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Effect of interface roughness on the density of states of finite barrier height quantum wells
Solid State Communications 145 (2008) 207–211 www.elsevier.com/locate/ssc
Effect of interface roughness on the density of states of finite barrier height quantum wells A. Thongnum a,∗ , U. Pinsook b , S. Khan-ngern a , V. Sa-yakanit b a Department of Physics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand b Center of Excellence in Forum for Theoretical Science, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand
Received 27 August 2007; received in revised form 16 October 2007; accepted 18 October 2007 by F. Peeters Available online 23 October 2007
Abstract We calculate the density of states of a 2D electron gas in finite barrier height quantum wells with the explicit inclusion of the interface roughness effect. By using Feynman path-integral method, the analytic expression is derived. The results show that the 2D density of states is dependent on the RMS of the fluctuation potential. The interface roughness causes localized states below the subband edge. We also apply the theory to model the finite barrier height quantum wells in Alx Ga1−x As/GaAs. Published by Elsevier Ltd PACS: 73.20.Dx Keywords: A. Finite barrier height quantum wells; D. Interface roughness; D. 2D density of states; D. Feynman path integrals
1. Introduction The study of quantum wells (QWs) is a currently active topic motivated by the progress of the epitaxial-growth technique. Especially, the transport and optical properties of Alx Ga1−x As/GaAs QWs have received much attention in the fields of fundamental physics and device applications [1]. As the density of states (DOS) determines all the basic characteristics of semiconductor heterostructures, the important information, such as, the carrier mobility [2] and the optical absorption [3] can also be derived. Thus, the determination of DOS is the crucial task in the theoretical studies. As suggested by experiments, the evaluation of QWs DOS is far from being a simple picture. Several refinements have been introduced. For example, it has been showed that the interface roughness scattering dominated the electron mobility [4–8] and the intersubband absorption linewidths [9–11] in some thin QWs. Furthermore, the barrier height of QWs showed significant impact on the electron mobility in some cases. The infinite barrier height model existed
∗ Corresponding author.
E-mail address: [email protected] (A. Thongnum). 0038-1098/$ - see front matter Published by Elsevier Ltd doi:10.1016/j.ssc.2007.10.024
and showed good agreement with the experimental electron mobility of AlAs/GaAs [4], AlSb/InAs QWs [5] and GaSb/InAs superlattices [8]. However, the experiments on Alx Ga1−x As/GaAs QWs [12–14] suggested a different picture. They showed that the finite barrier height caused noticeable changes in the electron mobility, i.e. the mobility is higher than that derived from the infinite barrier height model with the same width. On the theoretical side, several models of QWs have been proposed. For example, Takeshima [15] adopted a semiclassical approach for the 3D case. His model used a bulk-like screened Coulomb interaction for the electron–impurity interaction. However, it seemed to strongly overestimate the screening effect [16]. Serre et al. [17] applied a multiple scattering approach to evaluate the band-tail DOS. Many body effects (the screening and exchange correlation energy) were included in the self-energy. These methods would give a more accurate result. However, they invoked higher order approximations and numerical results, and were very difficult to handle, especially in analytic forms. Moreover, the existing theories seemed to be developed only for the case of charge impurities but the interface roughness effect was neglected. For these reasons, Quang et al. [18] adopted a path-integral technique, suggested by Sa-yakanit [19], to calculate the deep band-tail DOS.
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A. Thongnum et al. / Solid State Communications 145 (2008) 207–211
His calculation took into account the interface roughness effect. However, his theory considered only an infinite barrier height model. The purpose of this work is to develop the Feynman pathintegral theory and to calculate the 2D DOS of finite QWs. We include the effect of the interface roughness and the finite barrier height, as suggested by experiments. The advantages of this semiclassical approach are that it allows analytic solutions, where the link between the model’s parameters and the solutions can be clearly established. The paper is organized as follows; in Section 2, a brief overview on the model of the interface roughness effect in QWs with a finite barrier height is presented. The path integration method pioneered by Sayakanit [20] is explained. The results, compared with those of existing infinite barrier height model, are discussed in Section 3. 2. Theoretical framework 2.1. Interface roughness effect in a finite barrier height QW The effects of the interface roughness and the finite barrier height of QWs are the main objective of our investigation. The random variation in the well width causes fluctuation in the confinement energy. The barrier height is also an important factor, as suggested by experiments [12–14]. It is dependent on the alloy composition of the QWs. In this work, we are dealing with a semiclassical approach, and thus real-space interactions are needed. A brief overview of the real-space model of the interface roughness effect in QWs with a finite barrier height is presented, starting from the definition of the local fluctuation potential [13] U (r) =
∂ ∆(r), ∂a
(1)
where is the confinement energy, a is the well width, and ∆(r) characterizes the local height of a rough interface at the plane position. Our task is to identify as a function of a, and to model ∆(r). In finite QWs with given values of the barrier height V0 and the well width q a, the characteristic wave vector is represented by kw =
2m/ h 2
¯ inside the well, and the wave vector in q the barriers is kb = 2m(V0 − )/ h¯ 2 , satisfying the boundary conditions at the interfaces. The effective mass m is taken to be the same value within the well and the barriers. The fluctuation potential is limited to the local fluctuation within the neighborhood of the confinement energy . The lowest subband in the even parity case [21] is kb kw a = tan . (2) kw 2 By solving Eq. (2) in terms of , the relationship between the confinement energy and the barrier height can be expressed as r m 2 + tan a = V0 . (3) 2 h¯ 2
Insert this into Eq. (1), we get the local fluctuation potential as U (r) = − a+
2 q
2 h¯ 2 m(V0 −)
∆(r).
(4)
This represents the local interaction between an electron and the QWs with rough interfaces. However, the actual interaction should come from the average over all possible configurations of the rough interfaces. The statistical averaged potential is in the form of an autocorrelation function W (r − r0 ) = hU (r)U (r0 )i,
(5)
where the angular brackets denote the average over all possible configurations of the interface roughness. Ando, Fowler and Stern [16] suggested that the autocorrelation of ∆(r) can be described by a Gaussian function as |r − r0 |2 0 2 h∆(r)∆(r )i = ∆ exp − , (6) L2 where ∆ is the average height and L is the fluctuation correlation length. It is an economic model as it contains only a few important parameters, and yet gives reasonable description. Substituting Eqs. (4) and (6) into Eq. (5), we get |r − r0 |2 0 W (r − r ) = ξ exp − , (7) L2 where the variance of the correlation function has the dimension of the energy square. For the finite barrier (FB) height model, ξ F B is defined as 2 2∆ . q ξF B = (8) 2 h¯ 2 a + m(V 0 −) It is readily seen that Eqs. (7) and (8) are in a more general form than that of the infinite barrier height. In the limit V0 → ∞, these equations reduce to the expressions for the infinite barrier (IB) heights [4], which are U (r) = − where = ξI B =
h¯ 2 π 2 ∆(r), ma 3 h¯ 2 π 2 , 2ma 2
h¯ 2 π 2 ∆ ma 3
(9)
and the variance of the correlation function is !2 .
(10)
2.2. Path-integral approach for the calculation of the 2D DOS In order to calculate the 2D electron density of states in finite quantum wells, we use the path-integral approach. The density of states per unit volume [19] can be expressed in terms of the diagonal elements of the averaged propagator as Z ∞ 1 iEt ¯ n(E) = dt G(0, 0; t) exp , (11) 2π h¯ −∞ h¯
A. Thongnum et al. / Solid State Communications 145 (2008) 207–211
where the averaged propagator is defined by Z i ¯ 2 , r1 ; t) = S . G(r D[r(τ )] exp h¯
(12)
The boundary conditions are taken to be r(0) = r1 , r(t) = r2 . In this approach, the system is based on the one-electron interaction, and hence the system’s action can be written as Z t Z tZ t m 2 i dτ r˙ (τ ) + dτ dσ W (r(τ ) − r(σ )), (13) S= 2 2h¯ 0 0 0 where W is defined in Eq. (7). In general, this problem cannot be integrated out exactly. We must resort to a variational method. We introduce a nonlocal harmonic trial action Z t Z Z m ω2 t t dτ r˙ 2 (τ ) − dτ dσ (r(τ ) − r(σ ))2 , (14) S0 = 2 2t 0 0 0 where the variational parameter ω is used to adjust the energy function subjected to a variational principle [19,20]. This method gave very good results for the polaron problem, compared with similar methods and experiments. The key for solving this problem is that the actual propagator can be approximated in terms of the trial action by using the variational path-integral method [19,20]. By keeping only the first-order term in the cumulant expansion, the first-order approximation to G¯ is obtained as i ¯ G(r2 , r1 ; t) = G 0 (r2 , r1 ; t) exp hS − S0 i S0 , (15) h¯ where G 0 (r2 , r1 ; t) is the zero-order approximation associated with the trial action. For the fluctuation potential with a Gaussian form, Eq. (7), the generalized d-dimensional density of states [20] was derived as !d d/2 Z ∞ m 1 ωt n(E) = dt 2π h¯ −∞ 2πih¯ t 2 sin ωt 2 i d ωt ωt × exp Et + cot −1 h¯ 2 2 2 ! 2 d/2 Z t L ξ −d/2 t dx j (x, ω) , (16) − 2 h¯ 2 4 0 where j (x, ω) =
ω(t−x) L2 ih¯ sin ωx 2 sin 2 + . 4 mω sin ωt 2
(17)
Next, let us consider a semiclassical approximation, it corresponds to taking the limit t → 0 of the integrand in Eq. (16). In another words, this limit retains only high-energy electrons (near the subband edge) in the DOS. Thus, we get 2 limt→0 j (x, ω) ≈ L4 . This leads to the following expression for the 2D DOS as Z ∞ m i ξ 2 −1 dt (it) exp n(E) = Et − t . (18) h¯ (2π h¯ )2 −∞ 2 h¯ 2
209
This is a general form and independent of ω. As we turn off the interaction, i.e. ξ = 0, the 2D DOS becomes a step function as in the usual noninteracting 2D electron gas. At this stage, we perform time integration of Eq. (18). By using formula [22] Z ∞ dt (it) p exp(−β 2 t 2 − iqt) −∞
√
=
π q q2 , D exp − √ p 2 p/2 β p+1 8β 2 β 2
(19)
where D p (x) is the parabolic cylinder function, the analytic expression of the 2D DOS can be written as 2m E −E 2 /4ξ n(E) = . (20) e D − √ −1 ξ (2π )3/2 h¯ 2 In this expression, the spin degeneracy is included. In order to obtain the 2D DOS in the low-energy tail, the asymptotic limit of the parabolic cylinder function [23] is considered. Finally, the band-tail of 2D DOS can be expressed in terms of the error function as E m . (21) n(E) = 1 + erf √ ξ 2π h¯ 2 3. Results and discussion Our results mainly come from Eqs. (8) and (21). As we attempt to give a clue to experiments, we use the parameters derived from the experimental results of Al0.3 Ga0.7 As/GaAs QW. The values of the confinement energy were given by Bastard [24] who assumed the barrier height of 224.5 meV. In order to make compatible comparison, we use the effective mass of m = 0.067m 0 [1], and the average height of one monolayer of 0.283 nm [4]. Fig. 1 shows the RMS of the variance of the fluctuation potential created by the interface roughness, derived from Eq. (8). It exhibits that the variance of the fluctuation potential is strongly dependent on the barrier height and the well width. The magnitude of fluctuation potential becomes very sensitive to the barrier height in thin QWs, i.e. low values of a. Moreover, it can be seen that the interface roughness has larger effects on the infinite barrier QWs than on the finite barrier QWs, as shown in Fig. 2. Next, the 2D DOS in Eq. (21) with various barrier heights are shown in Fig. 3. In this figure, the well width is set to 8 nm. It shows that the 2D DOS has a very long tail in the low-energy region. The explanation is that the interface roughness causes fluctuation in the confinement energy and leads to the existence of some electron states below the subband edge. This is the so-called localized states where the subband edge is fixed to zero. The correspondence between the localized states and the reduction of mobility is explained by the theory of the mobility gap [2]. From Fig. 3, it can be readily seen that the density of the localized states is increased with the increase in the barrier height. In other words, the density of the localized states is sensitive to the variation in the barrier height. This is consistent
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A. Thongnum et al. / Solid State Communications 145 (2008) 207–211
Fig. 1. Plot of the RMS of the fluctuation potential as a function of barrier height, as in Eq. (8), with three different values of the well width. The solid line represents the well width of 15 nm, the dashed line represents a = 8 nm, and the dashed–dotted line represents a = 3 nm.
Fig. 3. Plot of the 2D DOS as a function of electron energy, as in Eq. (21), with four different values of the barrier height. The well width is fixed at the value of 8 nm. The solid line represents the barrier height of 25 meV, the dashed–dotted line represents the barrier height of 100 meV, the dashed line represents the barrier height of 225 meV, and the dotted line represents the infinite barrier height, respectively.
Fig. 2. Plot of the RMS of the fluctuation potential as a function of the well width. The solid line represents the barrier height of 224.5 meV, and the dashed line represents the infinite barrier height.
with the earlier results of the RMS of the fluctuation potential, where the interface roughness causes stronger fluctuation in the confinement energy with increasing barrier height. Fig. 4 shows the 2D DOS of QWs with various well widths. The barrier height is fixed at the value of 224.5 meV. We observed that the density of the localized states strongly depended on the decrease in the well width. The 2D DOS resembles the step-like characteristic of the 2D noninteracting electron gas when the well width is equal to 15 nm. Hence, the interface roughness causes only weak fluctuation in wide QWs. In conclusion, the 2D DOS of QWs with the effect of the interface roughness and the finite barrier height is derived by the path-integral method with the semiclassical approximation. The interface roughness causes the fluctuation in the confinement energy, and leads to the localized states below the subband edge. The analytic expression of 2D DOS shows explicit dependence on the RMS of the fluctuation potential. The nature of the finite barrier height is to reduce the strength of the fluctuation potential and the density of the localized states,
Fig. 4. Plot of the 2D DOS in Eq. (21) with three different values of the well width and the fixed barrier height of 224.5 meV. The solid line represents the well width of 15 nm, the dashed line represents a = 8 nm, and the dashed–dotted line represents a = 3 nm.
compared with that of the infinite barrier height. Thus the influence of the interface roughness has stronger impact on the infinite barrier height QWs than on the finite barrier height QWs. In addition, we can neglect the effect of the interface roughness in wide QWs. Thus, the electron mobility should be higher in the case of the finite barrier height QWs than in the infinite ones. This is consistent with the experimental findings. Acknowledgement A. Thongnum expresses his gratitude to the Thai government for the supporting scholarship, under the University Developing Commissions (UDC) Grant, Ministry of University Affairs.
A. Thongnum et al. / Solid State Communications 145 (2008) 207–211
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
S. Adachi, J. Appl. Phys. 58 (1985) R1. H.A. Fertig, S. Das Sarma, Phys. Rev. B 42 (1990) 1448. R. Kucharczyk, M. Ste¸s´licka, Solid State Commun. 84 (1992) 727. H. Sakaki, T. Noda, H. Hirakawa, M. Tanaka, T. Matsusue, Appl. Phys. Lett. 51 (1987) 1934. C.R. Bolognesi, H. Kroemer, J.H. English, Appl. Phys. Lett. 61 (1992) 213. J.R. Meyer, D.J. Arnold, C.A. Hoffman, F.J. Bartoli, Appl. Phys. Lett. 58 (1991) 2523. B.R. Nag, M. Das, Appl. Surf. Sci. 182 (2001) 35. F. Szmulowicz, S. Elhami, H.J. Haugan, G.J. Brown, W.C. Mitchel, J. Appl. Phys. 101 (2007) 43706. K.L. Campman, H. Schmidt, A. Imamoglu, A.C. Gossard, Appl. Phys. Lett. 69 (1996) 2554. T. Unuma, M. Yoshita, T. Noda, H. Sakaki, H. Akiyama, J. Appl. Phys. 93 (2003) 1586. T. Unuma, T. Takahashi, T. Noda, H. Sakaki, M. Baba, H. Akiyama, Appl. Phys. Lett. 78 (2001) 3448.
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[12] R. Gottinger, A. Gold, G. Abstreiter, G. Weimann, W. Schlapp, Europhys. Lett. 6 (1988) 183. [13] J.M. Li, J.J. Wu, X.X. Han, Y.W. Lu, X.L. Liu, Q.S. Zhu, Z.G. Wang, Semicond. Sci. Technol. 20 (2005) 1207. [14] B.R. Nag, Semicond. Sci. Technol. 19 (2004) 162. [15] M. Takeshima, Phys. Rev. B 33 (1986) 4054. [16] T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437. [17] J. Serre, A. Ghazali, A. Gold, Phys. Rev. B 39 (1989) 8499. [18] D.N. Quang, N.H. Tung, Phys. Status Solidi B 209 (1998) 375. [19] V. Sa-yakanit, Phys. Rev. B 19 (1979) 2266. [20] V. Sa-yakanit, J. Phys. C 11 (1978) L521. [21] P. Harrison, Quantum Wells, Wires, and Dots Theoretical and Computational Physics, John Wiley & Sons, Chichester, 2000. [22] I.S. Gradshteyn, I.M. Ryzik, Table of Integrals, Series and Products, Academic, New York, 1965. [23] M. Abramovitz, A.I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, 1965. [24] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les Editions de Physique, Les Ulis, 1988.
Effect of interface roughness on the density of states of finite barrier height quantum wells A. Thongnum a,∗ , U. Pinsook b , S. Khan-ngern a , V. Sa-yakanit b a Department of Physics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand b Center of Excellence in Forum for Theoretical Science, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand
Received 27 August 2007; received in revised form 16 October 2007; accepted 18 October 2007 by F. Peeters Available online 23 October 2007
Abstract We calculate the density of states of a 2D electron gas in finite barrier height quantum wells with the explicit inclusion of the interface roughness effect. By using Feynman path-integral method, the analytic expression is derived. The results show that the 2D density of states is dependent on the RMS of the fluctuation potential. The interface roughness causes localized states below the subband edge. We also apply the theory to model the finite barrier height quantum wells in Alx Ga1−x As/GaAs. Published by Elsevier Ltd PACS: 73.20.Dx Keywords: A. Finite barrier height quantum wells; D. Interface roughness; D. 2D density of states; D. Feynman path integrals
1. Introduction The study of quantum wells (QWs) is a currently active topic motivated by the progress of the epitaxial-growth technique. Especially, the transport and optical properties of Alx Ga1−x As/GaAs QWs have received much attention in the fields of fundamental physics and device applications [1]. As the density of states (DOS) determines all the basic characteristics of semiconductor heterostructures, the important information, such as, the carrier mobility [2] and the optical absorption [3] can also be derived. Thus, the determination of DOS is the crucial task in the theoretical studies. As suggested by experiments, the evaluation of QWs DOS is far from being a simple picture. Several refinements have been introduced. For example, it has been showed that the interface roughness scattering dominated the electron mobility [4–8] and the intersubband absorption linewidths [9–11] in some thin QWs. Furthermore, the barrier height of QWs showed significant impact on the electron mobility in some cases. The infinite barrier height model existed
∗ Corresponding author.
E-mail address: [email protected] (A. Thongnum). 0038-1098/$ - see front matter Published by Elsevier Ltd doi:10.1016/j.ssc.2007.10.024
and showed good agreement with the experimental electron mobility of AlAs/GaAs [4], AlSb/InAs QWs [5] and GaSb/InAs superlattices [8]. However, the experiments on Alx Ga1−x As/GaAs QWs [12–14] suggested a different picture. They showed that the finite barrier height caused noticeable changes in the electron mobility, i.e. the mobility is higher than that derived from the infinite barrier height model with the same width. On the theoretical side, several models of QWs have been proposed. For example, Takeshima [15] adopted a semiclassical approach for the 3D case. His model used a bulk-like screened Coulomb interaction for the electron–impurity interaction. However, it seemed to strongly overestimate the screening effect [16]. Serre et al. [17] applied a multiple scattering approach to evaluate the band-tail DOS. Many body effects (the screening and exchange correlation energy) were included in the self-energy. These methods would give a more accurate result. However, they invoked higher order approximations and numerical results, and were very difficult to handle, especially in analytic forms. Moreover, the existing theories seemed to be developed only for the case of charge impurities but the interface roughness effect was neglected. For these reasons, Quang et al. [18] adopted a path-integral technique, suggested by Sa-yakanit [19], to calculate the deep band-tail DOS.
208
A. Thongnum et al. / Solid State Communications 145 (2008) 207–211
His calculation took into account the interface roughness effect. However, his theory considered only an infinite barrier height model. The purpose of this work is to develop the Feynman pathintegral theory and to calculate the 2D DOS of finite QWs. We include the effect of the interface roughness and the finite barrier height, as suggested by experiments. The advantages of this semiclassical approach are that it allows analytic solutions, where the link between the model’s parameters and the solutions can be clearly established. The paper is organized as follows; in Section 2, a brief overview on the model of the interface roughness effect in QWs with a finite barrier height is presented. The path integration method pioneered by Sayakanit [20] is explained. The results, compared with those of existing infinite barrier height model, are discussed in Section 3. 2. Theoretical framework 2.1. Interface roughness effect in a finite barrier height QW The effects of the interface roughness and the finite barrier height of QWs are the main objective of our investigation. The random variation in the well width causes fluctuation in the confinement energy. The barrier height is also an important factor, as suggested by experiments [12–14]. It is dependent on the alloy composition of the QWs. In this work, we are dealing with a semiclassical approach, and thus real-space interactions are needed. A brief overview of the real-space model of the interface roughness effect in QWs with a finite barrier height is presented, starting from the definition of the local fluctuation potential [13] U (r) =
∂ ∆(r), ∂a
(1)
where is the confinement energy, a is the well width, and ∆(r) characterizes the local height of a rough interface at the plane position. Our task is to identify as a function of a, and to model ∆(r). In finite QWs with given values of the barrier height V0 and the well width q a, the characteristic wave vector is represented by kw =
2m/ h 2
¯ inside the well, and the wave vector in q the barriers is kb = 2m(V0 − )/ h¯ 2 , satisfying the boundary conditions at the interfaces. The effective mass m is taken to be the same value within the well and the barriers. The fluctuation potential is limited to the local fluctuation within the neighborhood of the confinement energy . The lowest subband in the even parity case [21] is kb kw a = tan . (2) kw 2 By solving Eq. (2) in terms of , the relationship between the confinement energy and the barrier height can be expressed as r m 2 + tan a = V0 . (3) 2 h¯ 2
Insert this into Eq. (1), we get the local fluctuation potential as U (r) = − a+
2 q
2 h¯ 2 m(V0 −)
∆(r).
(4)
This represents the local interaction between an electron and the QWs with rough interfaces. However, the actual interaction should come from the average over all possible configurations of the rough interfaces. The statistical averaged potential is in the form of an autocorrelation function W (r − r0 ) = hU (r)U (r0 )i,
(5)
where the angular brackets denote the average over all possible configurations of the interface roughness. Ando, Fowler and Stern [16] suggested that the autocorrelation of ∆(r) can be described by a Gaussian function as |r − r0 |2 0 2 h∆(r)∆(r )i = ∆ exp − , (6) L2 where ∆ is the average height and L is the fluctuation correlation length. It is an economic model as it contains only a few important parameters, and yet gives reasonable description. Substituting Eqs. (4) and (6) into Eq. (5), we get |r − r0 |2 0 W (r − r ) = ξ exp − , (7) L2 where the variance of the correlation function has the dimension of the energy square. For the finite barrier (FB) height model, ξ F B is defined as 2 2∆ . q ξF B = (8) 2 h¯ 2 a + m(V 0 −) It is readily seen that Eqs. (7) and (8) are in a more general form than that of the infinite barrier height. In the limit V0 → ∞, these equations reduce to the expressions for the infinite barrier (IB) heights [4], which are U (r) = − where = ξI B =
h¯ 2 π 2 ∆(r), ma 3 h¯ 2 π 2 , 2ma 2
h¯ 2 π 2 ∆ ma 3
(9)
and the variance of the correlation function is !2 .
(10)
2.2. Path-integral approach for the calculation of the 2D DOS In order to calculate the 2D electron density of states in finite quantum wells, we use the path-integral approach. The density of states per unit volume [19] can be expressed in terms of the diagonal elements of the averaged propagator as Z ∞ 1 iEt ¯ n(E) = dt G(0, 0; t) exp , (11) 2π h¯ −∞ h¯
A. Thongnum et al. / Solid State Communications 145 (2008) 207–211
where the averaged propagator is defined by Z i ¯ 2 , r1 ; t) = S . G(r D[r(τ )] exp h¯
(12)
The boundary conditions are taken to be r(0) = r1 , r(t) = r2 . In this approach, the system is based on the one-electron interaction, and hence the system’s action can be written as Z t Z tZ t m 2 i dτ r˙ (τ ) + dτ dσ W (r(τ ) − r(σ )), (13) S= 2 2h¯ 0 0 0 where W is defined in Eq. (7). In general, this problem cannot be integrated out exactly. We must resort to a variational method. We introduce a nonlocal harmonic trial action Z t Z Z m ω2 t t dτ r˙ 2 (τ ) − dτ dσ (r(τ ) − r(σ ))2 , (14) S0 = 2 2t 0 0 0 where the variational parameter ω is used to adjust the energy function subjected to a variational principle [19,20]. This method gave very good results for the polaron problem, compared with similar methods and experiments. The key for solving this problem is that the actual propagator can be approximated in terms of the trial action by using the variational path-integral method [19,20]. By keeping only the first-order term in the cumulant expansion, the first-order approximation to G¯ is obtained as i ¯ G(r2 , r1 ; t) = G 0 (r2 , r1 ; t) exp hS − S0 i S0 , (15) h¯ where G 0 (r2 , r1 ; t) is the zero-order approximation associated with the trial action. For the fluctuation potential with a Gaussian form, Eq. (7), the generalized d-dimensional density of states [20] was derived as !d d/2 Z ∞ m 1 ωt n(E) = dt 2π h¯ −∞ 2πih¯ t 2 sin ωt 2 i d ωt ωt × exp Et + cot −1 h¯ 2 2 2 ! 2 d/2 Z t L ξ −d/2 t dx j (x, ω) , (16) − 2 h¯ 2 4 0 where j (x, ω) =
ω(t−x) L2 ih¯ sin ωx 2 sin 2 + . 4 mω sin ωt 2
(17)
Next, let us consider a semiclassical approximation, it corresponds to taking the limit t → 0 of the integrand in Eq. (16). In another words, this limit retains only high-energy electrons (near the subband edge) in the DOS. Thus, we get 2 limt→0 j (x, ω) ≈ L4 . This leads to the following expression for the 2D DOS as Z ∞ m i ξ 2 −1 dt (it) exp n(E) = Et − t . (18) h¯ (2π h¯ )2 −∞ 2 h¯ 2
209
This is a general form and independent of ω. As we turn off the interaction, i.e. ξ = 0, the 2D DOS becomes a step function as in the usual noninteracting 2D electron gas. At this stage, we perform time integration of Eq. (18). By using formula [22] Z ∞ dt (it) p exp(−β 2 t 2 − iqt) −∞
√
=
π q q2 , D exp − √ p 2 p/2 β p+1 8β 2 β 2
(19)
where D p (x) is the parabolic cylinder function, the analytic expression of the 2D DOS can be written as 2m E −E 2 /4ξ n(E) = . (20) e D − √ −1 ξ (2π )3/2 h¯ 2 In this expression, the spin degeneracy is included. In order to obtain the 2D DOS in the low-energy tail, the asymptotic limit of the parabolic cylinder function [23] is considered. Finally, the band-tail of 2D DOS can be expressed in terms of the error function as E m . (21) n(E) = 1 + erf √ ξ 2π h¯ 2 3. Results and discussion Our results mainly come from Eqs. (8) and (21). As we attempt to give a clue to experiments, we use the parameters derived from the experimental results of Al0.3 Ga0.7 As/GaAs QW. The values of the confinement energy were given by Bastard [24] who assumed the barrier height of 224.5 meV. In order to make compatible comparison, we use the effective mass of m = 0.067m 0 [1], and the average height of one monolayer of 0.283 nm [4]. Fig. 1 shows the RMS of the variance of the fluctuation potential created by the interface roughness, derived from Eq. (8). It exhibits that the variance of the fluctuation potential is strongly dependent on the barrier height and the well width. The magnitude of fluctuation potential becomes very sensitive to the barrier height in thin QWs, i.e. low values of a. Moreover, it can be seen that the interface roughness has larger effects on the infinite barrier QWs than on the finite barrier QWs, as shown in Fig. 2. Next, the 2D DOS in Eq. (21) with various barrier heights are shown in Fig. 3. In this figure, the well width is set to 8 nm. It shows that the 2D DOS has a very long tail in the low-energy region. The explanation is that the interface roughness causes fluctuation in the confinement energy and leads to the existence of some electron states below the subband edge. This is the so-called localized states where the subband edge is fixed to zero. The correspondence between the localized states and the reduction of mobility is explained by the theory of the mobility gap [2]. From Fig. 3, it can be readily seen that the density of the localized states is increased with the increase in the barrier height. In other words, the density of the localized states is sensitive to the variation in the barrier height. This is consistent
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Fig. 1. Plot of the RMS of the fluctuation potential as a function of barrier height, as in Eq. (8), with three different values of the well width. The solid line represents the well width of 15 nm, the dashed line represents a = 8 nm, and the dashed–dotted line represents a = 3 nm.
Fig. 3. Plot of the 2D DOS as a function of electron energy, as in Eq. (21), with four different values of the barrier height. The well width is fixed at the value of 8 nm. The solid line represents the barrier height of 25 meV, the dashed–dotted line represents the barrier height of 100 meV, the dashed line represents the barrier height of 225 meV, and the dotted line represents the infinite barrier height, respectively.
Fig. 2. Plot of the RMS of the fluctuation potential as a function of the well width. The solid line represents the barrier height of 224.5 meV, and the dashed line represents the infinite barrier height.
with the earlier results of the RMS of the fluctuation potential, where the interface roughness causes stronger fluctuation in the confinement energy with increasing barrier height. Fig. 4 shows the 2D DOS of QWs with various well widths. The barrier height is fixed at the value of 224.5 meV. We observed that the density of the localized states strongly depended on the decrease in the well width. The 2D DOS resembles the step-like characteristic of the 2D noninteracting electron gas when the well width is equal to 15 nm. Hence, the interface roughness causes only weak fluctuation in wide QWs. In conclusion, the 2D DOS of QWs with the effect of the interface roughness and the finite barrier height is derived by the path-integral method with the semiclassical approximation. The interface roughness causes the fluctuation in the confinement energy, and leads to the localized states below the subband edge. The analytic expression of 2D DOS shows explicit dependence on the RMS of the fluctuation potential. The nature of the finite barrier height is to reduce the strength of the fluctuation potential and the density of the localized states,
Fig. 4. Plot of the 2D DOS in Eq. (21) with three different values of the well width and the fixed barrier height of 224.5 meV. The solid line represents the well width of 15 nm, the dashed line represents a = 8 nm, and the dashed–dotted line represents a = 3 nm.
compared with that of the infinite barrier height. Thus the influence of the interface roughness has stronger impact on the infinite barrier height QWs than on the finite barrier height QWs. In addition, we can neglect the effect of the interface roughness in wide QWs. Thus, the electron mobility should be higher in the case of the finite barrier height QWs than in the infinite ones. This is consistent with the experimental findings. Acknowledgement A. Thongnum expresses his gratitude to the Thai government for the supporting scholarship, under the University Developing Commissions (UDC) Grant, Ministry of University Affairs.
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