Experimental determination of effective mass within the energy gap

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Experimental

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determination

3 June IYYl: accepted

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‘I‘hc mass of an elementary particle is defined if and only if it is specified by a real energy level. This means that it is difficult to determine experimentally with what mass an electron with momentum p tunnels through the vacuum between kcvp2 + (mc)’ + IJ’(, (V,,: any potential barrier less than 2mcL). In contrast to this case, similar physics seems to be well specified in the solid state if an effective mass equation is relied on and a precise calculation is compared with spectroscopic data, especially, in the field of refined mesoscopic semiconductor physics. In solid state physics, it is usual to employ an effective mass approximation in order to project a complicated solid state onto a one-particle system. In this approximation, the envelope function obeys the one-dimensional Schriidinger-type equation with a mass m, within the well, whereas the mass within the barrier should be taken to be that of the band gap, e.g.. AlGaAs. It is ordinary to calculate the energy levels of the well states for 1” 1902

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1. Introduction

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electrons and holes by a priori taking for the effective masses, respectively. those of the conduction band bottom and the valence band top of the barrier material. (Hereafter the subscripts B and W refer to the barrier and well. respectively.) Consider the case where the electron energy of’ concern goes below the conduction band bottom. For a very thin well width, the lowest well state comes close to the bottom of the barrier conduction band, while for a considerably wide well it is below the bottom by 65% of the band offset between GaAs ( E, = IS 19 meV) and Al ,Ga , , As ( E, = 1904 meV) [ 1,2]. This amounts to some 250 meV for x = 0.30. There is no means of direct measurement of the effective mass pertinent to the energy within the band gap. However. it is conceivable that the mass must differ from both the electron mass at the conduction band bottom and the hole mass at the valence band top. In this paper the difference in effective mass is our main concern. From a theoretical viewpoint, Ando and Akera [.?I studied this problem based on a tightbinding calculation. They derived an effcctivc

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Y Murayama, .I. Kasai / Experimental determination of effective mass within the energy gap

ton binding energy which is not easy to definitely specify. In fig. 2, photoluminescence data are shown as a function of the well width L, on a sample with a precisely controlled composition prepared by MBE and measured at T = 4.2 K. Each well is separated by a 30 nm Al,,,,Ga,,,,As barrier layer.

3. Calculation

Fig. 1. A schematic display of the calculation scheme effective mass studied in ref. [3].

of an

mass in the barrier band gap from the equation: rnn = A2~ dK/dE, where l/~ = 6 is the penetration depth of the wavefunction into the barrier. They calculated K based on the tight-binding method. This kind of procedure to obtain the effective mass within the gap follows the original discussion which appeared in ref. [4]. In fig. 1, their idea is schematically plotted. Recently a tunneling cyclotron resonance method was proposed, also with the aim to obtain the tunneling effective mass in the barriers in superlattices, when a magnetic field is applied parallel to the well/barrier plane [5]. In that study they succeeded in reproducing cyclotron energies in good agreement with their observation, by taking the effective mass within the barriers mB to be rng -Q
First we calculate the energy levels within the well based on the effective mass equation with an appropriate boundary condition at the well-barrier interface. The effective mass within the well is well-defined, as long as the non-parabolicity effect is neglected. On the contrary, the effective mass within the barrier is taken as a parameter, which is to be adjusted so as to reproduce the observed spectrum. Regarding the boundary condition between the wavefunction solutions in the well and the barri-

840 820 800 E 780 C \ 5 760 c” z? 740 : g 720 % 0, a 700

2. Experiment Thanks to advances in MBE and OMVPE technology, it is a matter of prevailing measurements that photoluminescence from quantum wells with various well widths can be observed [6-81, although its analysis is sometimes difficult because luminescence occurs between the lowest electron and highest heavy hole level minus exci-

4 well

8

12

width

/ nm

16

Fig. 2. Photoluminescence data as a function of well width. These energies always correspond to the difference between the lowest electron and the highest heavy hole levels minus the exciton binding energy, as shown in the inset. The barrier composition x in AI,Ga, _,As was determined to be 0.30 by comparing it with other data [18]. Its measurement is at 4.2 K.

ers, they can bc solved separately and eventually continued. From investigations [9,10], it is concluded that both 4 and iV$/m* should be continued together at the well/barrier interfaces. since the original Bloch function is rcduccd to an envelope function satisfying an cffcctive mass equation not necessarily imposing boundary conditions independently on both quantities. In the case of a GaAs/AlGaAs system, it is known that the conditions arc decoupled approximately to separate equalities on I,!I and iV~/m”. Thus. in this system, it is allowed to solve the equation { - ~Vzm*(z)P’Vz + V(Z)}I,MZ) = E@(z) directly by discretizing it by means of the finite difference method in the real coordinate space. Calculated energy levels in this way rclativc to the conduction band bottom of the barrier are plotted in fig. 3 (with the negative sign on the vertical axis) as a function of the effective mass of the electron in the barrier m,, in units of the free electron mass nz,,, in addition to the heavy hole energy above the valence band top (with the positive sign in this case). In this calculation. the mass of the electron in the well is taken as 0.0665 [ 1 I], and that of the heavy hole in the well as 0.34

[?I. For the hole. an additional calculation tar comparison is also carrid out for m w,,,, = 0.02 [ 13.13]. However. the larger value of m w,,,, proves not to give reasonable final results, as will hc discussed Iatcr. The dependence is grcatcr the thinner the well width or the smaller the mass in the well. The reason for this mass dependcncc is the following. When the mass in the barrier i\ large. the exponential decay of the cnvelopc function thcrcin is also large. which means more localized nature of the well state. A larger degree of localization is equivalent to a dccpcr cncrgy level for the electron or a higher level lOr the hole. Comparing the data shown in fig. 3 with the thus calculated barrier-mass dcpcndcnt clcctron energy (fig. 3). the effective mass in the barrier band gap can be precisely determined. When determining it. the well-width dcpcndcnt binding energy of an exciton must be taken into account. The data of exciton binding energies, which arc determined from Is-2 splittings, arc KCLImulated by Nelson et al. [14]. Although those data cover lots of samples with various alloy compositions and considerably scatter. thcrc is no

I’. ~uray~mu, J. Kusai / Expe~ment~l det~rrn~n~ti~nof effectire mass &thin the energy gap

fact that the hole effective mass in the well is sufficiently large. Thus, this assumption introduces few errors on the results. The horizontal axis in fig. 4 is the well width. However, since the well width determines the energy depth below the barrier conduction band bottom, this figure implicity plots the energy dependence of the effective mass within the gap.

0

E”

mB,=0.0916(Al.&a.~As

LB.)

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0

5

607

10

15

LW / nm Fig. 4. Calculated well-width dependent barrier band-gap mass for electrons with parameters: mwe = 0.0665, m 0.0916, mWhh = 0.34, mBhh = 0.358 , tht= data of (BE) t:iei from ref. 1141and our experimented data of PL. TG open circles are calculated points and the solid line shows a gradual decrease in the mass within the energy gap versus the energy below the conduction band bottom. L, = 10 nm corresponds to E = -215 mcV.

knowledge of other data than these at the present moment, Theoretical studies on the binding energy [15,161 are not precise enough to be utilized in our analysis. Thus, the finally dete~i~ed mass in the barrier is ptotted in fig. 4. In the calculation, the heavy hole mass was fixed to the value at the barrier valence band top m,,, = 0.358, for simplicity. For thinner well widths around - 1 nm, the hole energies come close to the valence band top and, hence, one introduces few errors when assuming the barrier effective mass to be that at the valence band top. In the data in ref. [14], crossovers between heavy hole and light hole excitons are seen. These effects may modify the present determination scheme slightly according to the head-light hole mixing effect. However, it makes sense only for thin well samples, where the assumption of heavy hole mass equal to the band edge mass is well justified, as was discussed above. On the other hand, for thicker wells like - 10 nm, hole energies hardly depend on the parameterized m B, as is seen in fig. 3b. This insensitivity results from the fact that the band offset for the heavy hole is at the most 135 meV as well as the

4. Discussion and summa~ The plot in fig. 4 is based on the rather scattering data in ref. [14], so that it is quite difficult to state definitely that the effective mass decreases in this manner with energy from the conduction band bottom down to the valence band top, as was predicted in ref. [3]. First of all a more precise determination of the exciton binding energy is required as a function of the well width. However, it might be safe to say that the effective mass decrecrses with going deep in energy. At a width of L, = 10 nm (i.e., E = -215 meV), our calculation gives m Be =: 0.059, while Ando and Akera’s mass is equal to - 0.064. The agreement is not so bad. Here, it should be noted that this experimental determination of effective mass covers onfy 0 2 E 2 -230 meV, which amounts to around 12% of Eg. The most remarkable point in our way of mass determination is the following. Although the exciton binding energy varies drastically between L, = 0 and - 4 nm [14,17], the determined mass behaves very smoothly starting from the mass at the conduction band bottom. This is considered to justify our scheme for obtaining information on the effective mass: PL, exciton spectrum and calculation based on the effective mass equation. Here, we add a remark regarding the nonparabolic mass within the well. From Raman scattering in GaAs [19], the nonparabolic mass is estimated to be 0.09 for a vanishing well width (consequently, at a high enough energy level) in comparison to 0.0665 for an infinitely wide well (consequently, on the well bottom). For a larger mass within the well, the calculated barrier mass gives a smaller value (in fig. 3, rnw,,,, = 0.34 and 0.62 are compared). This causes a larger variation

in electron mass within the gap over the same range of energy presently concerned. Finally, the heavy hole mass of 0.62 was compared with 0.34. For the former mass, 0.62, too small a mass results for the barrier material (L w + 0). This means that mWhh = 0.34 for GaAs gives a better result. In the beginning of this paper, we referred to tunneling phenomena. The present topic is essentially equivalent to that of a tunneling mass; if there exists precise tunneling spectroscopy data to compare with the calculations, more direct conclusions will be obtained on tunneling mass problems, as well. A more interesting study than this seems to be of a quantum well sample having low enough well energy states to stand around the midgap of the barrier material. The fact that an electron effective mass is varied continuously to a hole mass should be experimentally tested. The authors are grateful to Professors T. Ando and Y. Shiraki as well as their colleagues: T. Uda, R. Sugano, T. Onogi, M. Hirao, for their valuable discussions.

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