Electron Transmission Through Nonabrupt GaAs/AlxGa1-xAs Double-Barriers Subjected to an Electric Field

Electron Transmission Through Nonabrupt GaAs/Al_xGa_1-xAs Double-Barriers Subjected to an Electric Field

M. C. A. Lima,¹ G. A. Farias,² and V. N. Freire2^*

1: Departamento de Física, Universidade Federal do Ceará, Campus do Pici, Caixa Postal 6030,

Centro de Ciências Exatas, 60455-760 Fortaleza, Ceará, Brazil

2: Departamento de Física, Universidade Federal do Maranhão, Campus do Bacanga,

65080-420 São Luís, Maranhão, Brazil

Received May 2, 1997

It is shown that the existence of nonabrupt interfaces modify electric field effects on the electron transmission through a GaAs/Al_xGa_1-xAs double-barrier. When the applied electric intensity is 25 kV/cm, and the abrupt well and barriers are 100 Å wide, interfaces as thin as two GaAs lattice parameters are responsible for shifts at least of 10 meV in the electron tunneling resonance energies. The type of interface potential and electron effective mass description changes significantly theoretical results related to the electric field influence on the electron transmission properties.

I. Introduction

It is a great pleasure for us to contribute with this paper to the special number of the Brazilian Journal of Physics in honor to Prof. Dr. Roberto Luzzi on the special occasion of his sixtieth anniversary. One of the authors, V. N. Freire, had the opportunity to be formally one of his PhD students at the Universidade Estadual de Campinas (UNICAMP), in Campinas, Brasil.

Although considerably attention has been given to the study of carrier tunneling phenomena through double-barrier semiconductor heterostructures (DBSH),^(1-4) only recently it was shown that interface effects are important to be take into account for an improved description of their tunneling characteristics. In the sequential tunneling picture, interfaces are considered as an important mechanism to obtain smaller peak-to-valley ratios in the tunneling current.^(5-7) The influence of interface roughness on coherent tunneling of single and double-barriers was also shown to be relevant.^(8-10) However, in both the sequential and coherent tunneling picture, interface effects are generally investigated within the assumption of the existence of islands at an otherwise abrupt semiconductor interface.^(5-10)

Graded interface effects, i.e., smooth interfacial variations of the semiconductor alloy are also important in the coherent tunneling picture of single⁽¹¹⁾ and double barriers.⁽¹²⁾ In this case, interface widths as small as two lattice parameters (LP) can change considerably the tunneling behavior of carriers through semiconductor barriers. Since experiments have estimated that GaAs/Al_xGa_1-xAs interfacial regions have widths of the order of two lattice parameters,^(13-14) it was suggested that some disagreement between theory and experiment in DBSH tunneling may be related with nonabrupt interface characteristics.⁽¹²⁾ In fact, the interfaces of the quantum well in a GaAs/Al_xGa_1-xAs DBSH determine the energy of the transmission resonance peaks and an increase of the interfacial widths of the quantum well shifts the transmission resonances.⁽¹²⁾

Since the existence of nonabrupt interfaces reduces (enhances) the Stark shift of the electron ground state energy (first excited electron energy level) in a single nonabrupt GaAs/Al_xGa_1-xAs quantum well,⁽¹⁵⁾ interface effects modify the transmission properties of carriers through DBSH subjected to an electric field. The electron transmission through a single nonabrupt GaAs/Al_xGa_1-xAs barrier subjected to an electric field was shown to depend strongly on the interface potential and carrier effective-mass description.⁽¹⁶⁾ In this case, the electron transmission is reduced both by the electric field induced asymmetry and the existence of nonabrupt interfaces.⁽¹⁶⁾

The purpose of this paper is to investigate the influence of nonabrupt interfaces in the transmission properties of an electron through a GaAs/Al_xGa_1-xAs double-barrier subjected to an electric field. The GaAs/Al_xGa_1-xAs nonabrupt interface model is described in the Section II, the results of the transmission calculations of electrons through nonabrupt GaAs/Al_xGa_1-xAs double-barriers are discussed in the Section III. The Section IV closes this work with the presentation of the main conclusions.

II. The nonabrupt GaAs/Al_xGa_1-xAs double-barrier model

The GaAs/Al_xGa_1-xAs interfaces are described according the Al_X(z)Ga_1-X(z)As alloy variation in the interface regions. The double-barrier growth is in the z direction and X(z) is a function that describes the aluminum molar fraction variation in the interfacial regions. Actually, the interfacial alloy variation profile depends on the conditions and growth techniques of the semiconductor sample. Interfacial growth patterns influence the properties of semiconductor heterostructures, as can be infered from the role of growth interface interruptions on the luminescense properties of semiconductor quantum wells.⁽¹⁷⁾ Transmission properties of carriers through nonabrupt GaAs/Al_xGa_1-xAs single barriers were shown to depend on the interfacial growth pattern of the aluminum molar fraction.⁽¹⁸⁾ However, a linear variation for X(z) is assumed here because: (i) it allows to obtain analytical expressions for the transmission coefficient in the constant interfacial effective-mass approximation; (ii) it is the simplest nontrivial approximation for any kind of semiconductor alloy variation.

With the method first proposed by Freire, Auto, and Farias,⁽¹⁹⁾ the inter-related interface potential and electron effective mass are obtained from X(z). In this case, the electron effective mass is position dependent, m = m(z), and a kinetic energy operator with position dependent effective mass has to be employed. Several forms for the kinetic energy operator were proposed,⁽²⁰⁾ and while controversy exist about the best choice,^(20-26) the most used is that proposed originally by BenDaniel and Duke,⁽²⁷⁾ (-²/2) (d/dz) [m(z)]^-1 (d/dz). With this operator, the quantum behavior of an electron tunneling through a nonabrupt GaAs/Al_xGa_1-xA double-barrier subjected to an electric field (see Fig. 1), is described by the following Schrödinger-like equations:

Figure 1. GaAs/Al_xGa_1-xAs double-barrier potential and electron effective mass. In the case of the potential: the abrupt potential picture (dotted); the parabolic potential variation (solid); the linear potential variation (dashed); the constant potential approximation (dotted dashed) - inclined in the figure by the action of the electric field E_F. In the case of the electron effective mass: the abrupt effective-mass picture (dotted); the linear effective-mass variation (solid); the constant effective-mass approximation (dotted dashed).

In the preceding equations,

V_i,j(z) = Q_e é
ë V₀(z_i,z_j) + V₁(z_i,z_j) z +

V₂(z_i,z_j) z² ù
û ; (10)

M_i,j(z) = m^* é
ë m₀(z_i,z_j) + m₁(z_i,z_j) z ù
û ; (11)

V_F(z) = q E_F (z - z₁) ; (12)

V_MAX = Q_e é
ë e₁ + e₂ x² ù
û ; (18)

M_MAX = m^* é
ë m₁ + m₂ x ù
û ; (19)

M_MIN = m^*m₁ , (20)

where E is the electron energy, z_i the interfaces coordinates, x the Al_xGa_1-xAs aluminum molar fraction, q the electron charge, E_F the intensity of the applied electric field, Q_e the electron band offset, and m^* the free-electron mass. The parameters e_i (m_i) are obtained from experiments, and are related with the compositional dependence of the Al_xGa_1-xAs electron effective mass (energy gap) in the G direction.⁽¹⁹⁾ According to Adachi,⁽²⁸⁾e₁ = 1.155, e₂ = 0.37, m₁ = 0.067, and m₂ = 0.083.

The ensemble of equations (1-9) can be solved analitically by disregarding the linear spatial dependence of the interfacial electron effective mass - the so called constant interfacial effective-mass (CIEM) approximation,^{(19, 29)} and by imposing matching conditions on Y(z) and [m(z)]^-1 dY(z)/dz at z_i, i = 1, ... ,8 (see Fig. (1)). Beyond the CIEM approximation, only a numerical solution of the Eqs. (1-9) is possible. In this case, the numerical method of Ando and Itoh⁽³⁰⁾ was used to solve the equations.

III. Numerical Results and Discussion

The electron transmission coefficient T_pl is calculated considering the parabolic interface potential and the linear dependence of the interfacial electron effective mass. In the CIEM approximation, both the linear and the constant potential approximations are used to obtain the transmission coefficients T_lc and T_cc, respectively. A comparison between these coefficients and the transmission coefficient T_ab, calculated in the abrupt interface picture, shows the significance of the inclusion and a good modelling of interfaces to the description of electron transmission properties through GaAs/Al_xGa_1-xAs double-barrier subjected to an electric field.

Figure 1 depicts the potential and electron effective mass associated to GaAs/Al_xGa_1-xAs double-barriers subjected to an electric field. Both the abrupt well and barriers are 100 Å wide. The AlGaAs aluminum molar fraction content is x = 0.35, the maximum height of the first barrier is V_MAX = 270 meV, the electron band offset is Q_e = 0.6, and R_G (R_A) are GaAs (Al_0.35Ga_0.65As) regions. According experimental results,^{(13,17,31,32)} interfacial regions of Al_xGa_1-xAs grown in GaAs (R_GA regions) have a roughness degree stronger than interfacial regions of GaAs grown in Al_xGa_1-xAs (R_AG regions). While this interfacial asymmetry is impossible to be take into account within the abrupt interface picture, it is described here by considering simply that R_GA regions are wider than R_AG regions. The interface widths are: 2 LP for R_GA regions and 1 LP for R_AG regions in Figs. 2 and 3; 4 LP for R_GA regions and 1 LP for R_AG regions in Figs. 4 and 5. For all the interfaces, the applied electric field intensities are E_F = 0, 25, 55, and 85 kV/cm.

Figure 2. Transmission coefficients of electrons through single GaAs/Al_0.35Ga_0.65As double-barriers subjected to electric field intensities of 0 kV/cm, 25 kV/cm, 55 kV/cm, and 85 kV/cm, considering: abrupt interfaces, T_ab (short dashed); the parabolic potential and the linear electron effective mass in the interfacial regions, T_pl (solid); the linear potential and the constant electron effective mass approximation in the interfacial regions, T_cc (long dashed) - it is always almost identical to T_pl in the figure; the constant potential and the electron effective mass approximation in the interfacial regions, T_lc (dotted dashed). A band offset Q_e = 0.6, an aluminium molar fraction x = 0.35, and interface widths of 2 LP were used. The abrupt barriers and well widths were of 100 Å, and the electron energy was 0 < E < V_MAX.

Figure 3. Transmission coefficients of electrons through single GaAs/Al_0.35Ga_0.65As double-barriers subjected to electric field intensities of 0 kV/cm, 25 kV/cm, 55 kV/cm, and 85 kV/cm, considering: abrupt interfaces, T_ab (short dashed); the parabolic potential and the linear electron effective mass in the interfacial regions, T_pl (solid); the linear potential and the constant electron effective mass approximation in the interfacial regions, T_cc (long dashed) - it looks like T_pl in the low energy region of the figure; the constant potential and the electron effective mass approximation in the interfacial regions, T_lc (dotted dashed). A band offset Q_e = 0.6, an aluminium molar fraction x = 0.35, and interface widths of 2 LP were used. The abrupt barriers and well widths were of 100 Å, and the electron energy was V_MAX < E < 2.

Figure 4. Transmission coefficients of electrons through single GaAs/Al_0.35Ga_0.65As double-barriers subjected to electric field intensities of 0 kV/cm, 25 kV/cm, 55 kV/cm, and 85 kV/cm, considering: abrupt interfaces, T_ab (short dashed); the parabolic potential and the linear electron effective mass in the interfacial regions, T_pl (solid); the linear potential and the constant electron effective mass approximation in the interfacial regions, T_cc (long dashed) - it is always almost identical to T_pl in the figure; the constant potential and the electron effective mass approximation in the interfacial regions, T_lc (dotted dashed). A band offset Q_e = 0.6, an aluminium molar fraction x = 0.35, and interface widths of 4 LP were used. The abrupt barriers and well widths were of 100 Å, and the electron energy was 0 < E < V_MAX.

Figure 5. Transmission coefficients of electrons through single GaAs/Al_0.35Ga_0.65As double-barriers subjected to electric field intensities of 0 kV/cm, 25 kV/cm, 55 kV/cm, and 85 kV/cm, considering: abrupt interfaces, T_ab (short dashed); the parabolic potential and the linear electron effective mass in the interfacial regions, T_pl (solid); the linear potential and the constant electron effective mass approximation in the interfacial regions, T_cc (long dashed) - it looks like T_pl in the low energy region of the figure; the constant potential and the electron effective mass approximation in the interfacial regions, T_lc (dotted dashed). A band offset Q_e = 0.6, an aluminium molar fraction x = 0.35, and interface widths of 4 LP were used. The abrupt barriers and well widths were of 100 Å, and the electron energy was V_MAX < E < 2.

When E < V_MAX, one could see in Fig. 2 that not only the number of tunneling resonances can be reduced for suficiently high electric fields, but that this number depends also on the widths of the nonabrupt double-barrier interfaces. Three tunneling resonances occur in the absence of an electric field, two exist when E_F = 25 kV/cm, and none is present in the case of high electric fiels, E_F 85 kV/cm. Electron tunneling resonance energies are always smaller in the case of nonabrupt interfaces. High order resonances are more influentiated by the existence of nonabrupt interfaces than low order resonances. When E_F = 25 kV/cm , the first (second) tunneling resonance energy of T_pl are shifted toward low energies in comparison to that of T_ab by as much as 13 meV (22 meV). By comparing T_ab, T_pl, T_lc, and T_cc, it is observed that always T_lc T_pl. On the other hand, the constant interface potential and electron effective-mass approximation is good for interface modelling when E_F 55 kV/cm and E < V_MAX, since in this case T_pl T_cc. If E_F 85 kV/cm, the differences between abrupt and nonabrupt interfaces are unimportant.

When E > V_MAX, the role of interfaces on the transmission properties of electrons through nonabrupt Al_0.35Ga_0.65As double-barriers are important even when E_F = 85 kV/cm (see Fig. 3). If no electric field is applied, the first transmission peak of T_ab, T_pl, T_lc, and T_cc are very similar, and occur approximately at the same electron energy. When an electric field is applied, the positions of the transmission peaks of T_pl, T_lc, and T_cc are very close, but are shifted toward low energies in comparison with that of T_ab. The CIEM approximation smooths the transmission peaks, and in general T_pl > T_lc > T_cc.

When the widths of the R_GA regions increase, the transmission phenomena related with interfaces are enhanced. By comparing Figs. 4 and 5 (obtained with R_GA interfaces of 4 LP) with Figs. 2 and 3 (obtained with R_GA interfaces of 2 LP), one verify that the differences between the tunneling resonance energies of T_ab and T_pl have increased. In Fig. 4, the first (second) tunneling resonances of T_ab and T_pl have a difference of 20 meV (37 meV) when E_F = 25 kV/cm. When E > V_MAX (see Fig. (5)), the differences between the transmission (E > V_MAX) coefficients T_pl, T_lc, and T_cc increase. All of them are more smooth. Finally, the reduction of T_pl, T_lc, and T_cc with the intensity of the electric field is smaller when the interfaces are wider.

Our results indicate that the existence of GaAs/Al_xGa_1-xAs interfaces as thick as 2 LP in double-barriers samples could be responsible for considerable disagreement between experiments and theoretical results obtained in the abrupt interface picture. Since the best grown GaAs/Al_xGa_1-xAs samples have nowadays interface widths of the order of 2 LP,^(13,14) it seems always necessary to consider the existence of nonabrupt interfaces for a better description of GaAs/Al_xGa_1-xAs double-barrier properties.

IV. Concluding remarks

Experiments on electron tunneling through semiconductor barriers are very scarce. The technique used by Choi, Newman, and Iafrate,⁽³³⁾ carefully designed to observe coherent tunneling, may be used to probe the influence of interfaces on the electron energy tunneling resonances of GaAs/Al_xGa_1-xAs double-barrier heterostructures. On the other hand, the measurements of the reflection and transmission coefficients of ballistic two-dimensional electrons by Ying et al.⁽³⁴⁾ indicate that they are very sensitive to the barrier shape, principally nonabrupt interfaces. However, experimental evidences of interface effects on the electron transmission through GaAs/Al_xGa_1-xAs double-barriers subjected to an electric field were not obtained up to now. The transmission calculations performed here are now being used by the present authors in the calculation of current × voltage curves of GaAs/Al_xGa_1-xAs double-barrier heterostructures to be compared with experimental measurements.

Since the main interest in this work is the study of the role of interfaces on the transmission properties of undoped GaAs/Al_xGa_1-xAs double-barrier heterostructures, the calculations were performed without considering charge accumulation in the emitter and well regions. On the other hand, accumulation layer and interface effects must be take into account when high doped nonabrupt GaAs/Al_xGa_1-xAs heterostructures are studied. Recently, Freire et al ⁽³⁵⁾ showed that interfacial effects on the electron energy levels of doped nonabrupt GaAs/Al_xGa_1-xAs heterojunctions are more (less) important for high doping levels (small band bending widths). They have calculated energy level corrections associated to the existence of band bending and interface effects that are almost one order higher than those obtained by Stern and Das Sarma⁽³⁶⁾ in the case of wide accumulation layers and small tickness of the GaAs/Al_xGa_1-xAs interfaces.

To conclude, it was shown that interface effects have to be considered for a better description of the electron transmission through GaAs/Al_xGa_1-xAs double-barriers subjected to an alectric field. Errors at least of the order of 10 meV in the tunneling resonances energies may be frequent when comparison between experimental results obtained with actual semiconductor samples and theoretical calculations based in the abrupt interface picture are done.

Acknowledgments

The authors would like to acknowledge the partial financial support received from the Brazilian National Research Council (CNPq), the Ministry of Planning (FINEP), and the Science Funding Foundation of the Ceará State (FUNCAP) for the realization of this work.

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Publication Dates

• Publication in this collection
  14 June 1999
• Date of issue
  Dec 1997

History

• Received
  02 May 1997