Optical and Quantum Electronics manuscript No. (will be inserted by the editor)
An Explicit FDM Calculation of Nonparabolicity Effects in Energy States of Quantum Wells Fahhad Alharbi
King Abdulaziz City for Science and Technology (KACST)
Received: date / Revised version: date
Abstract
An explicit finite difference method (FDM) to solve the non-
parabolic effective mass approximation of Schrodinger wave equation (SWE) for arbitrary quantum wells (QWs) is presented. The explicit nature of the presented method and its sparse matrices allow fast computation for energy states in QWs. The nonparabolicity effects are considered explicitly without iteration. This in turn results in faster and more stable calculations. The method is used to study the nonparabolicity effects in energy states and states overlapping in asymmetric AlGaAs/GaAs QWs.
[email protected]
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Fahhad Alharbi
1 Introduction
The industry of nano semiconductor structures has been advanced greatly over the past two decades in almost all aspects. This is mainly because of the continuous development of the crystal growth techniques, which allow the fabrication of high quality nano-structures such as thin QWs. To study and design QW structures, finding the quantized energy states and levels is an essential and a primary step. This can be done analytically and numerically. However, the numerical solutions are more practical and applicable. Therefore, different numerical methods have been used for this purpose such as variational method[1], exponential transfer matrix method[2, 3], transfer matrix method using Airy functions[4,5], finite element methods (FEM)[6], finite difference methods (FDM)[7], and the spectral grid method[8]. Each method has its advantages and disadvantages when compared to others. Two of the main comparison factors are the speed and the explicitness, which are very important for many applications like structure optimization and fast temporal analysis. With regard to nonparabolicity, most of these methods consider its effects (if considered) iteratively using self-consistent algorithms and usually a single state at a time. This lack of explicitness and comprehensiveness limits its use for extensive computations such as temporal analysis. In this paper, we present a simple and explicit FDM to solve the nonparabolic effective mass approximation of SWE for arbitrary multiple QWs. The problem is formulated as an eigenvalue problem and hence all the states are found at
Title Suppressed Due to Excessive Length
3
once. The nonparabolicity effects are usually neglected especially in wide and interband-transition QWs. However, the effects are more apparent and can’t be ignored in thin, small band gap, and intersubband-transition QWs. In the next section, a brief introduction of nonparabolicity origins, effects, and theoretical treatment is presented. Then, the FDM is derived and applied. The last section is the conclusion.
2 The Nonparabolicity Usually, the dispersion relation is assumed parabolic, where the energy (E) is related to the wave number (k) (i.e. the momentum) as follows: E=
¯ 2 k2 h 2m∗
(1)
where h ¯ is the normalized Plank constant and m∗ is the effective mass. m∗ is usually calculated directly from the above equation by: 1 d2 E 1 = 2 2 ∗ m ¯ dk h
(2)
This approximation is highly valid and accurate. However, it is not the most comprehensive dispersion relation. The complete relation is a power-series expansion in k as follows: E=
∞
ai k i
(3)
i=0
where a0 is the energy at k = 0. a0 is constant, and can take any value depending on the choice of the reference energy. Also, it is known that for crystalline structures with inversion symmetry, the dispersion relation is evenly symmetric in any direction regardless of
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Fahhad Alharbi
how complex the structure is. Then, only the even-power components of k contribute to the expansion, and hence Equation (3) is reduced to: E=
∞
a2i k 2i
(4)
i=0
Generally, semiconductor crystals in QWs are intrinsically inversion asymmetric. However, by decomposition, the asymmetric portion is very small in comparison with the symmetric part. There, inversion asymmetric effects can be ignored [21,22]. The low order components are the most significant, therefore, the above equation is approximated further to: E ≈ E +
¯ 2 k2 h 1 − γk 2 ∗ 2m
(5)
Moreover, k 2 ∝ (E − V ). So, Equation (5) becomes [9–11]: E ≈ E +
¯ 2 k2 h (1 − α (E − V )) 2m∗
(6)
where α is the nonparabolicity factor. The nonparabolicity is not a single-cause effect. Actually, many factors contribute to α such as band intermixing [11–13], QW subbands’ confinement [11], and carriers’ densities [9,13] and their degrees of influence are varied depending on the materials and the structure. For example, in small band-gap materials, band intermixing plays an important role in enhancing the nonparabolicity effects. Also, the subbands’ confinements increase the nonparabolicity in QWs when compared to bulk semiconductors. Theoretically, different models were used to account for the nonparabolicity effects. For example, Hiroshima suggested that the conduction band
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5
nonparabolicity can be treated fairly well using Luttinger-Kohn equation for slowly varying lattice periodic perturbation [10]. Also, Bastard’s two-band model correctly accounts for conduction band nonparabolicity in III-IVmaterial QWs [11]. In this paper, theoretical derivation of the nonparabolicity factor is beyond its scope and will not be addressed. An empirical formulation of the nonparabolic parameter will be used instead[7,9–11,14].
3 The Explicit FDM
The electronic states in QWs are described by the envelopes of the Bloch wave function. These are the solutions of the effective mass approximation of SWE, which is: −¯ h2 d 2 dz
1 dφ(z) m∗ dz
+ V (z)φ(z) = Eφ(z)
(7)
where V (z) is the QW potential in the growth direction, which is assumed to be the z-axis. At the beginning, the parabolic model is assumed to derive the FDM and the nonparabolic effect will be imposed at the end. By discretizing the z-axis into N points that are uniformly separated by a distance, d, SWE for each point becomes: −¯ h2 d 2 dz
1 dφi m∗i dz
+ Vi φi = Eφi
(8)
After that, the wave functions of all the points can be combined in one vector φ as follows: φ = φ1 φ2 · · · φN −1 φN
T (9)
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Fahhad Alharbi
Equation (7) can now be represented by a matrix-coefficient ordinary differential equation (ODE). To do so, the 2nd derivative and the potential must be represented in matrix forms. The potential matrix is found in a straightforward way. It is just a diagonal matrix where:
Vii = V (zi )
(10)
The 2nd derivative matrix D is found using a combination of 1st and 2nd order derivative approximations. Higher order approximations can also be used to find it. However, it was found that the effect of including them is very small and yet requires more computational time. These approximations are obtained by using Taylor series expansions where the boundary conditions and the fact that the envelope function is bounded in space are applied. The used boundary conditions are:
φi− = φi+
(11)
1 dφi− 1 dφi+ = ∗ m∗i− dz mi+ dz
(12)
and
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7
After some algebra, D is found to be tri-diagonal and a function of the effective masses at each point as follows: ⎡ ⎢ c1 ⎢ ⎢ ⎢b ⎢ 2 ⎢ ⎢ ⎢0 ⎢ D=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
¯2 h ci = 2 4d
c2 b3 ..
.
−¯ h2 4d2
1 1 + ∗ ∗ mi mi−1
−¯ h2 4d2
1 1 + ∗ m∗i+1 mi
(13)
(14)
1 2 1 + ∗+ ∗ m∗i+1 mi mi−1
and di =
⎥ ⎥ ⎥ .. ⎥ d2 . ⎥ ⎥ ⎥ .. .. ⎥ . c3 . ⎥ ⎥ ⎥ .. .. .. . . . 0 ⎥ ⎥ ⎥ ⎥ .. .. . . cN −1 dN −1 ⎥ ⎥ ⎥ ⎦ 0 b N cN
d1 0
where bi =
⎤
(15)
(16)
By applying these matrices and the φ vector in equation (7), the following matrix form is obtained: D + V φ = Eφ
(17)
The above equation is an eigenvalues equation. Therefore, the values of the energy levels (i.e. E) and their corresponding wave functions can be found directly and explicitly by solving this eigenfunction. The energy levels (En ) are just the eigenvalues and the wave functions are the corresponding normalized eigenvectors (φn ). Up to this point, there is nothing new and all the above formulation was presented before by many like Nag [14] and Burt [15]. The new contribution
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Fahhad Alharbi
is the explicit inclusion of the nonparabolicity effects. To do so, we need to modify the derived FDM. Equation (6) can be rewritten in the following form: E = E +
¯ 2 k2 h 2m∗ (E)
(18)
where 1 1 = ∗ (1 − α (E − V )) m∗ (E, z) m
(19)
The above equation is then inserted in the definitions of the coefficients of the 2nd derivative matrix D (Eqs. (13-16)). So, D becomes: D = A − EB + C
(20)
A, B, and C have the same tri-diagonal form of D; but with different elements. A contains the parabolic effective mass effects and equals the previously defined D (Eqs. (13-16)). B accounts for the energy level effects and its elements are:
−¯ h2 αi αi−1 = + ∗ 4d2 m∗i mi−1 αi+1 h2 ¯ 2αi αi−1 B + ∗ + ∗ ci = 2 4d m∗i+1 mi mi−1 bB i
and dB i =
−¯ h2 4d2
αi+1 αi + ∗ ∗ mi+1 mi
(21) (22)
(23)
α is calculated using [7,10,14,11,9]: α=
β Eg
2 m∗ 1− m0
(24)
where β=
1 + 4x + 2x2 1 + 5x + 2x2
(25)
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9
and x=
Eg
(26)
Eg is the energy gap and is the split-off potential. The last matrix, C accounts for the system potential effects and its elements are defined as: bC i =
cC i =
¯2 h 4d2
−¯ h2 4d2
αi Vi αi−1 Vi−1 + m∗i m∗i−1
(27)
2αi Vi αi−1 Vi−1 αi+1 Vi+1 + + m∗i+1 m∗i m∗i−1
(28)
and dC i
−¯ h2 = 4d2
αi+1 Vi+1 αi Vi + ∗ m∗i+1 mi
(29)
Using Equation (20), Equation (17) becomes: −1 I+B A + C + V φ = Eφ
(30)
Again, the above equation is an eigenvalue equation which are solved explicitly. The eigenvalues are the energy levels (En ) and the normalized eigenvectors are the corresponding wave functions (φn ).
4 Results and discussion
In this section, some results obtained using the presented method are discussed to study the effects of the nonparabolicity in AlGaAs quantum wells. The properties of the alloy are extracted by interpolation of published results [18–20].
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Fahhad Alharbi
In the first analysis, a single QW, which consists of GaAs well buried in Al0.3 Ga0.7 As is studied. Energy levels and their corresponding transitions are calculated with and without the nonparabolicity, where the width of the well is varied from 5 to 15 nm. The obtained results are shown in Figure-1.
(a)
0.15 0.1 0.05 0 5
10
(b)
0
Energy (eV)
Energy (eV)
0.2
15
−0.05 −0.1 −0.15
(d)
−0.2 5
QW width (μ m) (c)
10
15
QW width (μ m)
0 −0.05
Energy (eV)
Energy (eV)
e2−lh2
−0.1 −0.15 −0.2 5
10
15
QW width (μ m)
1.7 e2−hh2
e3−hh1 e3−lh1
1.6 e1−lh1 1.5
e1−hh1
1.4 5
10
15
QW width (μ m)
Fig. 1 Nonparabolicity effects in energy states and interband transitions of Al0.3 Ga0.7 As/GaAs single QW. (a) Electron states above conduction band edge (b) Heavy hole states below valence band edge (c) Light hole states below valence band edge (d) Some transition energies. The solid line is for parabolic model and the crossed line is for nonparabolic model
It is clear that the nonparabolicity reduces the energy for high order levels as expected [16]. Figure-1d shows that the transitions are actually
Title Suppressed Due to Excessive Length
11
less than predicted by the parabolic model. Some of the differences are about 20 meV. In the second analysis, the dipole moments for AlGaAs QWs are found for two structures used by Kocinac et al [17]. The first QW is 5.5/4.5 nm GaAs/ Al0.16 Ga0.84 As well in Al0.39 Ga0.61 As bulk. The second QW is 5.0/1.0/5.0 nm GaAs/Al0.41 Ga0.59 As/Al0.16 Ga0.84 As in Al0.41 Ga0.59 As bulk. Table-1 shows the obtained transitional and permanent dipole moments for the lowest three electron states and the energy differences between them. The permanent dipole moment arises from the spatial separation of the electron and hole in the growth direction, perpendicular to the QW planes while the transitional dipole moment is the electric dipole moment associated with the transition between the two states. The transitional dipole moments are almost the same. However, there are slight differences in the obtained permanent dipole moments and energies. This is because of the nonparabolicity. As stated in the first analysis, the nonparabolicity reduces the energy level of higher order states. The last analysis is for a biased graded quantum well where the Al composition is varied over 4 well layers buried in Al0.4 Ga0.6 As. Al compositions of the layers are 0.0, 0.1, 0.2, and 0.3 and their widths are 3, 2, 2, and 2 nm respectively. Figure-2 shows the obtained states (the first two electron states and the first heavy and light hole states) using the parabolic and nonparabolic models for -30, 0, and 30 kV /cm bias voltages. Table-2 shows the obtained subband energies, overlap integrals, and dipole moments.
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Fahhad Alharbi
Table 1 Dipole moments and energy differences in the two QW studied in the second analysis 1st QW (NP
2nd
(TMM [17])
FDM)
(TMM [17])
(NP FDM)
r12 (nm)
1.82
1.91
1.26
1.42
r13 (nm)
0.57
0.51
1.13
1.08
r23 (nm)
2.93
3.03
2.33
2.55
|r22 − r11 | (nm)
1.79
2.28
5.24
5.08
|r33 − r11 | (nm)
1.94
2.58
1.70
1.46
|r33 − r22 | (nm)
0.15
0.29
3.54
3.62
E21 (meV )
120.0
120.1
118.0
105.4
E31 (meV )
241.0
203.5
228.0
193.8
Parameter
1st
QW
QW
2nd
QW
This analysis illustrates the effect of the nonparabolicity in interband overlapping (i.e. interband transition rate). OLe2lh1 is severely changed. This changes the interband transition accordingly. In general, this effect depends on the shape of QW and states confinement. In the presented analysis, the second electron state energy is shifted down because of the nonparabolicity and because of the shape of the QW. Also, its wave function is confined more toward the left as shown in Figure-2.
5 Conclusion An explicit FDM to solve the effective mass approximation of SWE is formulated and presented. The presented method were tested against many
Title Suppressed Due to Excessive Length
−30 kV/cm
13
0 kV/cm
30 kV/cm
Energy (eV) Parabolic
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−1.5
−10 −5 0 5 z−axis (nm)
10
−10 −5 0 5 z−axis (nm)
10
Energy (eV) Nonparabolic
0.5
−10 −5 0 5 z−axis (nm)
10
−10 −5 0 5 z−axis (nm)
10
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−1.5
−10 −5 0 5 z−axis (nm)
10
−10 −5 0 5 z−axis (nm)
10
Fig. 2 The first two electron states and the first heavy and light hole states of an AlGaAs QW using the parabolic and nonparabolic models for -30, 0, and 30 kV /cm bias voltages
published results and exhibited excellent agreement and fast computation. Considering nonparabolicity explicitly shall enhance and improve the speed of many algorithms such as shape optimization and temporal analysis.
References 1. T Ando et al, IEEE Journal of Quantum Electronics, 38, (2002) 1372. 2. AK Ghatak, T Thyagarajan, and MR Shenoy, IEEE Journal of Quantum Electronics, 24, (1988) 1524. 3. B Jonsson and ST Eng, IEEE Journal of Quantum Electronics, 26, (1990) 2025.
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Fahhad Alharbi
Table 2 The obtained states’ energies, overlapping integrals, and dipole moments in the QW studied in the third analysis (V is in kV /cm V = −30
V = −30
V
Par
Non Par
Par
Non Par
Par
Non Par
Elh1 (eV )
-1.4883
-1.4887
-1.4945
-1.4949
-1.5002
-1.5006
Ehh1 (eV )
-1.4541
-1.4542
-1.4617
-1.4618
-1.4690
-1.4690
Ee1 (eV )
0.0983
0.0991
0.0926
0.0934
0.0864
0.0872
Ee2 (eV )
0.2316
0.2214
0.2347
0.2226
0.2355
0.2222
OLe2hh1
0.2645
0.2631
0.2047
0.1929
0.1173
0.1064
OLe1hh1
0.9334
0.9331
0.9510
0.9516
0.9633
0.9644
OLe2lh1
0.0793
0.0634
0.0010
0.0197
0.0829
0.1066
OLe1lh1
0.9963
0.9961
0.9999
1.0000
0.9962
0.9961
re1e2 (nm)
1.661
1.705
1.681
1.695
1.639
1.683
Parameter
=
00
V
=
00
V
=
30
4. DC Hutchings, Applied Physics Letters, 55, (1989) 1082. 5. AG Ghatak, IC Goyal, and RL Gallawa, IEEE Journal of Quantum Electronics, 26, (1990) 305. 6. K Nakamura et al, IEEE Journal of Quantum Electronics, 25, (1989) 889. 7. P Harrision, Quantum Wells, Wires, and Dots (John Wiley and Sons Ltd, Chichester 2000). 8. QH Liu, C Cheng, and HZ Massoud, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 23, (2004) 1200. 9. A Raymond, JL Robert, and C Bernard, Journal of Physics C: Solid State Physics, 12, (1979) 2289. 10. T Hiroshima and R Lang, Applied Physics Letters, 49, (1986) 456.
V
=
30
Title Suppressed Due to Excessive Length
15
11. DF Nelson, Physical Review B, 35, (1987) 7770. 12. I Kobori, T Nomura, and T Ohyama, in Proceedings of the International Conference on Microwave and Millimeter Wave Technology ICMMT’98, (1998) 269. 13. J Hader, JV Moloney, and SW Koch, IEEE Journal of Quantum Electronics, 35, (1999). 14. BR Nag, Physics of Quantum Well Devices (Kluwer Academic Publishers, Dordrecht 2000). 15. MG Burt, Journal of Physics: Condensed Matter, 5, (1993) 4091. 16. DF Nelson, Physical Review B, 35, (1987) 7770. 17. S Kocinac, Z Ikonic, and V Milanovic, Optics Communications, 140, (1997) 89. DF Nelson, Physical Review B, 35, (1987) 7770. 18. B Saleh and M Teich, Fundamental of Photonics (John Wiley and Sons, New York 1991). 19. J Singh, Physics of Semiconductor and Their Heterostructure (McGraw Hill, New York 1993). 20. H Wu, Numerical Analysis of the Nonlinear Optical Properties in Asymmetric Quantum Well Structures (University of Colorado, Boulder 1994). 21. JP Eisenstein et al, Physical Review Letters, 53, (1984) 2579. 22. M Dressel and G Gruner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge University Press, Cambridge 2002).
