonable agreement with the measured value -0.01 [S]. In conclusion it is worth to mention that the electron g factor is a much more sensitive parameter as compared, say, to the confinement energy or the in-plane effective mass of a quasi-two-dimensional electron. The reason is that usually the bulk values of g in the well and barrier compositional materials differ considerably and sometimes are even of opposite signs. As a result any interlayer redistribution and modification of electronic wave function leads to a remarkable change in the g factor and its anisotropy. Measurements of the g factor as a function of the magnetic field direction are promising for the analysis of the shape of electron envelopes and its modification in an external electric field in 1D and OD structures. In general the effective g factor tensor, g,p, can contain both symmetrical and antisymmetrical offdiagonal components and, therefore, for totally asymmetrical quantum dots the Zeeman effect is described by nine independent parameters. Acknowledgements-We would like to express our gratitude to A.A. Sirenko for valuable discussions. This work was supported in part by the Royal Swedish Academy of Science. E.L.I. acknowledges also the support of the NATO collaboration program (HTECCRG 950377).

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REFERENCES Ivchenko, E.L. and Kiselev, A.A., Fiz. Tekhn. Poluprovodn., 26, 1992, 1471 [Soviet Phys. Semicond., 26, 1992, 827. 2. Ivchenko, E.L., Kochereshko, V.P., Uraltsev, I.N. and Yakovlev, D.R., In HighMagnetic Fields in Semiconductor Physics III (Edited by G. Landwehr), p. 533. Springer Ser. Solid-State Sci., Vol. 101. Springer, Berlin, Heidelberg, 1992. 3. Kalevich, V.K. and Korenev, V.L., Pis’ma ZhETF, 56, 1992, 257 [JETP Lett., 56, 1992, 2531. 4. Kalevich, V.K., Zakharchenya, B.P. and Fedorova, O.M., Fiz. Tverd. Tela, 37, 1995, 287 [Phys. Solid State, 37, 1995, 1541. 5. Sirenko, A.A., Ruf, T., Eberl, K., Cardona, M., Kiselev, A.A., Ivchenko, E.L. and Ploog, K., Proc. 12th Int. Co@ on the Application of High Magnetic Fields in Semiconductor Physics, Wiirzburg, 1996 (to be published). 6. Le Jeune, P., Robart, D., Marie, X., Amand, T., Brousseau, M., Barrau, J., Kalevich, V. and Rodichev, D., Semicond. Sci. Technol., 1997 (to be published). 7. Kowalski, B., Omling, P., Meyer, B.K., Hofmann, D.M., Wetzel, C., Hirle, V., Scholz, F. and Sobkowicz, P., Phys. Rev., B49, 1994, 14786. S., Oestreich, M., Rtihle, W.W. and 8. Hallstein, Kohler, K., Proc. 12th Int. Co@ on the Application of High Magnetic Fields in Semiconductor Physics, Wiirzburg, 1996 (to be published). 9. Kalevich, V.K. and Korenev, V.L., Pis’ma ZhETF, 57, 1993, 557 [JETP Lett., 57, 1993, 5711. 10. Kane, E.O., J. Phys. Chem. Solids, 1, 1957, 249. 11. Kiselev, A.A. and Moiseev, L.V., Fiz. Tverd. Tela, 38, 1996, 1574 [Phys. Solid State, 38, 1996, 8661. 1.