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Spatially varying effective mass effects on energy level populations in semiconductor quantum devices
Superlattices and Microstructures, Vol. 32, No. 1, 2002 doi:10.1006/spmi.2002.1051 , Available online at http://www.idealibrary.com on
Spatially varying effective mass effects on energy level populations in semiconductor quantum devices N. I MAM , E. N. G LYTSIS , T. K. G AYLORD School of Electrical and Computer Engineering and Microelectronics Research Center, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. (Received 20 May 2002) For the analysis and design of semiconductor intersubband devices, accurate values for the Fermi energy and the subband electron population are needed. The effect of the positiondependent electron effective mass m ∗ (z) is commonly neglected in the determination of these two intersubband device parameters. This approach is nearly valid for single-well devices in the GaAs/Alx Ga1−x As material system. However, in multiple-coupled-well devices, the variable nature of the effective mass must be taken into account. In material systems other than the ones based on GaAs, failure to include the position-dependence of the electron mass may give rise to significant errors in the values of the Fermi energy and other device parameters such as the intersubband absorption coefficient. In this paper, the effects of m ∗ (z) on the Fermi level and the intersubband charge distribution are explored and quantified. Theoretical formulation for the intersubband Fermi energy and the subband electron distribution, with the inclusion of position-dependent electron mass, is presented. The eigenenergies of the intersubband structures are obtained by solving the single-band effective-mass Schroedinger equation using the argument principle method. The electron distribution and the Fermi energy are calculated using both the approximate method (m ∗0 ) and the rigorous formulation [m ∗ (z)], and the relative differences in the corresponding values are presented. It is demonstrated that these differences are small in the GaAs/Alx Ga1−x As material system, but can become very significant in other materials. In addition, the variable nature of carrier effective mass plays an important role in other types of devices such as interband quantum well photodetectors and lasers that employ optical transitions between the valence and the conduction bands. Electronic devices such as the resonant tunneling diode are also affected by the position-dependence of carrier mass and thus the results are applicable to both optoelectronic and electronic quantum devices. c 2002 Elsevier Science Ltd. All rights reserved.
Key words: intersubband, effective mass, fermi energy, absorption coefficient.
1. Background and motivation Since the pioneering proposal by Kazarinov and Suris [1], significant research efforts have been focused on exploiting intersubband transitions in quantum well superlattice structures for the purpose of achieving 0749–6036/02/070001 + 09
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c 2002 Elsevier Science Ltd. All rights reserved.
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light emission/lasing/detection. Superlattices are tailor-made semiconductors that are grown by molecular beam epitaxy (MBE) or metal organic chemical vapor deposition (MOCVD) and consist of alternating layers of semiconductor materials of similar lattice constants. The band structure and material parameters within each layer of the superlattice can be controlled over a relatively wide range. Where tens of micrometers were once the common size, superlattices formed by micro-fabrication technology now have feature sizes as small as a few nanometers. On these spatial scales, quantum confinement effects of the charge carriers are evident. In this regime, the dimensions of the structure become comparable with electron wavelength and charge transport is dominated by the quantum mechanical properties of the electron. Optical transitions between the conduction band states arising from quantum confinement in these structures can provide large oscillator strengths as was first observed by West and Eglash [2]. This revolutionary class of intersubband electron devices has found wide applications in optical and optoelectronic devices. A very successful intersubband device is the quantum cascade (QC) laser. Dual wavelength QC lasers with optically cascaded transitions, emitting simultaneously at two mid-infrared wavelengths (λ = 8 and 10 µm) have been reported [3]. The first GaAs quantum well infrared photodetector (QWIP) employing a photo-excited tunneling intersubband transition was fabricated by Levine [4]. After an accelerated developmental phase, multi-wavelength QWIPs with external voltage-tunable spectral response peaks [5] and very narrow-band detection QWIPs [6] have been developed. Other examples of intersubband devices are electron energy filters [7], resonant tunneling transistors [8], hot-electron spectrometers [9], and electron diffraction-grating switches [10, 11]. In order to optimize the performance of intersubband devices, an accurate knowledge of the intersubband absorption is crucial. Transition rates and optical absorption/gain coefficients for intersubband device operation are defined in terms of the electron population difference between the two states involved in the transition, which in turn is dependent on the Fermi energy [12]. The subband electron population n i of the ith energy state in a quantum heterostructure can be written as n i (z) =
k B T m ∗0 π h¯ 2
E F − E i ψi (z) 2 ln 1 + exp √L , kB T z
(1)
where T is the device temperature, L√z is the device length, and E F is the Fermi energy. The quantity E i (ψi (z) is dimensionless). The is the bound eigenenergy and ψi (z)/ L z is the corresponding eigenfunction P total subband population sheet density can be found as Nsubband = i n i L z , where the summation is over all bound states of the structure. After the evaluation of the number of electrons, the Fermi energy can be found from the charge neutrality condition, assuming either complete or incomplete ionization. It should be noted that in the above expression, a single value of the electron effective mass (m ∗0 ) is used for simplicity. However, the value of the electron mass changes from one hetero-layer to another, and this position-dependence [m ∗ (z)] is neglected in the conventional approach for calculating E F [13]. The effect of including m ∗ (z) can be either an increase or a decrease in the value of E F , depending on the material system. Through the values of Nsubband and E F , the variable nature of the effective mass m ∗ (z) also influences the conduction-band potential profile through space-charge effects [14]. In this paper, a numerical method for the exact determination of Fermi energy and intersubband absorption/gain coefficient by including position-dependent electron effective mass is presented. Section 2 outlines the theoretical formulation of the method. In Section 3, two quantum heterostructures are investigated: a wide and a narrow rectangular quantum well. A variable-mass-constant-potential (VMCP) material system is hypothesized where the errors in the approximate values of the Fermi energy E F , and the absorption coefficient α ji can become very large. GaAs/Alx Ga1−x As based systems with the same device parameters are also investigated and the corresponding errors in E F and αi j are presented. In GaAs/Alx Ga1−x As systems these errors are not significant. However, the results for the VMCP system clearly demonstrates the necessity of following the rigorous method of E F and α ji determination.
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2. Theoretical formulation The initial step in the present method is to find the eigenstates of the heterostructure. The eigenenergies of a given quantum structure are determined under the single-band effective-mass approximation using the argument principle method (APM) in conjunction with the transfer matrix approach as described in [15]. In quantum heterostructure modeling, it is common practice to employ the time-independent single-band effective-mass Schroedinger equation as the starting point. For the ith layer of the heterostructure with effective mass m i∗ , the Schroedinger equation can be written as 2m i∗ d2 ψ (z) + [E − E t − Vi − v(z)]ψi (z) = 0, i dz 2 h¯ 2
for z i−1 ≤ z ≤ z i
(2)
where ψi (z) represents the envelope wavefunction, E is the total electron energy, E t is the transverse electron energy, Vi is the potential energy in the ith layer, and v(z) is the externally applied bias (the quantity v(z) relates to the applied voltage bias u(z) by v(z) = −eu(z), where e is the magnitude of the electronic charge). The transverse energy E t is assumed to be zero for simplicity. The boundary conditions at the interface of the ith and the (i + 1)th layer are given by [16] ψi (z i ) = ψi+1 (z i ), and (1/m i∗ )(d/dz)ψi (z i ) = ∗ )(d/dz)ψ (1/m i+1 i+1 (z i ). For the unbiased case, v(z) = 0 and the wavefunction of each layer is the sum of two complex exponentials representing left and right propagating waves, ψi (z) = Ai exp[ jκi (z − z i−1 )] + Bi exp[− jκi (z − z i−1 )], where Ai and Bi are the amplitude coefficients and z i is a thickness constant that shifts the z-axis origin to the left-edge q of the ith region. The quantity κi represents the complex wavevector in
the ith layer and is given by κi = ± 2m i∗ (E − Vi )/h¯ 2 . In the case of an externally applied voltage bias, the envelope wavefunctions of the heterostructures are no longer linear combinations of exponentials. An analytical solution of the Schroedinger equation cannot be found for an arbitrary potential. In this case, the region is divided into a finite number of sufficiently small constant-potential steps. As can be seen, the positiondependent nature of the electron effective mass is taken into account in the formulation of the boundary conditions, and in the electron eigenfunction determination. However, this position dependence is ignored in the charge density formulation for eigenstates as calculated from the conventional expression of eqn (1). As such, there exists an inherent contradiction in the modeling of quantum heterostructure characteristics. The theoretical formulation presented here resolves this contradiction by incorporating the variable nature of the carrier effective mass. The carrier population in quantum heterostructure originates from two sources: electrons residing in bound states and the clasically-free propagating electrons. The most general expression for the le f t → right stream of propagating or continuum electron charge density can be written as Z ZZ ZZZ 2 dk z |ψl→r (z)|2 2 2 nl→r (z) = dk x dk y dk z f (E)|ψl→r (z)| = dk x dk y f (E). (3) 2π 2π 3 (2π )2 In a material system like GaAs, dk x dk y = kk dkk dθ and E = E k + E z = E z + h¯ 2 kk2 /2m ∗ (z). Here E is the total kinetic energy. After evaluating the integral over kk , some algebra, and transforming the integral over k z into an integral for E z , the propagating electron density nl→r (z) is Z d Ez [m ∗ (z)](3/2) k B T ∞ nl→r (z) = |ψl→r (z)|2 ln 1 + exp(E F −E z )/k B T √ . (4) √ 3 2 Ez 2 2π h¯ 0 The integration over kk can be done in any region since kk is conserved across heterostructure boundaries. However, the transverse energy E k is not conserved since the effective mass m ∗ (z) varies from one heterolayer to another. Therefore, in the above expression m ∗ (z) is a function of position and cannot merely be replaced by m ∗0 , where m ∗0 is the effective mass of the input left contact. Similar arguments hold for nr →l (z).
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Now let us examine a bound state. The bound state electron density for the ith eigenstate can be written as ZZ 1 2 m ∗ (z) n i (z) = dk x dk y f (E) |ψ(z, k zi )|2 = k B T ln 1 + exp(E F −E zi )/k B T |ψ(z, k zi )|2 , (5) 2 2 Lz (2π) π h¯ L z 2 /2m ∗ (z) = constant = ith eigenstate energy. In addition, from the normalization condiwhere E zi = h¯ 2 k zi R∞ −1 tion, L z −∞ |ψ(z, k zi )|2 dz = 1. The total number of electrons/area in the ith subband is Z ∞ Z ∞ kB T (E F −E zi )/k B T 1 ln 1 + exp m ∗ (z)|ψ(z, k zi )|2 dz. (6) ni = n i (z) dz = L z −∞ π h¯ 2 −∞
The eigenenergy E zi can be taken out of the integral since Rit has constant discrete values due to quantization ∞ in the z direction. From the normalization condition L z = −∞ |ψ(z, k zi )|2 dz, n i can be written as R∞ −∞ m ∗ (z)|ψ(z, k zi )|2 dz kB T (E F −E zi )/k B T R∞ ln 1 + exp ni = . (7) 2 π h¯ 2 −∞ |ψ(z, k zi )| dz For a piecewise constant m ∗ (z), n i becomes X ∗ kB T (E F −E zi )/k B T ni = ln 1 + exp m l ρl π h¯ 2 l R zl 2 zl−1 |ψ| dz ρl = R ∞ = probability of electron residing in the lth region. 2 −∞ |ψ| dz
(8)
(9)
Here zl − zl−1 = dl = width of the lth region. The above formulation for the subband electron population can be used to determine the Fermi energy E F according to the charge neutrality condition Z Z ∞ X + ND = n i (E F ) + nl→r (z, E zi , E F ) + nr →l (z, E zi , E F ) d E z dz , (10) E z −∞ i {z } | {z } | propagating
bound
+ where N D is the ionized donor sheet-density, and i n i is the total two-dimensional subband population for the bound states [12]. The last term in the above expression represents the charge density due to propagating carriers and can be significant if there are strongly localized quasi-bound states due to quantum interference effects in the heterostructure. The neutrality condition assumes that there are no acceptors or holes present in the quantum structure. This formulation can be used for both complete and incomplete ionization. The optical absorption coefficient for the transition between states i and j is [17]
P
α ji ∝ C|Z ji |2 (Ni − N j ).
(11)
Here C depends on fundamental constants and Z ji is the dipole matrix element between the ith and the jth states. Using the above formulation, the effect of energy-dependent effective mass can also be included following the method of Ref. [18].
3. Numerical results The method presented in Section 2 is applied to two quantum heterostructures: a wide and a narrow rectangular well. Two material systems are investigated. For demonstration purposes an artificial system is formulated, where the conduction band potential energy remains unchanged from one hetero-layer to another but the value of effective mass m i∗ changes. This system is labeled the VMCP system. Results for the well known GaAs/Alx Ga1−x As system are also presented for comparison. These two types of structures are illustrated in Fig. 1.
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Fig. 1. A, Schematic representation of quantum well structure in VMCP material system. The well thickness is divided into two equal parts with zero conduction band potential energy. The electron effective mass m ∗ is m ∗1 and m ∗2 for the two hetero-layers. The barrier regions have conduction band-edge energy Vb and electron effective mass m ∗b . B, Schematic representation of quantum well structure in GaAs/Alx Ga1−x As material system. The well thickness is divided into two equal parts with conduction band potential energies V1 and V2 . The electron mass m ∗ is m ∗1 and m ∗2 for the two hetero-layers. The barrier regions have conduction band energy Vb and electron effective mass m ∗b .
3.1. Narrow quantum well structure The narrow rectangular quantum well is 5.23 nm wide (W = 20 monolayers), divided into two equallength hetero-layers. In the VMCP system, the conduction band-edges of both layers are assumed to be equal. All other energies are measured from this band-edge taken as the reference point (V1 = 0). The barrier region has potential energy Vb = 347.91 meV and electron mass m ∗b = 0.10435m 0 (m 0 is the free electron mass), to be consistent with the GaAs based system. The electron mass of the 1st layer has a constant value m ∗1 . The mass m ∗2 of the 2nd layer is varied from 0.2 to 1.75m ∗1 . The large variation in the ratio of the effective mass (R = m ∗2 /m ∗1 ) is considered to magnify the effect of the variable electron mass on the intersubband device parameters. The well has two bound eigenstates for the entire range of R. For R = 1 and m ∗1 = 0.067m 0 , E 1 = 71.76 meV and E 2 = 274.46 meV, as calculated via APM [15]. As the ratio R is varied, the positions of the eigenstates, as well as the optical transition wavelength (λ21 ) changes (see Fig. 3). The well is assumed to have a constant donor doping profile of N D = 2.0 × 1018 cm−3 and the Fermi energy is calculated assuming complete ionization. The quantity E F is calculated for the entire range of R, using both the approximate eqn (1) and the rigorous eqn (8) methods. The percentage error in the values of E F is defined as 1E F = {E F [m ∗ (z)] − E F (m ∗0 )} × 100/E F [m ∗ (z)]. This parameter is plotted in Fig. 2A as a function of R. For these calculations m ∗1 is taken as a constant and four different values of m ∗1 = 0.063m 0 , 0.067m 0 , 0.071m 0 , and 0.075m 0 have been used, where m 0 represents the free electron mass. These particular values are in the range of electron masses for the GaAs system. In fact, 0.067m 0 and 0.063m 0 are the low temperature (77 K) and the room temperature (300 K) values of the electron effective mass in GaAs. Two plots are presented in Fig. 2A for m ∗1 = 0.063m 0 and 0.075m 0 . The plots for m ∗1 = 0.067m 0 and 0.071m 0 lie between these two. As can be seen from the plots, the error 1E F increases rapidly as the ratio R = m ∗2 /m ∗1 moves away from unity. Different values of m ∗1 do not make a large difference in the quantity 1E F . The optical absorption coefficient α21 is also calculated for this structure for the entire range of R. The dipole matrix element Z 21 has been calculated following the methodology described in [17]. The peak values of α P21 , calculated using the approximate and the rigorous methods, have been calculated and the error is presented as 1α P21 = {α P21 [m ∗ (z)] − α P21 (m ∗0 )}/α P21 (m ∗ (z)) and the results are plotted in percentage as a function of R (Fig. 4A). It is seen that the error 1α P21 remains small over the entire range of R even though the error 1E F varies significantly. For this narrow quantum well, almost all of the electrons (≈99%) reside in the first energy level (subband E 1 ). The second eigenlevel remains above the Fermi energy for both
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Fig. 2. A, Plot of 1E F as a function of the ratio R = m ∗2 /m ∗1 for a narrow quantum well in the VMCP system. The solid circles represent the values for the GaAs based system. B, Plot of 1E F as a function of the ratio R = m ∗2 /m ∗1 for a wide quantum well in the VMCP system. The solid circles represent the values for the GaAs based system.
methods of calculation for the entire range of R. The electron density decays exponentially as E 2 − E F increases, even though there is significant variation in the values of E F calculated by these two methods, the population difference N1 − N2 does not change significantly. The absorption coefficient is defined in terms of the population difference 1N = N1 − N2 and the dipole matrix element Z 21 . The variable nature of the effective mass is already included in the determination of Z 21 [17]. Therefore, the absorption coefficient α21 does not vary significantly between the approximate and the rigorous approach for this particular quantum structure. Similar calculations are also performed for the GaAs based system (Fig. 1B). The well thicknesses, the barrier region potential energy, and the barrier effective mass are identical for the two systems (VMCP and GaAs). However, the conduction band-edge energy of the two hetero-layers are determined according to the relation Vc = 0.77314x, and the effective mass is determined according to m ∗ = (0.067 + 0.083x)m 0 , where x is the aluminum fraction in Alx Ga1−x As [19]. The aluminum fraction of the 1st hetero-layer is zero and thus V1 = 0 and m ∗1 = 0.067m 0 . A few values of the ratio R were selected from its parameter space and the values of m ∗2 and V2 were calculated accordingly. The parameters 1E F and 1α P21 were calculated for the GaAs system and are represented as solid circles on Figs 1A and 2A. The optical transition wavelength λ21 is also represented in Fig. 3. As mentioned before, in the GaAs based system, the errors in the device parameter values are small even when the variable nature of the effective mass is not included in the calculations. The maximum 1α P21 error is 0.01%. 3.2. Wide quantum well structure The second quantum structure analyzed in this paper is a wide (W/2 = 20 monolayers) rectangular quantum well. The well is twice as thick as the well structure presented in Section 3.1. There are two hetero-layers with lengths of 5.23 nm each. The barrier region conduction band-edge energy and the barrier effective mass are left unchanged from the structure analyzed previously. Once again two material systems are considered: the VMCP system and the GaAs system. For the VMCP system, the carrier mass of the 1st layer m ∗1 is
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Fig. 3. The optical transition wavelength λ21 for the wide and narrow quantum wells as a function of R = m ∗2 /m ∗1 . The solid circles represent the values for the GaAs based system.
assumed constant where the value of m ∗2 is varied from 0.2 to 1.75m ∗1 . This well is wider and has three eigenenergy levels. For R = 1 and m ∗1 = 0.067m 0 , E 1 = 26.71 meV, E 2 = 106.69 meV, and E 3 = 236.37 meV, as calculated by APM [15]. The well is assumed to have a high donor doping level of N D = 3.0 × 1018 cm−3 and the Fermi energy is calculated assuming complete ionization. Figure 2B shows the quantity 1E F as defined in Section 1 as a function of R. As expected, the error due to not including the variable effective mass is significant for this material system. Figure 3 shows the optical transition wavelength λ21 for this structure. The energy level positions are such for the wide quantum well that both the 1st and the 2nd eigen-levels reside below the Fermi level E F and therefore have significant electron population. Hence it is expected that the error in the peak values of the absorption coefficient 1α P21 = {α P21 [m ∗ (z)] − α21 (m ∗0 )}/α P21 (m ∗ (z)) will be significant for this structure. This fact is evident from Fig. 4B, where 1α P21 is plotted as a function of R. Once again four different values of m ∗1 = 0.063m 0 , 0.067m 0 , 0.071m 0 , and 0.075m 0 have been used. Two plots are presented in Fig. 4B for m ∗1 = 0.063m 0 and 0.075m 0 . The plots for m ∗1 = 0.067m 0 and 0.071m 0 lie between these two. A GaAs based wide quantum well was also analyzed with the same hetero-layer thicknesses. The barrier region conduction band-edge potential and the barrier effective mass are left unchanged from the VMCP system. The 1st layer conduction band-edge energy V1 = 0 and m ∗1 = 0.067m 0 . The mass m ∗2 is varied according to a few points selected from the parameter space of R, and the conduction band-edge energy V2 is calculated accordingly [19]. The quantities 1E F , λ21 , and 1α P21 are represented as isolated points in Figs 2B, 3, and 4B. As expected, these quantities are small for the GaAs based system. The maximum 1α P21 error is 0.7%.
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Fig. 4. A, Plot of 1α P21 as a function of the ratio R = m ∗2 /m ∗1 for a narrow quantum well in the VMCP system. The solid circles represent the values for the GaAs based system. B, Plot of 1α P21 as a function of the ratio R = m ∗2 /m ∗1 for a wide quantum well in the VMCP system. The solid circles represent the values for the GaAs based system.
4. Discussion and summary To establish a complete theory for quantum device simulation, the inclusion of the variable nature of the carrier effective mass is essential. The need for an integrated analysis of the position-dependent carrier mass on (1) subband electron population, (2) Fermi energy, and (3) optical absorption/gain in intersubband quantum devices is demonstrated in the present work. A compact and efficient formulation has been presented for the determination of quantum device parameters dependent on the variable nature of the carrier mass. The sensitivity of the Fermi energy and the absorption coefficient to different material systems has been addressed. It is demonstrated that in certain material systems, the noninclusion of the variable carrier mass can lead to significantly erroneous results for device parameters such as the subband population, Fermi energy, and intersubband absorption/gain coefficient. Although the numerical results presented in this paper focus on n-type semiconductor intersubband optoelectronic devices, this method is applicable to a wide range of quantum devices such as p-type intersubband devices (in this case electrons are replaced by holes), interband photodetectors (valence band → conduction band transition), and lasers (conduction band → valence band transition). In these cases the same approach has to be followed for both electron and hole populations. The present approach is similarly applicable to semiconductor optoelectronic switches and modulators. The method is also pertinent for electronic quantum devices that modulate the quantum level carrier population such as in electronic resonant tunneling diodes (RTDs), semiconductor electronic switches and modulators. The work presented in this paper integrates the variable nature of the carrier effective mass with the determination of Fermi energy and absorption/gain coefficient, and thus possesses wide applicability. Beyond improving existing types of intersubband devices, the present method is clearly useful for the design of future novel semiconductor quantum devices. Acknowledgements—This work was supported by grant ERC-94-02723 from the National Science Foundation. E. N. Glytsis was supported by the State of Georgia’s Yamacraw project.
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References [1] R. F. Kazarinov and R. A. Suris, Sov. Phys. Semicond. 5, 797 (1971). [2] L. C. West and S. J. Eglash, Appl. Phys. Lett. 46, 1156 (1985). [3] C. Sirtori, A. Tredicucci, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Opt. Lett. 23, 1366 (1998). [4] B. F. Levine, J. Appl. Phys. 74, R1 (1993). [5] H. C. Liu, Opt. Eng. 33, 1461 (1994). [6] D. D. Coon, R. P. G. Karunasiri, and L. Z. Liu, Appl. Phys. Lett. 47, 89 (1985). [7] T. K. Gaylord, E. N. Glytsis, and K. F. Brennan, J. Appl. Phys. 65, 2535 (1989). [8] F. Capasso, K. Mohammed, and A. Y. Cho, IEEE J. Quantum Electron. 22, 1853 (1986). [9] A. F. J. Levi, J. R. Hayes, P. M. Platzman, and W. Wiegman, Physica B134, 480 (1985). [10] K. Furuya and K. Kurishima, IEEE J. Quantum Electron. 24, 1652 (1997). [11] G. N. Henderson, T. K. Gaylord, E. N. Glytsis, E. Anemogiannis, L. C. West, C. W. Roberts, and M. T. Asom, Microelectron. J. 24, 805 (1993). [12] S. L. Chuang, Physics of Optoelectronic Devices (Wiley, New York, 1995). [13] Y. Fu, N. Li, M. Karlsteen, and M. Willander, J. Appl. Phys. 87, 511 (2000). [14] N. Imam, E. N. Glytsis, and T. K. Gaylord, Superlatt. Microstruct. 29, 411 (2001). [15] E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, IEEE J. Quantum Electron. 29, 2731 (1993). [16] G. Bastard, Phys. Rev. B24, 5693 (1981). [17] N. Imam, E. N. Glytsis, and T. K. Gaylord, Superlatt. Microstruct. 28, 11 (2000). [18] J. S. Blakemore, J. Appl. Phys. 53, R123 (1982). [19] S. Adachi, J. Appl. Phys. 58, R1 (1985).
Spatially varying effective mass effects on energy level populations in semiconductor quantum devices N. I MAM , E. N. G LYTSIS , T. K. G AYLORD School of Electrical and Computer Engineering and Microelectronics Research Center, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. (Received 20 May 2002) For the analysis and design of semiconductor intersubband devices, accurate values for the Fermi energy and the subband electron population are needed. The effect of the positiondependent electron effective mass m ∗ (z) is commonly neglected in the determination of these two intersubband device parameters. This approach is nearly valid for single-well devices in the GaAs/Alx Ga1−x As material system. However, in multiple-coupled-well devices, the variable nature of the effective mass must be taken into account. In material systems other than the ones based on GaAs, failure to include the position-dependence of the electron mass may give rise to significant errors in the values of the Fermi energy and other device parameters such as the intersubband absorption coefficient. In this paper, the effects of m ∗ (z) on the Fermi level and the intersubband charge distribution are explored and quantified. Theoretical formulation for the intersubband Fermi energy and the subband electron distribution, with the inclusion of position-dependent electron mass, is presented. The eigenenergies of the intersubband structures are obtained by solving the single-band effective-mass Schroedinger equation using the argument principle method. The electron distribution and the Fermi energy are calculated using both the approximate method (m ∗0 ) and the rigorous formulation [m ∗ (z)], and the relative differences in the corresponding values are presented. It is demonstrated that these differences are small in the GaAs/Alx Ga1−x As material system, but can become very significant in other materials. In addition, the variable nature of carrier effective mass plays an important role in other types of devices such as interband quantum well photodetectors and lasers that employ optical transitions between the valence and the conduction bands. Electronic devices such as the resonant tunneling diode are also affected by the position-dependence of carrier mass and thus the results are applicable to both optoelectronic and electronic quantum devices. c 2002 Elsevier Science Ltd. All rights reserved.
Key words: intersubband, effective mass, fermi energy, absorption coefficient.
1. Background and motivation Since the pioneering proposal by Kazarinov and Suris [1], significant research efforts have been focused on exploiting intersubband transitions in quantum well superlattice structures for the purpose of achieving 0749–6036/02/070001 + 09
$35.00/0
c 2002 Elsevier Science Ltd. All rights reserved.
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Superlattices and Microstructures, Vol. 32, No. 1, 2002
light emission/lasing/detection. Superlattices are tailor-made semiconductors that are grown by molecular beam epitaxy (MBE) or metal organic chemical vapor deposition (MOCVD) and consist of alternating layers of semiconductor materials of similar lattice constants. The band structure and material parameters within each layer of the superlattice can be controlled over a relatively wide range. Where tens of micrometers were once the common size, superlattices formed by micro-fabrication technology now have feature sizes as small as a few nanometers. On these spatial scales, quantum confinement effects of the charge carriers are evident. In this regime, the dimensions of the structure become comparable with electron wavelength and charge transport is dominated by the quantum mechanical properties of the electron. Optical transitions between the conduction band states arising from quantum confinement in these structures can provide large oscillator strengths as was first observed by West and Eglash [2]. This revolutionary class of intersubband electron devices has found wide applications in optical and optoelectronic devices. A very successful intersubband device is the quantum cascade (QC) laser. Dual wavelength QC lasers with optically cascaded transitions, emitting simultaneously at two mid-infrared wavelengths (λ = 8 and 10 µm) have been reported [3]. The first GaAs quantum well infrared photodetector (QWIP) employing a photo-excited tunneling intersubband transition was fabricated by Levine [4]. After an accelerated developmental phase, multi-wavelength QWIPs with external voltage-tunable spectral response peaks [5] and very narrow-band detection QWIPs [6] have been developed. Other examples of intersubband devices are electron energy filters [7], resonant tunneling transistors [8], hot-electron spectrometers [9], and electron diffraction-grating switches [10, 11]. In order to optimize the performance of intersubband devices, an accurate knowledge of the intersubband absorption is crucial. Transition rates and optical absorption/gain coefficients for intersubband device operation are defined in terms of the electron population difference between the two states involved in the transition, which in turn is dependent on the Fermi energy [12]. The subband electron population n i of the ith energy state in a quantum heterostructure can be written as n i (z) =
k B T m ∗0 π h¯ 2
E F − E i ψi (z) 2 ln 1 + exp √L , kB T z
(1)
where T is the device temperature, L√z is the device length, and E F is the Fermi energy. The quantity E i (ψi (z) is dimensionless). The is the bound eigenenergy and ψi (z)/ L z is the corresponding eigenfunction P total subband population sheet density can be found as Nsubband = i n i L z , where the summation is over all bound states of the structure. After the evaluation of the number of electrons, the Fermi energy can be found from the charge neutrality condition, assuming either complete or incomplete ionization. It should be noted that in the above expression, a single value of the electron effective mass (m ∗0 ) is used for simplicity. However, the value of the electron mass changes from one hetero-layer to another, and this position-dependence [m ∗ (z)] is neglected in the conventional approach for calculating E F [13]. The effect of including m ∗ (z) can be either an increase or a decrease in the value of E F , depending on the material system. Through the values of Nsubband and E F , the variable nature of the effective mass m ∗ (z) also influences the conduction-band potential profile through space-charge effects [14]. In this paper, a numerical method for the exact determination of Fermi energy and intersubband absorption/gain coefficient by including position-dependent electron effective mass is presented. Section 2 outlines the theoretical formulation of the method. In Section 3, two quantum heterostructures are investigated: a wide and a narrow rectangular quantum well. A variable-mass-constant-potential (VMCP) material system is hypothesized where the errors in the approximate values of the Fermi energy E F , and the absorption coefficient α ji can become very large. GaAs/Alx Ga1−x As based systems with the same device parameters are also investigated and the corresponding errors in E F and αi j are presented. In GaAs/Alx Ga1−x As systems these errors are not significant. However, the results for the VMCP system clearly demonstrates the necessity of following the rigorous method of E F and α ji determination.
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2. Theoretical formulation The initial step in the present method is to find the eigenstates of the heterostructure. The eigenenergies of a given quantum structure are determined under the single-band effective-mass approximation using the argument principle method (APM) in conjunction with the transfer matrix approach as described in [15]. In quantum heterostructure modeling, it is common practice to employ the time-independent single-band effective-mass Schroedinger equation as the starting point. For the ith layer of the heterostructure with effective mass m i∗ , the Schroedinger equation can be written as 2m i∗ d2 ψ (z) + [E − E t − Vi − v(z)]ψi (z) = 0, i dz 2 h¯ 2
for z i−1 ≤ z ≤ z i
(2)
where ψi (z) represents the envelope wavefunction, E is the total electron energy, E t is the transverse electron energy, Vi is the potential energy in the ith layer, and v(z) is the externally applied bias (the quantity v(z) relates to the applied voltage bias u(z) by v(z) = −eu(z), where e is the magnitude of the electronic charge). The transverse energy E t is assumed to be zero for simplicity. The boundary conditions at the interface of the ith and the (i + 1)th layer are given by [16] ψi (z i ) = ψi+1 (z i ), and (1/m i∗ )(d/dz)ψi (z i ) = ∗ )(d/dz)ψ (1/m i+1 i+1 (z i ). For the unbiased case, v(z) = 0 and the wavefunction of each layer is the sum of two complex exponentials representing left and right propagating waves, ψi (z) = Ai exp[ jκi (z − z i−1 )] + Bi exp[− jκi (z − z i−1 )], where Ai and Bi are the amplitude coefficients and z i is a thickness constant that shifts the z-axis origin to the left-edge q of the ith region. The quantity κi represents the complex wavevector in
the ith layer and is given by κi = ± 2m i∗ (E − Vi )/h¯ 2 . In the case of an externally applied voltage bias, the envelope wavefunctions of the heterostructures are no longer linear combinations of exponentials. An analytical solution of the Schroedinger equation cannot be found for an arbitrary potential. In this case, the region is divided into a finite number of sufficiently small constant-potential steps. As can be seen, the positiondependent nature of the electron effective mass is taken into account in the formulation of the boundary conditions, and in the electron eigenfunction determination. However, this position dependence is ignored in the charge density formulation for eigenstates as calculated from the conventional expression of eqn (1). As such, there exists an inherent contradiction in the modeling of quantum heterostructure characteristics. The theoretical formulation presented here resolves this contradiction by incorporating the variable nature of the carrier effective mass. The carrier population in quantum heterostructure originates from two sources: electrons residing in bound states and the clasically-free propagating electrons. The most general expression for the le f t → right stream of propagating or continuum electron charge density can be written as Z ZZ ZZZ 2 dk z |ψl→r (z)|2 2 2 nl→r (z) = dk x dk y dk z f (E)|ψl→r (z)| = dk x dk y f (E). (3) 2π 2π 3 (2π )2 In a material system like GaAs, dk x dk y = kk dkk dθ and E = E k + E z = E z + h¯ 2 kk2 /2m ∗ (z). Here E is the total kinetic energy. After evaluating the integral over kk , some algebra, and transforming the integral over k z into an integral for E z , the propagating electron density nl→r (z) is Z d Ez [m ∗ (z)](3/2) k B T ∞ nl→r (z) = |ψl→r (z)|2 ln 1 + exp(E F −E z )/k B T √ . (4) √ 3 2 Ez 2 2π h¯ 0 The integration over kk can be done in any region since kk is conserved across heterostructure boundaries. However, the transverse energy E k is not conserved since the effective mass m ∗ (z) varies from one heterolayer to another. Therefore, in the above expression m ∗ (z) is a function of position and cannot merely be replaced by m ∗0 , where m ∗0 is the effective mass of the input left contact. Similar arguments hold for nr →l (z).
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Now let us examine a bound state. The bound state electron density for the ith eigenstate can be written as ZZ 1 2 m ∗ (z) n i (z) = dk x dk y f (E) |ψ(z, k zi )|2 = k B T ln 1 + exp(E F −E zi )/k B T |ψ(z, k zi )|2 , (5) 2 2 Lz (2π) π h¯ L z 2 /2m ∗ (z) = constant = ith eigenstate energy. In addition, from the normalization condiwhere E zi = h¯ 2 k zi R∞ −1 tion, L z −∞ |ψ(z, k zi )|2 dz = 1. The total number of electrons/area in the ith subband is Z ∞ Z ∞ kB T (E F −E zi )/k B T 1 ln 1 + exp m ∗ (z)|ψ(z, k zi )|2 dz. (6) ni = n i (z) dz = L z −∞ π h¯ 2 −∞
The eigenenergy E zi can be taken out of the integral since Rit has constant discrete values due to quantization ∞ in the z direction. From the normalization condition L z = −∞ |ψ(z, k zi )|2 dz, n i can be written as R∞ −∞ m ∗ (z)|ψ(z, k zi )|2 dz kB T (E F −E zi )/k B T R∞ ln 1 + exp ni = . (7) 2 π h¯ 2 −∞ |ψ(z, k zi )| dz For a piecewise constant m ∗ (z), n i becomes X ∗ kB T (E F −E zi )/k B T ni = ln 1 + exp m l ρl π h¯ 2 l R zl 2 zl−1 |ψ| dz ρl = R ∞ = probability of electron residing in the lth region. 2 −∞ |ψ| dz
(8)
(9)
Here zl − zl−1 = dl = width of the lth region. The above formulation for the subband electron population can be used to determine the Fermi energy E F according to the charge neutrality condition Z Z ∞ X + ND = n i (E F ) + nl→r (z, E zi , E F ) + nr →l (z, E zi , E F ) d E z dz , (10) E z −∞ i {z } | {z } | propagating
bound
+ where N D is the ionized donor sheet-density, and i n i is the total two-dimensional subband population for the bound states [12]. The last term in the above expression represents the charge density due to propagating carriers and can be significant if there are strongly localized quasi-bound states due to quantum interference effects in the heterostructure. The neutrality condition assumes that there are no acceptors or holes present in the quantum structure. This formulation can be used for both complete and incomplete ionization. The optical absorption coefficient for the transition between states i and j is [17]
P
α ji ∝ C|Z ji |2 (Ni − N j ).
(11)
Here C depends on fundamental constants and Z ji is the dipole matrix element between the ith and the jth states. Using the above formulation, the effect of energy-dependent effective mass can also be included following the method of Ref. [18].
3. Numerical results The method presented in Section 2 is applied to two quantum heterostructures: a wide and a narrow rectangular well. Two material systems are investigated. For demonstration purposes an artificial system is formulated, where the conduction band potential energy remains unchanged from one hetero-layer to another but the value of effective mass m i∗ changes. This system is labeled the VMCP system. Results for the well known GaAs/Alx Ga1−x As system are also presented for comparison. These two types of structures are illustrated in Fig. 1.
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Fig. 1. A, Schematic representation of quantum well structure in VMCP material system. The well thickness is divided into two equal parts with zero conduction band potential energy. The electron effective mass m ∗ is m ∗1 and m ∗2 for the two hetero-layers. The barrier regions have conduction band-edge energy Vb and electron effective mass m ∗b . B, Schematic representation of quantum well structure in GaAs/Alx Ga1−x As material system. The well thickness is divided into two equal parts with conduction band potential energies V1 and V2 . The electron mass m ∗ is m ∗1 and m ∗2 for the two hetero-layers. The barrier regions have conduction band energy Vb and electron effective mass m ∗b .
3.1. Narrow quantum well structure The narrow rectangular quantum well is 5.23 nm wide (W = 20 monolayers), divided into two equallength hetero-layers. In the VMCP system, the conduction band-edges of both layers are assumed to be equal. All other energies are measured from this band-edge taken as the reference point (V1 = 0). The barrier region has potential energy Vb = 347.91 meV and electron mass m ∗b = 0.10435m 0 (m 0 is the free electron mass), to be consistent with the GaAs based system. The electron mass of the 1st layer has a constant value m ∗1 . The mass m ∗2 of the 2nd layer is varied from 0.2 to 1.75m ∗1 . The large variation in the ratio of the effective mass (R = m ∗2 /m ∗1 ) is considered to magnify the effect of the variable electron mass on the intersubband device parameters. The well has two bound eigenstates for the entire range of R. For R = 1 and m ∗1 = 0.067m 0 , E 1 = 71.76 meV and E 2 = 274.46 meV, as calculated via APM [15]. As the ratio R is varied, the positions of the eigenstates, as well as the optical transition wavelength (λ21 ) changes (see Fig. 3). The well is assumed to have a constant donor doping profile of N D = 2.0 × 1018 cm−3 and the Fermi energy is calculated assuming complete ionization. The quantity E F is calculated for the entire range of R, using both the approximate eqn (1) and the rigorous eqn (8) methods. The percentage error in the values of E F is defined as 1E F = {E F [m ∗ (z)] − E F (m ∗0 )} × 100/E F [m ∗ (z)]. This parameter is plotted in Fig. 2A as a function of R. For these calculations m ∗1 is taken as a constant and four different values of m ∗1 = 0.063m 0 , 0.067m 0 , 0.071m 0 , and 0.075m 0 have been used, where m 0 represents the free electron mass. These particular values are in the range of electron masses for the GaAs system. In fact, 0.067m 0 and 0.063m 0 are the low temperature (77 K) and the room temperature (300 K) values of the electron effective mass in GaAs. Two plots are presented in Fig. 2A for m ∗1 = 0.063m 0 and 0.075m 0 . The plots for m ∗1 = 0.067m 0 and 0.071m 0 lie between these two. As can be seen from the plots, the error 1E F increases rapidly as the ratio R = m ∗2 /m ∗1 moves away from unity. Different values of m ∗1 do not make a large difference in the quantity 1E F . The optical absorption coefficient α21 is also calculated for this structure for the entire range of R. The dipole matrix element Z 21 has been calculated following the methodology described in [17]. The peak values of α P21 , calculated using the approximate and the rigorous methods, have been calculated and the error is presented as 1α P21 = {α P21 [m ∗ (z)] − α P21 (m ∗0 )}/α P21 (m ∗ (z)) and the results are plotted in percentage as a function of R (Fig. 4A). It is seen that the error 1α P21 remains small over the entire range of R even though the error 1E F varies significantly. For this narrow quantum well, almost all of the electrons (≈99%) reside in the first energy level (subband E 1 ). The second eigenlevel remains above the Fermi energy for both
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Fig. 2. A, Plot of 1E F as a function of the ratio R = m ∗2 /m ∗1 for a narrow quantum well in the VMCP system. The solid circles represent the values for the GaAs based system. B, Plot of 1E F as a function of the ratio R = m ∗2 /m ∗1 for a wide quantum well in the VMCP system. The solid circles represent the values for the GaAs based system.
methods of calculation for the entire range of R. The electron density decays exponentially as E 2 − E F increases, even though there is significant variation in the values of E F calculated by these two methods, the population difference N1 − N2 does not change significantly. The absorption coefficient is defined in terms of the population difference 1N = N1 − N2 and the dipole matrix element Z 21 . The variable nature of the effective mass is already included in the determination of Z 21 [17]. Therefore, the absorption coefficient α21 does not vary significantly between the approximate and the rigorous approach for this particular quantum structure. Similar calculations are also performed for the GaAs based system (Fig. 1B). The well thicknesses, the barrier region potential energy, and the barrier effective mass are identical for the two systems (VMCP and GaAs). However, the conduction band-edge energy of the two hetero-layers are determined according to the relation Vc = 0.77314x, and the effective mass is determined according to m ∗ = (0.067 + 0.083x)m 0 , where x is the aluminum fraction in Alx Ga1−x As [19]. The aluminum fraction of the 1st hetero-layer is zero and thus V1 = 0 and m ∗1 = 0.067m 0 . A few values of the ratio R were selected from its parameter space and the values of m ∗2 and V2 were calculated accordingly. The parameters 1E F and 1α P21 were calculated for the GaAs system and are represented as solid circles on Figs 1A and 2A. The optical transition wavelength λ21 is also represented in Fig. 3. As mentioned before, in the GaAs based system, the errors in the device parameter values are small even when the variable nature of the effective mass is not included in the calculations. The maximum 1α P21 error is 0.01%. 3.2. Wide quantum well structure The second quantum structure analyzed in this paper is a wide (W/2 = 20 monolayers) rectangular quantum well. The well is twice as thick as the well structure presented in Section 3.1. There are two hetero-layers with lengths of 5.23 nm each. The barrier region conduction band-edge energy and the barrier effective mass are left unchanged from the structure analyzed previously. Once again two material systems are considered: the VMCP system and the GaAs system. For the VMCP system, the carrier mass of the 1st layer m ∗1 is
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Fig. 3. The optical transition wavelength λ21 for the wide and narrow quantum wells as a function of R = m ∗2 /m ∗1 . The solid circles represent the values for the GaAs based system.
assumed constant where the value of m ∗2 is varied from 0.2 to 1.75m ∗1 . This well is wider and has three eigenenergy levels. For R = 1 and m ∗1 = 0.067m 0 , E 1 = 26.71 meV, E 2 = 106.69 meV, and E 3 = 236.37 meV, as calculated by APM [15]. The well is assumed to have a high donor doping level of N D = 3.0 × 1018 cm−3 and the Fermi energy is calculated assuming complete ionization. Figure 2B shows the quantity 1E F as defined in Section 1 as a function of R. As expected, the error due to not including the variable effective mass is significant for this material system. Figure 3 shows the optical transition wavelength λ21 for this structure. The energy level positions are such for the wide quantum well that both the 1st and the 2nd eigen-levels reside below the Fermi level E F and therefore have significant electron population. Hence it is expected that the error in the peak values of the absorption coefficient 1α P21 = {α P21 [m ∗ (z)] − α21 (m ∗0 )}/α P21 (m ∗ (z)) will be significant for this structure. This fact is evident from Fig. 4B, where 1α P21 is plotted as a function of R. Once again four different values of m ∗1 = 0.063m 0 , 0.067m 0 , 0.071m 0 , and 0.075m 0 have been used. Two plots are presented in Fig. 4B for m ∗1 = 0.063m 0 and 0.075m 0 . The plots for m ∗1 = 0.067m 0 and 0.071m 0 lie between these two. A GaAs based wide quantum well was also analyzed with the same hetero-layer thicknesses. The barrier region conduction band-edge potential and the barrier effective mass are left unchanged from the VMCP system. The 1st layer conduction band-edge energy V1 = 0 and m ∗1 = 0.067m 0 . The mass m ∗2 is varied according to a few points selected from the parameter space of R, and the conduction band-edge energy V2 is calculated accordingly [19]. The quantities 1E F , λ21 , and 1α P21 are represented as isolated points in Figs 2B, 3, and 4B. As expected, these quantities are small for the GaAs based system. The maximum 1α P21 error is 0.7%.
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Fig. 4. A, Plot of 1α P21 as a function of the ratio R = m ∗2 /m ∗1 for a narrow quantum well in the VMCP system. The solid circles represent the values for the GaAs based system. B, Plot of 1α P21 as a function of the ratio R = m ∗2 /m ∗1 for a wide quantum well in the VMCP system. The solid circles represent the values for the GaAs based system.
4. Discussion and summary To establish a complete theory for quantum device simulation, the inclusion of the variable nature of the carrier effective mass is essential. The need for an integrated analysis of the position-dependent carrier mass on (1) subband electron population, (2) Fermi energy, and (3) optical absorption/gain in intersubband quantum devices is demonstrated in the present work. A compact and efficient formulation has been presented for the determination of quantum device parameters dependent on the variable nature of the carrier mass. The sensitivity of the Fermi energy and the absorption coefficient to different material systems has been addressed. It is demonstrated that in certain material systems, the noninclusion of the variable carrier mass can lead to significantly erroneous results for device parameters such as the subband population, Fermi energy, and intersubband absorption/gain coefficient. Although the numerical results presented in this paper focus on n-type semiconductor intersubband optoelectronic devices, this method is applicable to a wide range of quantum devices such as p-type intersubband devices (in this case electrons are replaced by holes), interband photodetectors (valence band → conduction band transition), and lasers (conduction band → valence band transition). In these cases the same approach has to be followed for both electron and hole populations. The present approach is similarly applicable to semiconductor optoelectronic switches and modulators. The method is also pertinent for electronic quantum devices that modulate the quantum level carrier population such as in electronic resonant tunneling diodes (RTDs), semiconductor electronic switches and modulators. The work presented in this paper integrates the variable nature of the carrier effective mass with the determination of Fermi energy and absorption/gain coefficient, and thus possesses wide applicability. Beyond improving existing types of intersubband devices, the present method is clearly useful for the design of future novel semiconductor quantum devices. Acknowledgements—This work was supported by grant ERC-94-02723 from the National Science Foundation. E. N. Glytsis was supported by the State of Georgia’s Yamacraw project.
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