- Email: [email protected]
Enhancement of multisubband electron mobility in asymmetrically doped coupled double quantum well structure
Author’s Accepted Manuscript Enhancement of multisubband electron mobility in asymmetrically doped coupled double quantum well structure S. Das, R.K. Nayak, T. Sahu, A.K. Panda www.elsevier.com/locate/physb
PII: DOI: Reference:
S0921-4526(15)30130-7 http://dx.doi.org/10.1016/j.physb.2015.07.014 PHYSB309062
To appear in: Physica B: Physics of Condensed Matter Received date: 18 November 2014 Revised date: 2 July 2015 Accepted date: 15 July 2015 Cite this article as: S. Das, R.K. Nayak, T. Sahu and A.K. Panda, Enhancement of multisubband electron mobility in asymmetrically doped coupled double quantum well structure, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2015.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Enhancement of multisubband electron mobility in asymmetrically doped coupled double quantum well structure S. Das, R.K. Nayak, T. Sahu* and A.K. Panda Department of Electronics and Communication Engineering National Institute of Science and Technology Palur Hills, Berhampur-761 008, Odisha, India * e-mail: [email protected] Tel: 91 680 249 2421 Fax: 91 680 249 2627
Abstract We study the effect of coupling of subband wave functions on the multisubband electron mobility in a barrier delta doped GaAs/AlxGa1-xAs asymmetric double quantum well structure. We use selfconsistent solution of the coupled Schrödinger equation and Poisson’s equation to calculate the subband wave functions and energy levels. The low temperature mobility is considered by using scatterings due to ionized impurities, interface roughness and alloy disorder. We show that variation of the width of the central barrier considerably affect the interplay of different scattering mechanisms on electron mobility through intersubband effects. Under single subband occupancy, the mobility increases with decrease in the barrier width as functions of doping concentration as well as function of well width. However, in case of double subband occupancy, effect of intersubband interaction yields opposite trend, i.e., increase in mobility with increase in barrier width. It is gratifying to show that in case of asymmetric variation of well widths the mobility shows nonmonotonic behavior which varies with change in the width of the central barrier under double subband occupancy.
Key words: Asymmetric quantum wells, Coupled Quantum Well Structures, Intersubband scatterings, Multisubband electron mobility. PACS classification No. 73.63.Hs, 73.50.Bk.
1. Introduction In recent years, a great deal of attempts has been made to study asymmetric quantum well structures because of their potential applications in electronic and optical devices [1-10]. Efforts have been made to utilize asymmetrically doped coupled quantum well GaAs/AlGaAs structures
1
to develop and characterize tunable mid-infrared photodetectors [1]. Suitably engineered GaAs/AlGaAs double quantum well structures have been shown to exhibit coherent terahertz emission due to intersubband excitation between two lowest conduction subbands [2]. InGaN based light emitting devices with asymmetric coupled quantum wells yield improved light emission intensity compared to that of a symmetric coupled quantum well device [3]. Attempts have been made to study the magnetic-field-induced charged excitonic transitions in modulationdoped AlGaAs/GaAs asymmetric coupled double quantum wells [4]. Recently it has been shown that in an asymmetric double InGaAs quantum well structure the coherent excitonic interwell coupling originates from many-body effects [5]. Further it has been shown that in a coupled asymmetric quantum well system the eigenstate probability distribution localizes exclusively either in the wide or the narrow parts of the well pair [6]. Attempts have also been made to investigate electron spin dynamics by the time-resolved Kerr rotation technique in a pair of special GaAs/AlGaAs asymmetric quantum well samples to explore the possibility of developing spin based electronic devices [7]. A direct determination of nonradiative lifetimes in Si/SiGe asymmetric quantum well structures shows that the barrier and well widths play an important role in the determination of the carrier life time. Further by comparing with experimental and theoretical data obtained for mid-infrared GaAs/AlxGa1-xAs quantum well systems, it has been shown that interface roughness scattering plays a dominant role for a wide range of semiconductors at low temperature [8]. Attempts have also been made to study electron transport in asymmetric quantum well structures by considering lowest subband occupancy [9, 10]. However, in a quantum well system, where more than one subband is occupied, the intersubband interaction plays a vital role on the subband mobility [11-15]. Intersubband effects create additional intersubband scattering rates and also modify the intrasubband scattering rate through the dielectric screening of the scattering potentials [11-15]. 2
In the present paper we study multisubband electron mobility of a coupled asymmetric double quantum well structure. Variation of the central barrier width (b) changes the coupling of the subband wave functions of individual wells thereby changing the subband energy levels and wave functions which in turn affect the occupation of subbands of the coupled structure. As a result the strength of the scattering potentials also amended leading to change in the mobility. We consider GaAs/AlGaAs double quantum well in which the asymmetry of the structure is employed by introducing delta doped layer in one of the side barriers only and also taking different well widths. We obtain the subband electron energy levels and wave functions for the coupled structure by adopting selfconsistent solution of the Schrodinger equation and Poisson’s equation. We calculate the low temperature electron mobility µ by considering ionized impurity (imp-), interface roughness (ir-) and alloy disorder (al-) scatterings. We adopt static dielectric response function formalism to calculate the screened scattering potentials by using the random phase approximation (RPA) [11-15]. We show that µ increases with increase in the doping concentration (Nd). The mobility is mostly governed by imp- and ir-scatterings. The effect of ir-scattering, which is more dominating at low values of Nd, decreases with increase in Nd so that the effect of imp-scattering on mobility enhances. The doping dependence of µ for different b also shows interesting results. As long as single subband is occupied, the mobility decreases with increase in b. A deviation in the trend occurs for small b at small Nd due to strong influence of ir-scattering. However, once double subband is occupied, the trend is reversed. The drop in mobility at the onset of occupation of the second subband and also the doping concentration at which the drop occurs, decrease with increase in b. The effect of b on the well width dependence of mobility also shows interesting results. The increasing trend in µ with increase in well width slowly reverses as the barrier width increases due to gradual dominance of the intrasubband scattering rate matrix element of the 3
higher occupied subband. We show that the mobility can be enhanced considerably at small well widths having double subband occupancy by increasing the barrier width. We further show that by increasing well width, the occupation of subbands goes on changing viz, from single subband to double subband and then again single subband occupancy. Accordingly the mobility shows a drop and then a rise near the change in occupancy of subbands. Our analysis therefore will help in choosing suitable structure parameters so that the mobility becomes a maximum. Our results of mobility of a single side barrier delta doped asymmetric double quantum well structure can be utilized for low temperature device applications.
2. Theory We consider a GaAs/AlxGa1-xAs double quantum well structure in which the left barrier is delta doped with a layer of Si of width d and doping concentrations Nd at a spacer distance s from the nearest interface (Fig. 1). The wells, of widths w1 and w2, are separated by a thin barrier of width b. Diffusion of electrons cause band bending due to Coulomb interaction with the ionized donors in the barrier. The concentration distribution of impurities and electrons nD (z) and n(z) along the growth axis, viz., z-axis can be written as:
Nd (d s w1 b / 2) z ( s w1 b / 2) nD ( z ) otherwise 0 n( z ) nn n ( z )
2
z
(1) (2)
n
n (z) are the subband wave functions and summation n is over the subband levels. nn is the number of electrons per unit area in the nth subband which can be written as [11]:
4
nn mk B T /
2
ln 1 exp E
F
En / kBT
(3)
En are the subband energy levels and EF is the Fermi energy. At temperature T = 0 K, the surface electron density Ns and the Fermi energy can be related as [11]: N s m /
E N
2
n 1
F
En EF En
(4)
N is the number of filled levels, is the Heaviside step function. We obtain the subband energy levels En and wave functionsn (z) by adopting selfconsistent solution of the one dimensional Schrödinger equation and the Poisson’s equation. In Fig. 1, we present a schematic diagram of the potential profile, subband energy levels and wave functions of the single side doped asymmetric double quantum well structure. Since at T = 0 K, the electrons on the Fermi surface take part in the conduction process, the subband transport life time can be expressed as n = n (EFn). Here EFn = EF - En is the Fermi energy of nth subband. The relaxation time n for a multisubband occupied system can be derived from the Boltzmann transport equation containing the intrasubband and intersubband scattering rate matrix elements in a mixed way [11-14]. For the lowest occupied subband (n = 0), one can write 0 in terms of intrasubband scattering rate matrix element b00 as [14]: 1
0
b00
(5)
Similarly, for a double subband is occupied structure (n=0, 1), both 0 and 1 contain intrasubband and intersubband scattering rate matrix elements (c01 and d01) as shown below [14]:
1 b00 c01 b11 c10 d01d10 0 b c E / E 12 d 11 10 F1 F0 01
5
1 b00 c01 b11 c10 d01d10 1 b c E / E 12 d 00 01 F0 F1 10
(6)
The scattering rate matrix elements bnn, cnm and dnm can be expressed in terms of scattering potential matrix elements Vnmeff(q) as [14]: bnn m /
3
d 1 cos V q eff nn
2
nn
0
cnm m /
3
d
eff Vnm qnm
2
0
d nm m /
3
d cos V q eff nm
2
(7)
nm
0
2 qnl kFn kFl2 2kFn kFl cos
1
2
and kFn=(2mEFn/ħ2)1/2. The screened scattering potential Vnmeff(q)
can be written as [11]: -1 Vnmeff (q) nm,n'm' (q)Vn ' m ' (q) ,
(8)
n'm'
1 where nm , n ' m ' is the element of the inverse matrix of the dielectric response function. n, m, n’, m’
are the subband indices. We use the random phase approximation (RPA) for the screening of the scattering potentials which can be written as [11]:
nm,n ' m ' (q) nn ' mm ' (qs / q) Fnm,n ' m ' (q) n'm' (q)
(9)
where Fnm,n 'm' (q) is the Coulomb form factor, qs 2me2 / 0
2
and is the static electron
density-density correlation functions without electron-electron interaction [11]. Vnm(q) is the element of Fourier transform of the external potential. We consider ionized impurity scattering, interface roughness scattering and alloy disorder scattering to calculate the subband mobility. The screened ionized impurity scattering potential for the present structure can be written as: 6
imp nm
V
4 2 e4 N0 (s+w1+b / 2) (q) dz 02 q 2 (d+s+w1+b / 2) i 2
2 -1 nm,n'm'
(q) Pn ' m ' (q, zi )
n'm'
(10)
where
Pn ' m ' q, zi
dz
n'
( z ) m ' ( z ) exp(-q z - zi )
(11)
The screened interface roughness scattering potential is:
V (q) V exp(q / 4) ir nm
2
2 b
2
2
2
2
I
2 n'
*( z ) m ' ( z ) z z I
n 'm'
-1 nm,n'm'
( q)
(12)
Similarly, the alloy disorder scattering potential can be written as:
V (q) a ( V ) x(1 x) / 4 dz al nm
2
3
2
j
2 n'
( z ) m ' ( z )
-1 nm,n'm'
(q)
(13)
n'm'
The ionized impurity scattering occurs from the delta-doped layer lying in the left side barrier. The alloy disorder scattering is considered from the alloy (AlGaAs) regions. The summation j stands for both the side barriers as well as the central barrier. ‘a’, V and x are the lattice constant, alloy scattering potential and alloy fraction respectively of the barrier layers. The summation I is over the interfaces lying towards the left side of the wells, i.e., (I1, I3), which are considered to be rough. The roughness is assumed to exist in the interfaces lying towards the substrate side only [12, 14, 16]. The subband electron mobility np is related to the subband transport life time np by:
np ( EF )
e p n ( EF ) m
(14)
7
where np is the subband transport life time described in Eqs. 5-6 for Pth (imp-, ir-, al-) scattering event. The mobility µP due to a scattering mechanism P can be written as
p
n n n
p n
n
(15)
n
n
The total mobility µ is calculated using Matthiessen’s relation.
3. Results and discussion We analyze the electron mobility µ of a single side barrier delta doped asymmetric GaAs/AlxGa1-xAs coupled double quantum well structure. We consider the potential barrier height Vb = 228 meV (x = 0.3) [17]. The interface roughness parameters = 2.83 Å and = 100 Å and the alloy disorder scattering potential V = 1.56 eV [12, 18]. The delta-doping layer width d = 20 Å and spacer width s = 80 Å. In Fig. 2, we present total mobility µ along with the mobility due to individual scattering mechanisms, i.e., ionized impurity scattering µimp, interface roughness scattering µir and alloy scattering µal as a function of doping concentration Nd for central barrier width b = 10 Å, taking well widths w1 = w2 = 100 Å. Up to Nd = 3.1×1018 cm-3, only lowest subband is occupied. Thereafter double subband is occupied. The mobility µ, which is dominated by imp- and irscatterings, increases with increase in doping concentration Nd. The al-scattering has least effect on µ as it occurs from the AlGaAs barrier where the amplitudes of the subband electron wave functions are less in magnitude. We note that with increase in Nd, the triangular like potential well near interface I1 (Fig. 1) becomes deeper. As a result the subband electron wave function distribution shifts towards the impurities lying in the left barrier thereby increasing the strength of the imp-scattering potential Vimp(q) (Eq. 10). The strength of the ir-scattering potential Vir(q)
8
(Eq. 12) is also increased due to increase in the amplitudes of the subband wave functions at the interface I1 even though the amplitudes of the wave function at I3 decrease slightly. However, it is interesting to note that the change in the potential profile of the quantum well due to increase in the doping concentration also affects the subband energy levels leading to increase in subband Fermi wave vector q that reduces the strengths of imp- and ir- scattering potentials (Eq.10 and 12). The later effect being predominating, both µimp and µir increase giving rise to enhanced µ with increase in Nd. Once the second subband is occupied, the imp-scattering is more influenced by the intersubband interactions so that the drop in µir becomes less than that of µimp, resulting µir slightly greater than µimp. In Fig. 3, we present mobility as a function of doping concentration Nd for different central barrier width (b = 10, 14, 20, 26, 30 and 40 Å) taking well widths w1 = w2 = 100 Å. As b increases, the overlapping of the subband wave functions of the individual wells through the central barrier decreases leading to less splitting in the subband energy levels. Accordingly the subband energy levels and wave functions of the coupled structure vary. We show that as b increases, the transition from single subband to double subband occupancy occurs at less value of Nd due to change in the coupling of the subband wave functions through the central barrier. The drop in mobility at that point also decreases. In the inset of Fig. 3, we present µir/µimp as a function of Nd from which the relative impact of imp- and ir- scatterings on µ can be derived. Under single subband occupancy, the mobility decreases with increase in b. The deviation in this trend for b up to 20 Å can be attributed to the stronger effect of ir-scattering on mobility for Nd < 2.4×1018 cm-3. Once double subband is occupied, the trend is reversed due to enhanced effect of imp-scattering mediated by intersubband effect. In Fig. 4, we analyze the functional dependence of mobility on well width w (w1 = w2 = w) for different central barrier width b taking doping concentration Nd = 2×1018 cm-3. As well width 9
increases, the occupation of subbands changes from single subband occupancy to double and then again to single subband occupancy. The drop in mobility at the transition from single to double subband occupancy is because of additional intersubband effect as described earlier. Similarly, the rise in mobility near the transition from double subband to single subband occupancy is due to suppression of intersubband effects. In the inset of Fig. 4, we present µir/µimp as a function of w for different b. The mobility is mostly governed by imp-scattering except at small well width (w < 80 Å) where the ir-scattering dominates the mobility. The enhancement in mobility with increase in well width is because of increase in the effective distance between the ionized impurities and the subband electrons through the ionized impurity scattering potential Vimp(q). We note that the interplay of different scattering mechanisms on mobility depends upon the occupation of subbands.through the intersubband effect. When the central barrier width b = 20 Å, only single subband is occupied. As b increases (b = 26, 30, 36 and 40 Å), transition from single subband occupancy to double subband occupancy starts at lower values of w (w = 84, 72, 56 and 50 Å, respectively). On the other hand, transition from double subband to single subband occupancy occurs almost at a single well width (w = 136 Å) for different b, indicating independence on the barrier width. At large well widths w >136 Å, when only a single subband is occupied, the mobility for different b remain almost unchanged since the dominant imp-scattering hardly depends on b. Whereas at small well widths, we show that the difference in mobility for different b is mostly because of the change in µir. A change in b affect the coupling of subband wave functions leading to a variation of the electron subband energy levels. Accordingly the Fermi wave vector (q) decreases and the amplitude of the subband wave function at I1 increases causing enhancement of Vnmir (q) (Eq. 12). In Fig. 5, we present mobility µ as a function of well width w1 taking w1 + w2 = 400 Å such that by varying w1, both the well widths w1 and w2 change. We take doping concentration Nd = 10
1×1018 cm-3 and vary central barrier width b (b = 10, 16, 20, 30 and 40 Å). We note that the mobility is mostly governed by imp-scattering. To start with, double subband is occupied. The nonmonotonic variation of mobility as a function of w1 for different b under double subband occupancy is governed by the intersubband effects. For b ≤ 20 Å, the mobility increases, while for b > 20 Å there is substantial reduction in µ with increase in w1 so that near the transition from double subband to single subband occupancy, the difference in µ for different b becomes marginal. Once the transition from double to single subband occupancy occurs, the mobility monotonically increases with increase in w1 showing negligible dependence on b. The change in the barrier width marginally affects the effective distance between the ionized impurities and the subband electron wave function resulting negligible dependence of mobility on barrier width. Further, we show that the nonmonotonic variation of mobility as a function of w1 for different b under double subband occupancy is attributed to the contribution of both µ0imp and µ1imp. In Fig. 6, we present the results of subband mobilities µ0imp and µ1imp as a function of well width w1 for b = 10, 20 and 40 Å taking Nd = 1×1018 cm-3 and w1 + w2 = 400 Å. There is not much variation of µ0imp with increase in the well width w1. However, it is interesting to note that large variation of µ1imp for different b is exhibited due to the difference in the intrasubband scattering rate b11imp. Since the subband electron densities n1 and n2 are of same order, the change in µ1imp causes the resulting trend in µ. shown in Fig.5.
4. Summary and Conclusion In the present work we analyze the effect of barrier width on the functional dependence of low temperature multisubband electron mobility on doping concentration and well widths in an asymmetric coupled double quantum well structure. We consider a GaAs/AlxGa1-xAs double quantum well in which one of the side barriers is delta-doped with Si. We obtain subband energy 11
levels and wave functions by using selfconsistent solution of the Schrödinger equation and Poisson’s equation. We consider scatterings due to ionized impurities (imp-), interface roughness (ir-) and alloy disorder (al-) to study low temperature electron mobility. The screening of scattering potentials is obtained using random phase approximation. We show that coupling of subband electron wave functions through the central barrier affect the interplay of different scattering mechanisms on mobility. The importance of our work is to relate the enhancement of electron mobility with the variation of barrier width mediated by intersubband effects. The doping concentration as well as well width dependence of mobility show that by reducing the central barrier width, the mobility can be enhanced when only single subband is occupied. Whereas, under double subband occupancy, the trend is reversed, i.e., mobility increases with increase in barrier width mediated by intersubband effects. Further, we show that variation of well widths asymmetrically gives rise to nonmonotonic variation of mobility for different central barrier width. Our work can be utilized for low temperature device application.
12
Reference [1]
J.K. Choi, N. Vagidov, A. Sergeev, S. Kalchmair, G. Strasser, F. Vasko, V. Mitin, Asymmetrically doped GaAs/AlGaAs double-quantum-well structure for voltage-tunable infrared detection, Japan. J. Appl. Phys. 51 (2012) 074004.
[2]
P.G. Huggard, C.J. Shaw, S.R. Andrews, J.A. Cluff, R. Grey, Mechanism of THz emission from asymmetric double quantum wells, Phys. Rev. Lett. 84 (2000) 1023-1026.
[3]
J.Y. Zhang, L. Cai, B.P. Zhang, X.L. Hu, F. Jiang, J.Z. Yu, Q.M. Wang, Efficient hole transport in asymmetric coupled InGaN multiple quantum wells, Appl. Phys. Lett. 95 (2009) 161110-161113.
[4]
K.S. Lee, S.K. Noh, S.K. Chang, Photoluminescence study of magnetic-field-induced excitonic transitions in AlxGa1-xAs/GaAs asymmetric double quantum wells, Phys. Rev. B 76 (2007) 073305.
[5]
G. Nardin, G. Moody, R. Singh, T. M. Autry, H. Li, F.M. Genoud, S.T. Cundiff, Coherent Excitonic Coupling in an Asymmetric Double InGaAs Quantum Well Arises from Many-Body Effects, Phys. Rev. Lett. 112 (2014) 046402.
[6]
A. Mialitsin, S. Schmult, I.A. Solov’yov, B. Fluegel, A. Mascarenhas, Eigenstate localization in an asymmetric coupled quantum well pair, Superlatt. Microstruct. 51 (2012) 834-841.
[7]
W. Gang, Y.E. Hui-Qi, S.H.I. Zhen-Wu, W. Wen-Xin, M. Xavier, B. Andrea, A. Thierry, L. Bao-Li, Spin Dynamics in (111) GaAs/AlGaAs Undoped Asymmetric Quantum Wells, Chin. Phys. Lett. 29 (2012) 097204.
[8]
M. Califano, N.Q. Vinh, P.J. Phillips, Z. Ikonic, R.W. Kelsall, P. Harrison, C.R. Pidgeon, B.N. Murdin, D.J. Paul, P. Townsend, J. Zhang, I.M. Ross, Interwell relaxation times in p-Si/SiGe asymmetric quantum well structures: Role of interface roughness, Phys. Rev. B 75 (2007) 045338.
[9]
F.M.S. Lima, F. Qu, O.A.C. Nunes, A.L.A. Fonseca, Electron mobility in one (two)-side modulation-doped GaAs/AlxGa1-xAs asymmetric quantum wells, Phys. Status Solidi B 225 (2001) 43-61.
13
[10]
M. Jung, T.W. Kim, D.U. Lee, D. C. Choo, K. H. Yoo, D. L. Kim, M. D. Kim, H. Lim, The dependence of the carrier density and the mobility on the spacer-layer thickness in modulation-doped AlxGa1−xAs/InyGa1−yAs/GaAs asymmetric single quantum wells, Appl. Surf. Sci. 177 (2001) 1-7.
[11]
T. Ando, A.B. Fowler, F. Stern, Electronic properties of two dimensional systems, Rev. Mod. Phys. 54 (1982) 437-672.
[12]
K. Inoue, T. Matsuno, Electron mobilities in modulation-doped AlxGa1-xAs/GaAs and pseudomorphic AlxGa1-xAs/InyGa1-yAs quantum well structures, Phys. Rev. B 47 (1993) 3771-3778.
[13]
G.Q. Hai, N. Studart, F.M. Peeters, P.M. Koenraad, J.H. Wolter, Intersubband-coupling and screening effects on the electron transport in a quasi-two-dimensional δ-doped semiconductor system, J. Appl. Phys. 80 (1996) 5809-5814.
[14]
T. Sahu, K.A. Shore, Multi-interface roughness effects on electron mobility in a Ga0.5In0.5P/GaAs multisubband coupled quantum well structure, Semicond. Sci. Technol. 24 (2009) 095021.
[15]
S. Das, R.K. Nayak, A.K. Panda, T. Sahu, Multisubband electron mobility in asymmetric GaAs/AlGaAs quantum well structures, Superlatt. Microstruct. 66 (2014) 39-47.
[16]
S.K. Lyo, Real-space and energy representations for the interface-roughness scattering in quantum-well structures, J. Phys.:Condens. Matter. 13 (2001) 1259-1264.
[17]
S. Adachi, GaAs, AlAs and AlxGa1-xAs material parameters for use in research and device applications, J. Appl. Phys. 58 (1985) R1-R29.
[18]
A.K. Saxena, A.D. Adams, Determination of alloy scattering potential in Ga1−xAlxAs alloys, J. Appl. Phys. 58 (1985) 2640-2645.
14
Collected Figure Captions Fig. 1. Schematic diagram of the one side barrier doped asymmetric double quantum well structure along with the potential energy profile, subband energy levels and wave functions. Fig. 2. Subband electron mobility for the ionized impurity scattering µimp, alloy scattering µal and interface roughness scattering µir and total mobility µ as a function of the doping concentration Nd (× 1018 cm-3) by taking b = 10 Å, s = 80 Å, w1 = w2 = 100 Å. Fig. 3. Mobility µ as a function of doping concentration Nd (cm-3) for different central barrier width b = 10, 14, 20, 26, 30 and 40 Å taking spacer widths s = 80 Å and well widths w1 = w2 = 100 Å. Inset shows the ratio µir/µimp for the above structure parameters.
Fig. 4. Mobility µ as a function of well width w (Å) for central barrier width b = 20, 26, 30, 36, and 40 Å taking doping concentration Nd = 2×1018 cm-3 and spacer widths s = 80 Å. Inset shows the ratio µir/µimp for the above structure parameters.
Fig. 5. Mobility µ (105 cm2/Vs) as a function of well width w1 (Å) for different central barrier width b = 10, 16, 20, 30 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å and w1 + w2 = 400 Å which depicts considerable influence of coupling of subband wave functions on mobility when more than one subband is occupied. Fig. 6. Subband electron mobility for the ionized impurity scattering µ0imp and µ1imp as a function of well width w1 for central barrier width b = 10, 20 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å taking w1 + w2 = 400 Å. In caption µnimp (N) implies µnimp for b = N Å.
15
Fig. 1. Schematic diagram of the one side barrier doped asymmetric double quantum well structure along with the potential energy profile, subband energy levels and wave functions.
16
Fig. 2. Subband electron mobility for the ionized impurity scattering µimp, alloy scattering µal and interface roughness scattering µir and total mobility µ as a function of the doping concentration Nd (× 1018 cm-3) by taking b = 10 Å, s = 80 Å, w1 = w2 = 100 Å.
17
Fig. 3. Mobility µ as a function of doping concentration Nd (cm-3) for different central barrier width b = 10, 14, 20, 26, 30 and 40 Å taking spacer widths s = 80 Å and well widths w1 = w2 = 100 Å. Inset shows the ratio µir/µimp for the above structure parameters.
18
Fig. 4. Mobility µ as a function of well width w (Å) for central barrier width b = 20, 26, 30, 36, and 40 Å taking doping concentration Nd = 2×1018 cm-3 and spacer widths s = 80 Å. Inset shows the ratio µir/µimp for the above structure parameters.
19
Fig. 5. Mobility µ (105 cm2/Vs) as a function of well width w1 (Å) for different central barrier width b = 10, 16, 20, 30 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å and w1 + w2 = 400 Å which depicts considerable influence of coupling of subband wave functions on mobility when more than one subband is occupied.
20
Fig. 6. Subband electron mobility for the ionized impurity scattering µ0imp and µ1imp as a function of well width w1 for central barrier width b = 10, 20 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å taking w1 + w2 = 400 Å. In caption µnimp (N) implies µnimp for b = N Å.
21
PII: DOI: Reference:
S0921-4526(15)30130-7 http://dx.doi.org/10.1016/j.physb.2015.07.014 PHYSB309062
To appear in: Physica B: Physics of Condensed Matter Received date: 18 November 2014 Revised date: 2 July 2015 Accepted date: 15 July 2015 Cite this article as: S. Das, R.K. Nayak, T. Sahu and A.K. Panda, Enhancement of multisubband electron mobility in asymmetrically doped coupled double quantum well structure, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2015.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Enhancement of multisubband electron mobility in asymmetrically doped coupled double quantum well structure S. Das, R.K. Nayak, T. Sahu* and A.K. Panda Department of Electronics and Communication Engineering National Institute of Science and Technology Palur Hills, Berhampur-761 008, Odisha, India * e-mail: [email protected] Tel: 91 680 249 2421 Fax: 91 680 249 2627
Abstract We study the effect of coupling of subband wave functions on the multisubband electron mobility in a barrier delta doped GaAs/AlxGa1-xAs asymmetric double quantum well structure. We use selfconsistent solution of the coupled Schrödinger equation and Poisson’s equation to calculate the subband wave functions and energy levels. The low temperature mobility is considered by using scatterings due to ionized impurities, interface roughness and alloy disorder. We show that variation of the width of the central barrier considerably affect the interplay of different scattering mechanisms on electron mobility through intersubband effects. Under single subband occupancy, the mobility increases with decrease in the barrier width as functions of doping concentration as well as function of well width. However, in case of double subband occupancy, effect of intersubband interaction yields opposite trend, i.e., increase in mobility with increase in barrier width. It is gratifying to show that in case of asymmetric variation of well widths the mobility shows nonmonotonic behavior which varies with change in the width of the central barrier under double subband occupancy.
Key words: Asymmetric quantum wells, Coupled Quantum Well Structures, Intersubband scatterings, Multisubband electron mobility. PACS classification No. 73.63.Hs, 73.50.Bk.
1. Introduction In recent years, a great deal of attempts has been made to study asymmetric quantum well structures because of their potential applications in electronic and optical devices [1-10]. Efforts have been made to utilize asymmetrically doped coupled quantum well GaAs/AlGaAs structures
1
to develop and characterize tunable mid-infrared photodetectors [1]. Suitably engineered GaAs/AlGaAs double quantum well structures have been shown to exhibit coherent terahertz emission due to intersubband excitation between two lowest conduction subbands [2]. InGaN based light emitting devices with asymmetric coupled quantum wells yield improved light emission intensity compared to that of a symmetric coupled quantum well device [3]. Attempts have been made to study the magnetic-field-induced charged excitonic transitions in modulationdoped AlGaAs/GaAs asymmetric coupled double quantum wells [4]. Recently it has been shown that in an asymmetric double InGaAs quantum well structure the coherent excitonic interwell coupling originates from many-body effects [5]. Further it has been shown that in a coupled asymmetric quantum well system the eigenstate probability distribution localizes exclusively either in the wide or the narrow parts of the well pair [6]. Attempts have also been made to investigate electron spin dynamics by the time-resolved Kerr rotation technique in a pair of special GaAs/AlGaAs asymmetric quantum well samples to explore the possibility of developing spin based electronic devices [7]. A direct determination of nonradiative lifetimes in Si/SiGe asymmetric quantum well structures shows that the barrier and well widths play an important role in the determination of the carrier life time. Further by comparing with experimental and theoretical data obtained for mid-infrared GaAs/AlxGa1-xAs quantum well systems, it has been shown that interface roughness scattering plays a dominant role for a wide range of semiconductors at low temperature [8]. Attempts have also been made to study electron transport in asymmetric quantum well structures by considering lowest subband occupancy [9, 10]. However, in a quantum well system, where more than one subband is occupied, the intersubband interaction plays a vital role on the subband mobility [11-15]. Intersubband effects create additional intersubband scattering rates and also modify the intrasubband scattering rate through the dielectric screening of the scattering potentials [11-15]. 2
In the present paper we study multisubband electron mobility of a coupled asymmetric double quantum well structure. Variation of the central barrier width (b) changes the coupling of the subband wave functions of individual wells thereby changing the subband energy levels and wave functions which in turn affect the occupation of subbands of the coupled structure. As a result the strength of the scattering potentials also amended leading to change in the mobility. We consider GaAs/AlGaAs double quantum well in which the asymmetry of the structure is employed by introducing delta doped layer in one of the side barriers only and also taking different well widths. We obtain the subband electron energy levels and wave functions for the coupled structure by adopting selfconsistent solution of the Schrodinger equation and Poisson’s equation. We calculate the low temperature electron mobility µ by considering ionized impurity (imp-), interface roughness (ir-) and alloy disorder (al-) scatterings. We adopt static dielectric response function formalism to calculate the screened scattering potentials by using the random phase approximation (RPA) [11-15]. We show that µ increases with increase in the doping concentration (Nd). The mobility is mostly governed by imp- and ir-scatterings. The effect of ir-scattering, which is more dominating at low values of Nd, decreases with increase in Nd so that the effect of imp-scattering on mobility enhances. The doping dependence of µ for different b also shows interesting results. As long as single subband is occupied, the mobility decreases with increase in b. A deviation in the trend occurs for small b at small Nd due to strong influence of ir-scattering. However, once double subband is occupied, the trend is reversed. The drop in mobility at the onset of occupation of the second subband and also the doping concentration at which the drop occurs, decrease with increase in b. The effect of b on the well width dependence of mobility also shows interesting results. The increasing trend in µ with increase in well width slowly reverses as the barrier width increases due to gradual dominance of the intrasubband scattering rate matrix element of the 3
higher occupied subband. We show that the mobility can be enhanced considerably at small well widths having double subband occupancy by increasing the barrier width. We further show that by increasing well width, the occupation of subbands goes on changing viz, from single subband to double subband and then again single subband occupancy. Accordingly the mobility shows a drop and then a rise near the change in occupancy of subbands. Our analysis therefore will help in choosing suitable structure parameters so that the mobility becomes a maximum. Our results of mobility of a single side barrier delta doped asymmetric double quantum well structure can be utilized for low temperature device applications.
2. Theory We consider a GaAs/AlxGa1-xAs double quantum well structure in which the left barrier is delta doped with a layer of Si of width d and doping concentrations Nd at a spacer distance s from the nearest interface (Fig. 1). The wells, of widths w1 and w2, are separated by a thin barrier of width b. Diffusion of electrons cause band bending due to Coulomb interaction with the ionized donors in the barrier. The concentration distribution of impurities and electrons nD (z) and n(z) along the growth axis, viz., z-axis can be written as:
Nd (d s w1 b / 2) z ( s w1 b / 2) nD ( z ) otherwise 0 n( z ) nn n ( z )
2
z
(1) (2)
n
n (z) are the subband wave functions and summation n is over the subband levels. nn is the number of electrons per unit area in the nth subband which can be written as [11]:
4
nn mk B T /
2
ln 1 exp E
F
En / kBT
(3)
En are the subband energy levels and EF is the Fermi energy. At temperature T = 0 K, the surface electron density Ns and the Fermi energy can be related as [11]: N s m /
E N
2
n 1
F
En EF En
(4)
N is the number of filled levels, is the Heaviside step function. We obtain the subband energy levels En and wave functionsn (z) by adopting selfconsistent solution of the one dimensional Schrödinger equation and the Poisson’s equation. In Fig. 1, we present a schematic diagram of the potential profile, subband energy levels and wave functions of the single side doped asymmetric double quantum well structure. Since at T = 0 K, the electrons on the Fermi surface take part in the conduction process, the subband transport life time can be expressed as n = n (EFn). Here EFn = EF - En is the Fermi energy of nth subband. The relaxation time n for a multisubband occupied system can be derived from the Boltzmann transport equation containing the intrasubband and intersubband scattering rate matrix elements in a mixed way [11-14]. For the lowest occupied subband (n = 0), one can write 0 in terms of intrasubband scattering rate matrix element b00 as [14]: 1
0
b00
(5)
Similarly, for a double subband is occupied structure (n=0, 1), both 0 and 1 contain intrasubband and intersubband scattering rate matrix elements (c01 and d01) as shown below [14]:
1 b00 c01 b11 c10 d01d10 0 b c E / E 12 d 11 10 F1 F0 01
5
1 b00 c01 b11 c10 d01d10 1 b c E / E 12 d 00 01 F0 F1 10
(6)
The scattering rate matrix elements bnn, cnm and dnm can be expressed in terms of scattering potential matrix elements Vnmeff(q) as [14]: bnn m /
3
d 1 cos V q eff nn
2
nn
0
cnm m /
3
d
eff Vnm qnm
2
0
d nm m /
3
d cos V q eff nm
2
(7)
nm
0
2 qnl kFn kFl2 2kFn kFl cos
1
2
and kFn=(2mEFn/ħ2)1/2. The screened scattering potential Vnmeff(q)
can be written as [11]: -1 Vnmeff (q) nm,n'm' (q)Vn ' m ' (q) ,
(8)
n'm'
1 where nm , n ' m ' is the element of the inverse matrix of the dielectric response function. n, m, n’, m’
are the subband indices. We use the random phase approximation (RPA) for the screening of the scattering potentials which can be written as [11]:
nm,n ' m ' (q) nn ' mm ' (qs / q) Fnm,n ' m ' (q) n'm' (q)
(9)
where Fnm,n 'm' (q) is the Coulomb form factor, qs 2me2 / 0
2
and is the static electron
density-density correlation functions without electron-electron interaction [11]. Vnm(q) is the element of Fourier transform of the external potential. We consider ionized impurity scattering, interface roughness scattering and alloy disorder scattering to calculate the subband mobility. The screened ionized impurity scattering potential for the present structure can be written as: 6
imp nm
V
4 2 e4 N0 (s+w1+b / 2) (q) dz 02 q 2 (d+s+w1+b / 2) i 2
2 -1 nm,n'm'
(q) Pn ' m ' (q, zi )
n'm'
(10)
where
Pn ' m ' q, zi
dz
n'
( z ) m ' ( z ) exp(-q z - zi )
(11)
The screened interface roughness scattering potential is:
V (q) V exp(q / 4) ir nm
2
2 b
2
2
2
2
I
2 n'
*( z ) m ' ( z ) z z I
n 'm'
-1 nm,n'm'
( q)
(12)
Similarly, the alloy disorder scattering potential can be written as:
V (q) a ( V ) x(1 x) / 4 dz al nm
2
3
2
j
2 n'
( z ) m ' ( z )
-1 nm,n'm'
(q)
(13)
n'm'
The ionized impurity scattering occurs from the delta-doped layer lying in the left side barrier. The alloy disorder scattering is considered from the alloy (AlGaAs) regions. The summation j stands for both the side barriers as well as the central barrier. ‘a’, V and x are the lattice constant, alloy scattering potential and alloy fraction respectively of the barrier layers. The summation I is over the interfaces lying towards the left side of the wells, i.e., (I1, I3), which are considered to be rough. The roughness is assumed to exist in the interfaces lying towards the substrate side only [12, 14, 16]. The subband electron mobility np is related to the subband transport life time np by:
np ( EF )
e p n ( EF ) m
(14)
7
where np is the subband transport life time described in Eqs. 5-6 for Pth (imp-, ir-, al-) scattering event. The mobility µP due to a scattering mechanism P can be written as
p
n n n
p n
n
(15)
n
n
The total mobility µ is calculated using Matthiessen’s relation.
3. Results and discussion We analyze the electron mobility µ of a single side barrier delta doped asymmetric GaAs/AlxGa1-xAs coupled double quantum well structure. We consider the potential barrier height Vb = 228 meV (x = 0.3) [17]. The interface roughness parameters = 2.83 Å and = 100 Å and the alloy disorder scattering potential V = 1.56 eV [12, 18]. The delta-doping layer width d = 20 Å and spacer width s = 80 Å. In Fig. 2, we present total mobility µ along with the mobility due to individual scattering mechanisms, i.e., ionized impurity scattering µimp, interface roughness scattering µir and alloy scattering µal as a function of doping concentration Nd for central barrier width b = 10 Å, taking well widths w1 = w2 = 100 Å. Up to Nd = 3.1×1018 cm-3, only lowest subband is occupied. Thereafter double subband is occupied. The mobility µ, which is dominated by imp- and irscatterings, increases with increase in doping concentration Nd. The al-scattering has least effect on µ as it occurs from the AlGaAs barrier where the amplitudes of the subband electron wave functions are less in magnitude. We note that with increase in Nd, the triangular like potential well near interface I1 (Fig. 1) becomes deeper. As a result the subband electron wave function distribution shifts towards the impurities lying in the left barrier thereby increasing the strength of the imp-scattering potential Vimp(q) (Eq. 10). The strength of the ir-scattering potential Vir(q)
8
(Eq. 12) is also increased due to increase in the amplitudes of the subband wave functions at the interface I1 even though the amplitudes of the wave function at I3 decrease slightly. However, it is interesting to note that the change in the potential profile of the quantum well due to increase in the doping concentration also affects the subband energy levels leading to increase in subband Fermi wave vector q that reduces the strengths of imp- and ir- scattering potentials (Eq.10 and 12). The later effect being predominating, both µimp and µir increase giving rise to enhanced µ with increase in Nd. Once the second subband is occupied, the imp-scattering is more influenced by the intersubband interactions so that the drop in µir becomes less than that of µimp, resulting µir slightly greater than µimp. In Fig. 3, we present mobility as a function of doping concentration Nd for different central barrier width (b = 10, 14, 20, 26, 30 and 40 Å) taking well widths w1 = w2 = 100 Å. As b increases, the overlapping of the subband wave functions of the individual wells through the central barrier decreases leading to less splitting in the subband energy levels. Accordingly the subband energy levels and wave functions of the coupled structure vary. We show that as b increases, the transition from single subband to double subband occupancy occurs at less value of Nd due to change in the coupling of the subband wave functions through the central barrier. The drop in mobility at that point also decreases. In the inset of Fig. 3, we present µir/µimp as a function of Nd from which the relative impact of imp- and ir- scatterings on µ can be derived. Under single subband occupancy, the mobility decreases with increase in b. The deviation in this trend for b up to 20 Å can be attributed to the stronger effect of ir-scattering on mobility for Nd < 2.4×1018 cm-3. Once double subband is occupied, the trend is reversed due to enhanced effect of imp-scattering mediated by intersubband effect. In Fig. 4, we analyze the functional dependence of mobility on well width w (w1 = w2 = w) for different central barrier width b taking doping concentration Nd = 2×1018 cm-3. As well width 9
increases, the occupation of subbands changes from single subband occupancy to double and then again to single subband occupancy. The drop in mobility at the transition from single to double subband occupancy is because of additional intersubband effect as described earlier. Similarly, the rise in mobility near the transition from double subband to single subband occupancy is due to suppression of intersubband effects. In the inset of Fig. 4, we present µir/µimp as a function of w for different b. The mobility is mostly governed by imp-scattering except at small well width (w < 80 Å) where the ir-scattering dominates the mobility. The enhancement in mobility with increase in well width is because of increase in the effective distance between the ionized impurities and the subband electrons through the ionized impurity scattering potential Vimp(q). We note that the interplay of different scattering mechanisms on mobility depends upon the occupation of subbands.through the intersubband effect. When the central barrier width b = 20 Å, only single subband is occupied. As b increases (b = 26, 30, 36 and 40 Å), transition from single subband occupancy to double subband occupancy starts at lower values of w (w = 84, 72, 56 and 50 Å, respectively). On the other hand, transition from double subband to single subband occupancy occurs almost at a single well width (w = 136 Å) for different b, indicating independence on the barrier width. At large well widths w >136 Å, when only a single subband is occupied, the mobility for different b remain almost unchanged since the dominant imp-scattering hardly depends on b. Whereas at small well widths, we show that the difference in mobility for different b is mostly because of the change in µir. A change in b affect the coupling of subband wave functions leading to a variation of the electron subband energy levels. Accordingly the Fermi wave vector (q) decreases and the amplitude of the subband wave function at I1 increases causing enhancement of Vnmir (q) (Eq. 12). In Fig. 5, we present mobility µ as a function of well width w1 taking w1 + w2 = 400 Å such that by varying w1, both the well widths w1 and w2 change. We take doping concentration Nd = 10
1×1018 cm-3 and vary central barrier width b (b = 10, 16, 20, 30 and 40 Å). We note that the mobility is mostly governed by imp-scattering. To start with, double subband is occupied. The nonmonotonic variation of mobility as a function of w1 for different b under double subband occupancy is governed by the intersubband effects. For b ≤ 20 Å, the mobility increases, while for b > 20 Å there is substantial reduction in µ with increase in w1 so that near the transition from double subband to single subband occupancy, the difference in µ for different b becomes marginal. Once the transition from double to single subband occupancy occurs, the mobility monotonically increases with increase in w1 showing negligible dependence on b. The change in the barrier width marginally affects the effective distance between the ionized impurities and the subband electron wave function resulting negligible dependence of mobility on barrier width. Further, we show that the nonmonotonic variation of mobility as a function of w1 for different b under double subband occupancy is attributed to the contribution of both µ0imp and µ1imp. In Fig. 6, we present the results of subband mobilities µ0imp and µ1imp as a function of well width w1 for b = 10, 20 and 40 Å taking Nd = 1×1018 cm-3 and w1 + w2 = 400 Å. There is not much variation of µ0imp with increase in the well width w1. However, it is interesting to note that large variation of µ1imp for different b is exhibited due to the difference in the intrasubband scattering rate b11imp. Since the subband electron densities n1 and n2 are of same order, the change in µ1imp causes the resulting trend in µ. shown in Fig.5.
4. Summary and Conclusion In the present work we analyze the effect of barrier width on the functional dependence of low temperature multisubband electron mobility on doping concentration and well widths in an asymmetric coupled double quantum well structure. We consider a GaAs/AlxGa1-xAs double quantum well in which one of the side barriers is delta-doped with Si. We obtain subband energy 11
levels and wave functions by using selfconsistent solution of the Schrödinger equation and Poisson’s equation. We consider scatterings due to ionized impurities (imp-), interface roughness (ir-) and alloy disorder (al-) to study low temperature electron mobility. The screening of scattering potentials is obtained using random phase approximation. We show that coupling of subband electron wave functions through the central barrier affect the interplay of different scattering mechanisms on mobility. The importance of our work is to relate the enhancement of electron mobility with the variation of barrier width mediated by intersubband effects. The doping concentration as well as well width dependence of mobility show that by reducing the central barrier width, the mobility can be enhanced when only single subband is occupied. Whereas, under double subband occupancy, the trend is reversed, i.e., mobility increases with increase in barrier width mediated by intersubband effects. Further, we show that variation of well widths asymmetrically gives rise to nonmonotonic variation of mobility for different central barrier width. Our work can be utilized for low temperature device application.
12
Reference [1]
J.K. Choi, N. Vagidov, A. Sergeev, S. Kalchmair, G. Strasser, F. Vasko, V. Mitin, Asymmetrically doped GaAs/AlGaAs double-quantum-well structure for voltage-tunable infrared detection, Japan. J. Appl. Phys. 51 (2012) 074004.
[2]
P.G. Huggard, C.J. Shaw, S.R. Andrews, J.A. Cluff, R. Grey, Mechanism of THz emission from asymmetric double quantum wells, Phys. Rev. Lett. 84 (2000) 1023-1026.
[3]
J.Y. Zhang, L. Cai, B.P. Zhang, X.L. Hu, F. Jiang, J.Z. Yu, Q.M. Wang, Efficient hole transport in asymmetric coupled InGaN multiple quantum wells, Appl. Phys. Lett. 95 (2009) 161110-161113.
[4]
K.S. Lee, S.K. Noh, S.K. Chang, Photoluminescence study of magnetic-field-induced excitonic transitions in AlxGa1-xAs/GaAs asymmetric double quantum wells, Phys. Rev. B 76 (2007) 073305.
[5]
G. Nardin, G. Moody, R. Singh, T. M. Autry, H. Li, F.M. Genoud, S.T. Cundiff, Coherent Excitonic Coupling in an Asymmetric Double InGaAs Quantum Well Arises from Many-Body Effects, Phys. Rev. Lett. 112 (2014) 046402.
[6]
A. Mialitsin, S. Schmult, I.A. Solov’yov, B. Fluegel, A. Mascarenhas, Eigenstate localization in an asymmetric coupled quantum well pair, Superlatt. Microstruct. 51 (2012) 834-841.
[7]
W. Gang, Y.E. Hui-Qi, S.H.I. Zhen-Wu, W. Wen-Xin, M. Xavier, B. Andrea, A. Thierry, L. Bao-Li, Spin Dynamics in (111) GaAs/AlGaAs Undoped Asymmetric Quantum Wells, Chin. Phys. Lett. 29 (2012) 097204.
[8]
M. Califano, N.Q. Vinh, P.J. Phillips, Z. Ikonic, R.W. Kelsall, P. Harrison, C.R. Pidgeon, B.N. Murdin, D.J. Paul, P. Townsend, J. Zhang, I.M. Ross, Interwell relaxation times in p-Si/SiGe asymmetric quantum well structures: Role of interface roughness, Phys. Rev. B 75 (2007) 045338.
[9]
F.M.S. Lima, F. Qu, O.A.C. Nunes, A.L.A. Fonseca, Electron mobility in one (two)-side modulation-doped GaAs/AlxGa1-xAs asymmetric quantum wells, Phys. Status Solidi B 225 (2001) 43-61.
13
[10]
M. Jung, T.W. Kim, D.U. Lee, D. C. Choo, K. H. Yoo, D. L. Kim, M. D. Kim, H. Lim, The dependence of the carrier density and the mobility on the spacer-layer thickness in modulation-doped AlxGa1−xAs/InyGa1−yAs/GaAs asymmetric single quantum wells, Appl. Surf. Sci. 177 (2001) 1-7.
[11]
T. Ando, A.B. Fowler, F. Stern, Electronic properties of two dimensional systems, Rev. Mod. Phys. 54 (1982) 437-672.
[12]
K. Inoue, T. Matsuno, Electron mobilities in modulation-doped AlxGa1-xAs/GaAs and pseudomorphic AlxGa1-xAs/InyGa1-yAs quantum well structures, Phys. Rev. B 47 (1993) 3771-3778.
[13]
G.Q. Hai, N. Studart, F.M. Peeters, P.M. Koenraad, J.H. Wolter, Intersubband-coupling and screening effects on the electron transport in a quasi-two-dimensional δ-doped semiconductor system, J. Appl. Phys. 80 (1996) 5809-5814.
[14]
T. Sahu, K.A. Shore, Multi-interface roughness effects on electron mobility in a Ga0.5In0.5P/GaAs multisubband coupled quantum well structure, Semicond. Sci. Technol. 24 (2009) 095021.
[15]
S. Das, R.K. Nayak, A.K. Panda, T. Sahu, Multisubband electron mobility in asymmetric GaAs/AlGaAs quantum well structures, Superlatt. Microstruct. 66 (2014) 39-47.
[16]
S.K. Lyo, Real-space and energy representations for the interface-roughness scattering in quantum-well structures, J. Phys.:Condens. Matter. 13 (2001) 1259-1264.
[17]
S. Adachi, GaAs, AlAs and AlxGa1-xAs material parameters for use in research and device applications, J. Appl. Phys. 58 (1985) R1-R29.
[18]
A.K. Saxena, A.D. Adams, Determination of alloy scattering potential in Ga1−xAlxAs alloys, J. Appl. Phys. 58 (1985) 2640-2645.
14
Collected Figure Captions Fig. 1. Schematic diagram of the one side barrier doped asymmetric double quantum well structure along with the potential energy profile, subband energy levels and wave functions. Fig. 2. Subband electron mobility for the ionized impurity scattering µimp, alloy scattering µal and interface roughness scattering µir and total mobility µ as a function of the doping concentration Nd (× 1018 cm-3) by taking b = 10 Å, s = 80 Å, w1 = w2 = 100 Å. Fig. 3. Mobility µ as a function of doping concentration Nd (cm-3) for different central barrier width b = 10, 14, 20, 26, 30 and 40 Å taking spacer widths s = 80 Å and well widths w1 = w2 = 100 Å. Inset shows the ratio µir/µimp for the above structure parameters.
Fig. 4. Mobility µ as a function of well width w (Å) for central barrier width b = 20, 26, 30, 36, and 40 Å taking doping concentration Nd = 2×1018 cm-3 and spacer widths s = 80 Å. Inset shows the ratio µir/µimp for the above structure parameters.
Fig. 5. Mobility µ (105 cm2/Vs) as a function of well width w1 (Å) for different central barrier width b = 10, 16, 20, 30 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å and w1 + w2 = 400 Å which depicts considerable influence of coupling of subband wave functions on mobility when more than one subband is occupied. Fig. 6. Subband electron mobility for the ionized impurity scattering µ0imp and µ1imp as a function of well width w1 for central barrier width b = 10, 20 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å taking w1 + w2 = 400 Å. In caption µnimp (N) implies µnimp for b = N Å.
15
Fig. 1. Schematic diagram of the one side barrier doped asymmetric double quantum well structure along with the potential energy profile, subband energy levels and wave functions.
16
Fig. 2. Subband electron mobility for the ionized impurity scattering µimp, alloy scattering µal and interface roughness scattering µir and total mobility µ as a function of the doping concentration Nd (× 1018 cm-3) by taking b = 10 Å, s = 80 Å, w1 = w2 = 100 Å.
17
Fig. 3. Mobility µ as a function of doping concentration Nd (cm-3) for different central barrier width b = 10, 14, 20, 26, 30 and 40 Å taking spacer widths s = 80 Å and well widths w1 = w2 = 100 Å. Inset shows the ratio µir/µimp for the above structure parameters.
18
Fig. 4. Mobility µ as a function of well width w (Å) for central barrier width b = 20, 26, 30, 36, and 40 Å taking doping concentration Nd = 2×1018 cm-3 and spacer widths s = 80 Å. Inset shows the ratio µir/µimp for the above structure parameters.
19
Fig. 5. Mobility µ (105 cm2/Vs) as a function of well width w1 (Å) for different central barrier width b = 10, 16, 20, 30 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å and w1 + w2 = 400 Å which depicts considerable influence of coupling of subband wave functions on mobility when more than one subband is occupied.
20
Fig. 6. Subband electron mobility for the ionized impurity scattering µ0imp and µ1imp as a function of well width w1 for central barrier width b = 10, 20 and 40 Å taking doping concentration Nd = 1×1018 cm-3, spacer width s = 80 Å taking w1 + w2 = 400 Å. In caption µnimp (N) implies µnimp for b = N Å.
21