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Conduction-band nonparabolicity and inter-subband absorption profile in quantum wires
Solid State Communications, Vol. 75, No. 3, pp. 259-261, 1990. Printed.in Great Britain.
0038-1098/90 $3.00 + .00 Pergamon Press plc
CONDUCTION-BAND NONPARABOLICITY AND INTER-SUBBAND ABSORPTION PROFILE IN QUANTUM WIRES D. Chattopadhyay and P.C. Rakshit Institute of Radio Physics and Electronics, 92 Acharya Prafulla Chandra Road, Calcutta 700009, India
(Received 15 January 1990, in revised form 9 March 1990 by B. Miihlschlegel) The conduction-band nonparabolicity effects on the inter-subband absorption in quantum wires are investigated. The nonparabolicity is found to shift the absorption peak energy, broaden it, and reduce its amplitude. The effects are larger for In0.53Ga047As quantum wire structures than those for GaAs structures. The peak asymmetry is significant at 300 K. THE INTRIGUING electrical and optical properties expanded to fourth order in the wavevector k, can be of quantum wires (QW's) have caused much interest in written as [9] the one-dimensional (1D) electronic systems in such structures [1-5]. In this communication, we study the E(k) = %k 4 + 2m, + (2% + flo)(k 2 + ~ ) k 2 effects of the conduction-band nonparabolicity on the h2 absorption of electromagnetic radiation producing + + AD + (2 0 + transitions of electrons from the ground subband to the first excited subband in a QW. In a parabolic + Cto(k~ + k~). (1) band, the inter-subband energy AEI does not depend on the longitudinal wave vector kx in the QW, so that Here h is Dirac's constant, ml is the effective mass of for small line widths the absorption is a maximum the QW material, % and fl0 are the nonparabolicity when the energy of the incident photon equals AE~. parameters, and kx, Icy, kz are the components of k The nonparabolicity of the conduction band changes along x, y, and z directions. For simplicity, the barrier potentials for y and z the inter-subband energy difference at k~ = 0 and causes AE1 to decrease with increasing k~, thus influ- directions are taken to be the same here. For too encing the absorption profile. We study such effects narrow wires, the quantum mechanical problem is here for GaAs and In0.53Ga0.47As QW's. Our results nonseparable due to the mixing of the y and z terms are potentially useful in interpreting absorption data [10]. For large barrier heights and relatively wide and in modelling performances of devices based on wires, however, the condition can be relaxed to investigate the nonparabolicity effects. The exact treatment QW's. Nonparabolicity effects in a two-dimensional of mixing including nonparabolicity will modify the electron system in a quantum well have been studied results slightly quantitatively, and is not attempted [6] and used to interpret experimental data [7]. The here in view of the complexity involved. Thus the subeffects in a 1D system need a separate investigation band energies at kx = 0, obtained from equation (1), because the density of states and the nonparabolicity are h2 are altered due to the double confinement in the QW. Strictly speaking, the problem needs to be solved E,m(O) = 2m--"~(k2~" + ~m) + %(kz~ + k4ym) numerically considering the self-consistent filling of + (2% + flo)k2.~m • (2) the confined states with electrons. But such treatments are tedious computationally. As our basic aim here is The remaining terms in equation (1) give the longito provide an insight into the nonparabolicity effects, tudinal dispersion: we rather concentrate on an analytic approach in this h 2k x2 4 initial work. Enm(kx) = Enm(O) q- ~ q- %kx We assume a QW of square cross-section of confinement length L along y and z directions, and take 2 + (2% + flo)(kz2,, + kym)~. (3) the longitudinal direction along x. Neglecting the spin splitting [8], the bulk conduction-band energy E, In the barrier material, the nonparabolicity can be 259
ABSORPTION
260
PROFILE
IN QUANTUM
WIRES
Vol. 75, No. 3
ignored and the band-edge effective mass m2 can be used but for very narrow QW’s [8]. With the boundary conditions that the wave function and its derivative divided by the effective mass are continuous across the interfaces, the confinement wave vectors kym and kZ, are obtained by solving the Schrodinger wave equation with the known conduction band offset V,, between the QW and the barrier material. The ground and the first subband energies at k, = 0 are determined from equation (2) by putting respectively n = m = 0, and n = 1, m = 0. The energy difference between these subbands at k, = 0 is AE,(O) = ;
I
(k;, -
k;)
+ a,(k:,
-
k:‘,)
@a0+ Bo)k#, - kz,).
+
(4)
Equation (3) gives the functional dependence on k, of the energy difference between the two subbands: AE, (k,)
= AE,(0)+ (2x,, + PO>(k:, - k;)k:. (5)
The absorption coefficient a, defined as the ratio of the absorped photon energy per unit volume per second and the incident power per unit area is given by 1111 a =
1
w,,/v
ho
n,A&0*/(2~c) ’
(6)
where ho is the photon energy, Wfi is the transition rate, V is volume, n, is the index of refraction, A, is the amplitude of the magnetic vector potential, p is the material permeability, and c is the velocity of light in free space. The summation ‘c extends over all possible states. Incorporating the dipole approximation together with the line broadening [11], one obtains a =
xr
=
For GaAs QW’s in Al,,Ga,., barriers the following material parameters are used [8]: m, = 0.0665 m,, a,, = B,, = -2.3 x IO-” eVm4, - 1.97 x 10-37eVm4 m, = 0.0999 m,, V,, =’ 0.324 eV, n, = 3.29. The parameter values for In0,,,Ga,,47As QW’s in In,52A10,4,As barriers are [12]: m, = 0.041 m,, m2 = 0.075 m,, V, = 0.52eV, n, = 3.37. The values of a0 and /&,for InGaAs are not exactly known. But, as a0 and /$, are approximately the same, we put them equal so that their values, estimated from the bulk nonparabolicity parameter [13], are a,, = /?,, = - 7.56 x 10m3’eVm4. The Fermi level EF is calculated for a typical linear density N,, = 8 x 10’ m-’ in the QW. AE,(O) is found to be more than 7E, at 0 K so that, in all the
O”W,(O) + ykZ,l*f(k)[l - g(k)1 dk
Io
[AE, (0) + yk: - f~w]~ + (l-/2)*
where x
Fig. 1. Plot of absorption coefficient versus the incident photon energy for GaAs QW. (a), (b), and (b’) are for L = 8 nm; (c), (d) and (d’) are for L = 11 nm. (a) and (c): parabolic band; (b), (b’), (d) and (d’): nonparabolic band. (b’) and (d’) are for 300 K; other curves are for OK.
’
”
(7)
8
I\
lb1
pee* cos*tl oven
h*
<~fIz14i>*~
r
and Y =
@;!;&k
,[
(2a0 + P0)(k3i - ks).
Here 8 is the angle between the electric field vector and the normal to the QW, C#J~ and & are respectively the wave functions of the initial and the final states,f(k,) and g(k,) are respectively the Fermi-Dirac distribution functions in the ground and the first excited subbands, and F is the line width.
D’
, 60
100
120
140
160
I
160
200
220
hw ImrVl Fig. 2. Plot of the absorption coefficient versus the incident photon energy for InGaAs QW. The labels of the curves have the same significance as in Fig. 1.
Vol. 75, No. 3
A B S O R P T I O N P R O F I L E IN Q U A N T U M WIRES
cases studied here, the occupancy of the first excited subband is negligible at 0 K. We calculate ~ for 0 = 0 ° taking a typical value of 10meV for F, as done for two-dimensional systems [6, 11]. Figure 1 shows the absorption profile for GaAs QW's of L = 8 and 11 nm at 0 and 300 K. The nonparabolicity is found to broaden and decrease the absorption peak by 1 to 2%. Though the peak for the parabolic band occurs practically at AE~ (0), the same for the nonparabolic band occurs about 1 meV below the corresponding AE~ (0) owing to the term 7 (which is negative) in equation (7). On the whole at 0 K, the nonparabolicity shifts the peak height to a lower energy by 23.4 meV for L = 8 nm and to a higher energy by 6.5 meV for L = 11 nm owing to the changes in the confinement energies with L. Note the sharp fall in the peak height as L is increased from 8 to 11 nm. At 300 K, the peak position shifts to lower energies by ! to 2 meV; its height reduces and its width increases compared to the profile for 0 K. The asymmetry in the absorption curve is also remarkable. Band nonparabolicity has a larger effect on the absorption for InGaAs QW's, as shown in Fig. 2. At 0 K, the peak is considerably broadened and its maximum is reduced by 19% for L = 8 n m and by 9% for L = 11 nm owing to nonparabolicity. The peak position is lowered below the nonparabolic AE~(0) by about 2 meV because of the wave vector dependence of the subband separation. For both L = 8 nm and L = 11 nm the nonparabolicity shifts the absorption peak to lower energies at 0 K, the shift being from 195.5 to l l l m e V in the former, and from 106 to 90 meV in the latter case. The changes in the absorption profile at 300 K are noticeable.
261
In conclusion, the primary effects of nonparabolicity are to decrease the maximum absorption, broaden the maximum, and to shift the peak position. The effects are stronger for narrow QW's of low bandgap materials at large carrier densities. The peak is degraded as the temperature rises.
REFERENCES 1.
P.M. Petroff, A.C. Gossard, R.A. Logan & W. Wiegmann, Appl. Phys. Lett. 41,635 (1982). 2. A. Warren, D. Antoniadis & H. Smith, Phys. Rev. Lett. 56, 1858 (1986). 3. A. Kabasi, D. Chattopadhyay & C.K. Sarkar, J. Appl. Phys. 65, 1598 (1989). 4. M. Tsuchiya, J.M. Gaines, R.H. Yan, R.J. Simes, P.O. Holtz & L.A. Coldren, J. Vac. Sci. Technol. B7, 315 (1989). 5. S. Briggs, D. Jovanovic & J.P. Leburton, Appl. Phys. Lett. 54, 2012 (1989). 6. D.J. Newson & A. Kurobe, Semicon. Sci. Technol. 3, 786 (1988). 7. B.F. Levine, A.Y. Cho, J. Walker, R.J. Mabik, D.A. Kleinman & D.L. Sivco, Appl. Phys. Lett. 52, 1481 (1988). 8. U. Ekenberg, Phys. Rev. B36, 6152 (1987). 9. M. Braun & U. R6ssler, J. Phys. C 18, 3365 (1985). 10. W.Y. Lai & S. Das Sarma, Phys. Rev. B33, 8874 (1986). 11. D. Ahn & S.L. Chuang, Phys. Bey. B35, 4149 (1987). 12. S. Hiyamizu, T. Fujii, S. Muto, T. Inata, Y. Nakata, Y. Sugiyama & S. Sasa, J. Cryst. Growth 81, 349 (1987). 13. J.H. Marsh, Appl. Phys. Lett. 41, 732 (1982).
0038-1098/90 $3.00 + .00 Pergamon Press plc
CONDUCTION-BAND NONPARABOLICITY AND INTER-SUBBAND ABSORPTION PROFILE IN QUANTUM WIRES D. Chattopadhyay and P.C. Rakshit Institute of Radio Physics and Electronics, 92 Acharya Prafulla Chandra Road, Calcutta 700009, India
(Received 15 January 1990, in revised form 9 March 1990 by B. Miihlschlegel) The conduction-band nonparabolicity effects on the inter-subband absorption in quantum wires are investigated. The nonparabolicity is found to shift the absorption peak energy, broaden it, and reduce its amplitude. The effects are larger for In0.53Ga047As quantum wire structures than those for GaAs structures. The peak asymmetry is significant at 300 K. THE INTRIGUING electrical and optical properties expanded to fourth order in the wavevector k, can be of quantum wires (QW's) have caused much interest in written as [9] the one-dimensional (1D) electronic systems in such structures [1-5]. In this communication, we study the E(k) = %k 4 + 2m, + (2% + flo)(k 2 + ~ ) k 2 effects of the conduction-band nonparabolicity on the h2 absorption of electromagnetic radiation producing + + AD + (2 0 + transitions of electrons from the ground subband to the first excited subband in a QW. In a parabolic + Cto(k~ + k~). (1) band, the inter-subband energy AEI does not depend on the longitudinal wave vector kx in the QW, so that Here h is Dirac's constant, ml is the effective mass of for small line widths the absorption is a maximum the QW material, % and fl0 are the nonparabolicity when the energy of the incident photon equals AE~. parameters, and kx, Icy, kz are the components of k The nonparabolicity of the conduction band changes along x, y, and z directions. For simplicity, the barrier potentials for y and z the inter-subband energy difference at k~ = 0 and causes AE1 to decrease with increasing k~, thus influ- directions are taken to be the same here. For too encing the absorption profile. We study such effects narrow wires, the quantum mechanical problem is here for GaAs and In0.53Ga0.47As QW's. Our results nonseparable due to the mixing of the y and z terms are potentially useful in interpreting absorption data [10]. For large barrier heights and relatively wide and in modelling performances of devices based on wires, however, the condition can be relaxed to investigate the nonparabolicity effects. The exact treatment QW's. Nonparabolicity effects in a two-dimensional of mixing including nonparabolicity will modify the electron system in a quantum well have been studied results slightly quantitatively, and is not attempted [6] and used to interpret experimental data [7]. The here in view of the complexity involved. Thus the subeffects in a 1D system need a separate investigation band energies at kx = 0, obtained from equation (1), because the density of states and the nonparabolicity are h2 are altered due to the double confinement in the QW. Strictly speaking, the problem needs to be solved E,m(O) = 2m--"~(k2~" + ~m) + %(kz~ + k4ym) numerically considering the self-consistent filling of + (2% + flo)k2.~m • (2) the confined states with electrons. But such treatments are tedious computationally. As our basic aim here is The remaining terms in equation (1) give the longito provide an insight into the nonparabolicity effects, tudinal dispersion: we rather concentrate on an analytic approach in this h 2k x2 4 initial work. Enm(kx) = Enm(O) q- ~ q- %kx We assume a QW of square cross-section of confinement length L along y and z directions, and take 2 + (2% + flo)(kz2,, + kym)~. (3) the longitudinal direction along x. Neglecting the spin splitting [8], the bulk conduction-band energy E, In the barrier material, the nonparabolicity can be 259
ABSORPTION
260
PROFILE
IN QUANTUM
WIRES
Vol. 75, No. 3
ignored and the band-edge effective mass m2 can be used but for very narrow QW’s [8]. With the boundary conditions that the wave function and its derivative divided by the effective mass are continuous across the interfaces, the confinement wave vectors kym and kZ, are obtained by solving the Schrodinger wave equation with the known conduction band offset V,, between the QW and the barrier material. The ground and the first subband energies at k, = 0 are determined from equation (2) by putting respectively n = m = 0, and n = 1, m = 0. The energy difference between these subbands at k, = 0 is AE,(O) = ;
I
(k;, -
k;)
+ a,(k:,
-
k:‘,)
@a0+ Bo)k#, - kz,).
+
(4)
Equation (3) gives the functional dependence on k, of the energy difference between the two subbands: AE, (k,)
= AE,(0)+ (2x,, + PO>(k:, - k;)k:. (5)
The absorption coefficient a, defined as the ratio of the absorped photon energy per unit volume per second and the incident power per unit area is given by 1111 a =
1
w,,/v
ho
n,A&0*/(2~c) ’
(6)
where ho is the photon energy, Wfi is the transition rate, V is volume, n, is the index of refraction, A, is the amplitude of the magnetic vector potential, p is the material permeability, and c is the velocity of light in free space. The summation ‘c extends over all possible states. Incorporating the dipole approximation together with the line broadening [11], one obtains a =
xr
=
For GaAs QW’s in Al,,Ga,., barriers the following material parameters are used [8]: m, = 0.0665 m,, a,, = B,, = -2.3 x IO-” eVm4, - 1.97 x 10-37eVm4 m, = 0.0999 m,, V,, =’ 0.324 eV, n, = 3.29. The parameter values for In0,,,Ga,,47As QW’s in In,52A10,4,As barriers are [12]: m, = 0.041 m,, m2 = 0.075 m,, V, = 0.52eV, n, = 3.37. The values of a0 and /&,for InGaAs are not exactly known. But, as a0 and /$, are approximately the same, we put them equal so that their values, estimated from the bulk nonparabolicity parameter [13], are a,, = /?,, = - 7.56 x 10m3’eVm4. The Fermi level EF is calculated for a typical linear density N,, = 8 x 10’ m-’ in the QW. AE,(O) is found to be more than 7E, at 0 K so that, in all the
O”W,(O) + ykZ,l*f(k)[l - g(k)1 dk
Io
[AE, (0) + yk: - f~w]~ + (l-/2)*
where x
Fig. 1. Plot of absorption coefficient versus the incident photon energy for GaAs QW. (a), (b), and (b’) are for L = 8 nm; (c), (d) and (d’) are for L = 11 nm. (a) and (c): parabolic band; (b), (b’), (d) and (d’): nonparabolic band. (b’) and (d’) are for 300 K; other curves are for OK.
’
”
(7)
8
I\
lb1
pee* cos*tl oven
h*
<~fIz14i>*~
r
and Y =
@;!;&k
,[
(2a0 + P0)(k3i - ks).
Here 8 is the angle between the electric field vector and the normal to the QW, C#J~ and & are respectively the wave functions of the initial and the final states,f(k,) and g(k,) are respectively the Fermi-Dirac distribution functions in the ground and the first excited subbands, and F is the line width.
D’
, 60
100
120
140
160
I
160
200
220
hw ImrVl Fig. 2. Plot of the absorption coefficient versus the incident photon energy for InGaAs QW. The labels of the curves have the same significance as in Fig. 1.
Vol. 75, No. 3
A B S O R P T I O N P R O F I L E IN Q U A N T U M WIRES
cases studied here, the occupancy of the first excited subband is negligible at 0 K. We calculate ~ for 0 = 0 ° taking a typical value of 10meV for F, as done for two-dimensional systems [6, 11]. Figure 1 shows the absorption profile for GaAs QW's of L = 8 and 11 nm at 0 and 300 K. The nonparabolicity is found to broaden and decrease the absorption peak by 1 to 2%. Though the peak for the parabolic band occurs practically at AE~ (0), the same for the nonparabolic band occurs about 1 meV below the corresponding AE~ (0) owing to the term 7 (which is negative) in equation (7). On the whole at 0 K, the nonparabolicity shifts the peak height to a lower energy by 23.4 meV for L = 8 nm and to a higher energy by 6.5 meV for L = 11 nm owing to the changes in the confinement energies with L. Note the sharp fall in the peak height as L is increased from 8 to 11 nm. At 300 K, the peak position shifts to lower energies by ! to 2 meV; its height reduces and its width increases compared to the profile for 0 K. The asymmetry in the absorption curve is also remarkable. Band nonparabolicity has a larger effect on the absorption for InGaAs QW's, as shown in Fig. 2. At 0 K, the peak is considerably broadened and its maximum is reduced by 19% for L = 8 n m and by 9% for L = 11 nm owing to nonparabolicity. The peak position is lowered below the nonparabolic AE~(0) by about 2 meV because of the wave vector dependence of the subband separation. For both L = 8 nm and L = 11 nm the nonparabolicity shifts the absorption peak to lower energies at 0 K, the shift being from 195.5 to l l l m e V in the former, and from 106 to 90 meV in the latter case. The changes in the absorption profile at 300 K are noticeable.
261
In conclusion, the primary effects of nonparabolicity are to decrease the maximum absorption, broaden the maximum, and to shift the peak position. The effects are stronger for narrow QW's of low bandgap materials at large carrier densities. The peak is degraded as the temperature rises.
REFERENCES 1.
P.M. Petroff, A.C. Gossard, R.A. Logan & W. Wiegmann, Appl. Phys. Lett. 41,635 (1982). 2. A. Warren, D. Antoniadis & H. Smith, Phys. Rev. Lett. 56, 1858 (1986). 3. A. Kabasi, D. Chattopadhyay & C.K. Sarkar, J. Appl. Phys. 65, 1598 (1989). 4. M. Tsuchiya, J.M. Gaines, R.H. Yan, R.J. Simes, P.O. Holtz & L.A. Coldren, J. Vac. Sci. Technol. B7, 315 (1989). 5. S. Briggs, D. Jovanovic & J.P. Leburton, Appl. Phys. Lett. 54, 2012 (1989). 6. D.J. Newson & A. Kurobe, Semicon. Sci. Technol. 3, 786 (1988). 7. B.F. Levine, A.Y. Cho, J. Walker, R.J. Mabik, D.A. Kleinman & D.L. Sivco, Appl. Phys. Lett. 52, 1481 (1988). 8. U. Ekenberg, Phys. Rev. B36, 6152 (1987). 9. M. Braun & U. R6ssler, J. Phys. C 18, 3365 (1985). 10. W.Y. Lai & S. Das Sarma, Phys. Rev. B33, 8874 (1986). 11. D. Ahn & S.L. Chuang, Phys. Bey. B35, 4149 (1987). 12. S. Hiyamizu, T. Fujii, S. Muto, T. Inata, Y. Nakata, Y. Sugiyama & S. Sasa, J. Cryst. Growth 81, 349 (1987). 13. J.H. Marsh, Appl. Phys. Lett. 41, 732 (1982).