2 Introduction their interaction intensity with electrons (Fq) are calculated by using the dielectric continuum model [7,8].
The alternate change (increase and decrease) of elec- The electron–PO-phonon scattering rates were calcutron mobility as a function of quantum well (QW) width lated at T = 100 K for the symmetric rectangular and doping level in modulation-doped AlGaAs/GaAs and Al0.25Ga0.75As/GaAs/Al0.25Ga0.75As QW with sheet electron AlGaAs/InGaAs QW’s was recently observed [1–3]. The concentration in the range of 5 · 1015 < ns < 6 · 1016 m-2 predominant mechanism which determines the electron and QW widths 15 < L < 35 nm. The parameters used in mobility in modulation-doped GaAs QW at T > 77 K is the calculations were: m/m0 = 0.67, 0 = 36.2meV, electron scattering by confined and interface polar optical = 10.9, s = 12.9 for GaAs and m/m0 = 0.09, (PO) phonons. 0 = 46.8meV, = 10.1, s = 13.0 for AlGaAs.
The Al0.25Ga0.75As/GaAs barrier is assumed U0 = 0.3eV.
In this paper the separate contribution to electron mobility of definite type intra- and intersubband electron transitions The nonelastic electron–PO-phonon scattering with a with absorption or emission of PO phonons is considered. large change of scattered electron energy requires to take The scattering mechanisms which are responsible for alter- into account the different occupation of electrons in the initial and final states. Taking into account the electron nate mobility changes with a QW width and doping level are state occupation, the mean frequency of electron transitions determined.
(scattering rates) from the initial state in subband i to the final one in subband f can be written in the form 1. Confined electron–PO-phonon scattering rate and mobility 1 - f (E ± ) Wia,e = wa,e (E) dE f (E) dE, f i f in Al0.25Ga0.75As/GaAs/Al0.25Ga0.75As 1 - f (E) Esi Esi quantum wells (2) where f (E) is the electron Fermi–Dirac distribution funcThe transition frequency of electrons, confined in a tion, and the plus sign is for phonon absorption and the quantum well (QW), from an initial state in subband i with minus one is for phonon emission.
momentum ki and subband electron energy Esi, to any final The electron mobility in pure GaAs at T > 77 K is state, kf, Esf, in subband f by emission (absorption) of determined by electron–PO-phonon scattering. In this paper -mode PO-phonon can be written as [4–7] the estimation of the separate contrubution to electron mobility of various type electron–PO-phonon scattering is 4me2 2 1 wa,e (ki, Esi) = |G|2Fq Nq + ± done assuming the inverse mean frequency of electron trani f 2 kf sitions by PO-phonon absorption (emission) as a momentum relaxation time k2 - k2 + ±, (1) f i i f = 1/Wi f. (3) where G is the overlap integral of the normalized electron This relaxation time approximation gives only a crude and phonon wave functions, Nq is the occupation number of estimation of the mobility limited by PO-phonon scattering, 2m -mode phonons, and ± = (Esf -Esi± ). The upper but it is expected that this approximation is sufficient both sign (plus) is for phonon emission and the lower (minus) for estimation of relative difference between the mobilities in one is for phonon absorption. The phonon potentials and QW’s with different widths and sheet electron concentrations 6 1362 K. Poela and for estimation the relative contribution to electron mo- conductivity (µns) with increasing ns takes place. This is bility of various electron scattering mechanisms by various shown in Fig. 1, b. When ns > 3 · 1016 m-2, the third phonon modes. Note that the values of mobility calculated subband electrons with the increased mobility, due to the e e within the used relaxation time approximation in the GaAs decrease of W31, W32, gives the enhancement of the total QW are near to the values observed experimentally. QW conductivity in spite of decreasing the first and second In this approximation, the i-subband electron mobility is subband mobilities.
The increase of µ3 is limited at ns > 4.5 · 1016 m-2 due e to the increase of electron scattering by phonon absorption µi = (Wie + Wia ), (4) f f m in the third subband, and the second decrease of the f conductivity µns is observed (Fig. 1, b).
and the total electron mobility in the QW is It is worth noting that the electron mobility in the second subband exceeds two times the first subband mobility at µ = µini, (5) ns = 1.5 · 1016 m-2. There are two types of electrons ns i in the QW: low mobility electrons in the first subband and two times faster electrons in the second subband.
where ns and At ns > 3 · 1016 m-2, the faster electrons are in the third m subband (see Fig. 1, b).
ni = f (E) dE (6) Therefore, for the mobility decrease with increasing ns in Esi the Al0.25Ga0.75As/GaAs/Al0.25Ga0.75As QW, the increase of the intra- and intersubband scattering rates by PO-phonon are the sheet electron concentrations in the QW and in absorption in degenerate electron gas is responsible. For subband i, respectively.
the alternate increase of conductivity, the decrease of the intersubband scattering by PO-phonon emission in the upper 2. The dependence of electron subband e e e subbands, W31, W32, W21, is responsible (because of electron mobility on sheet electron gas degeneration in the lower subbands).
concentration The calculated intra- and intersubband electron–PO-phonon scattering rates as functions of sheet electron concentration ns in the Al0.25Ga0.75As/GaAs/Al0.25Ga0.75As QW of width L = 20 nm are presented in Fig. 1, a.
The significant enhancement of the intra- and intersubband scattering rates by PO-phonon absorption with increasing ns a a a is observed in the lower subbands (W11, W12, W22). The enhancement takes place in the subbands with degenerate electron gas. This is largest in the lowest (first) electron a subband (W11), where the electron gas is most degenerated (see Fig. 1, a). On the contrary, the scattering rates by phonon emission from the upper subband to the lower one e e e (W31, W32, W21) decrease with electron gas degeneration in the lower subband. The scattering rate by phonon emission in the lower subband can be neglected.
The increase of scattering rates by PO-phonon absorption a a (W11 and W12) is responsible for the strong decrease of electron mobility in the lowest (first) electron subband, µ1, with increasing ns, as it is shown in Fig. 1, b. The second subband mobility µ2 increases due to decreasing the second subband electron scattering by phonon emission when ns changes in the range of (5-15) · 1015 m-2. At ns > 15 · 1015 m-2, µ2 decreases very fast because of the strong increase of a a a electron scattering rates W22, W21, W23 by phonon absorption.
Figure 1. The electron–PO-phonon scattering rates Wif with The contribution of the second subband electrons to the electron transitions from the initial state in subband i to the final total QW mobility increases with increasing second subband one in subband f by phonon emission and absorption (labeled electron population.
by i f e and i f a, respectively) (a) and the electron mobilities in This decrease of the mobilities in the first and secsubbands i = 1, 2, 3 (labeled by µ1, µ2, µ3), the total mobility µ ond subbands exceeds the increase of ns in the range and conductivity (µ · ns) (b) as functions of sheet electron of (2-3) · 1016 m-2. As a result, the decrease of QW concentration ns in the QW of width L = 20 nm.
Физика и техника полупроводников, 2001, том 35, вып. Electron nonelastic scattering by confined and interface polar optical phonons in a modulation-doped... 3. The dependence of electron mobility on a QW width Fig. 2, a shows the QW width dependencies of electron intra- and intersubband scattering rates in the Al0.25Ga0.75As/GaAs/Al0.25Ga0.75As QW at sheet electron concentration ns = 5 · 1015 m-2. At all QW widths in the range of 15 < L < 35 nm, the main electron scattering a a mechanism in the lowest subband remains W11 and W12.
a a At L > 18 nm, the scattering rates W11 and W12 decrease slowly with increasing QW width. In the second and the third subbands, the intersubband transitions by PO-phonon e e e emission are permitted and the scattering rates W31, W32, Ware very high and exceed 1012 s-1.
Figure 3. The subband and total electron mobilities as functions Such behavior of electron–PO-phonon scattering explains of QW width at ns = 3 · 1016 m-2. Notations are as in Fig. 1.
the alternate dependence of the mobility on GaAs QW width shown in Fig. 2, b.
We see that the increase of the first subband electron At QW widths L = 15-25 nm, the intersubband scatmobility µ1 with increasing QW width L due to the decrease e a tering rate with PO-phonon emission, W21, decreases due of W11 is limited at two QW widths: at L = 15-to filing the electron states in the first subband. Thereand L = 25-30 nm by the increase of the intersubband a a fore, the second subband electron mobility µ2 increases.
scattering rates W12, and W13, respectively. Between these At L > 27 nm, µ2 becomes larger than µ1 (see Fig. 2, b).
two limited scattering (in the interval of L = 18-25 nm) the mobility monotonically increases with a QW width L. When the sheet electron concentration in the QW increases until ns = 3·1016 m-2, the electron-phonon intersubband scattering rates changes significantly due to occupation of electron states in the lower subbands (see Fig. 1, b).
This gives a ten times increase of the scattering rate by a PO-phonon absorption in the lowest subband, W11, and decrease of the scattering rate by PO-phonon emission in the upper subbands (see Fig. 1, a).
As a result, the electron mobility decreases strongly in the lower QW subband, and increases in the upper (third) subband. At ns = 3 · 1016 m-2, the population of electrons in the upper subbands is large, and their contribution to the total mobility is significant.
Fig. 3 shows the QW width dependence of electron mobilities at ns = 3 · 1016 m-2.
We see that the second subband electrons give the main contribution to the total mobility. The decrease of the mobility is observed only at L = 22 nm. It is explained by the increase of the scattering of second subband a electrons to the third subband by phonon absorption (W23).
At L > 22 nm, the electron scattering rates in all subbands both with phonon emission and absorption decrease slowly with increasing a QW width. This circumstance gives the increase of the total mobility with increasing a QW width in the range of 22 < L < 35 nm. Note that at L > 25 nm the third subband mobility is seven times larger than the mobility in the first subband.
Therefore, the total electron mobility increases with a QW width. This increase limits the electron intersubband scatterFigure 2. The electron–PO-phonon scattering rates Wi f (a) and a a ing rates by phonon absorption: W12 and W13 in the QW with the subband and total electron mobilities (b) as functions of QW a width at ns = 5 · 1015 m-2. Notations are as in Fig. 1. ns = 5·1015 m-2 and W23 in the QW with ns = 6·1016 m-2.
6 Физика и техника полупроводников, 2001, том 35, вып. 1364 K. Poela Conclusions The electron–PO-phonon scattering mechanisms which are responsible for the dependencies of electron mobility on a QW width and doping level are determined. In is shown that:
1. The degeneration of subband electron gas decreases the electron scattering by PO-phonon emission and increases the scattering by phonon absorption. In the QW of width L = 20 nm, the scattering by PO-phonon absorption exceeds the scattering by PO-phonon emission in the first subband at ns > 5 · 1015 m-2, in the second subband at ns 1.5 · 1015 m-2 and in the third subband at ns 4.5 · 1015 m-2. At these ns, the mobility in the upper subbands exceeds the mobility in the lower ones.
The increase of the scattering by PO-phonon absorption is responsible for the decrease of QW conductivity with increasing the sheet electron concentration in the range of ns =(2-3) · 1016 m-2 and ns > 5.5 · 1016 m-2.
2. The enhancement of the mobility with increasing a QW width because of reduction of the intrasubband a a scattering rates by PO-phonon absorption (W11, W22) at ns = 5 · 1015 m-2 is limited by the increase of the a intersubband scattering of the first subband electrons (Wa and W13), and at ns = 3 · 1016 m-2, by the increase of a the scattering of the second subband electrons (W23). The competition of these scattering mechanisms explains the alternate changes of the mobility with a QW width.
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