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Complex dielectric function of InP-InGaAsP quantum wells
Volume 151, number 1,2
PHYSICS LETI’ERS A
26 November 1990
Complex dielectric function of InP—InGaAsP quantum wells B. Sauer Lehrstuhlfir Technische Elektrophysik, Technische Universiläz München, Arcisstrasse 21, 8000 Munich 2, FRG Received 2 May 1990; revised manuscript received 6 September 1990; accepted for publication 21 September 1990 Communicated by D. Bloch
The complex dielectricfunction
t=
+ i~
2of a lattice-matched InP—InGaAsP quantum well is calculated by the envelope function approximation and a Kramers—Kronig transformation. It is shown that both ~ and ~2 exhibit an exciton-induced fine structure at the band gap and that the refractive index, which is related to ~, exceeds the bulk values.
Recently both theoretical [1] and experimental [21 investigations have paid great attention to the optical absorption properties of semiconductor quantum wells (QWs). Less attention was given to the real part of the dielectric function. Spector and Hassan [3] calculated the complex dielectric function =~ +i 2 in GaAs—GaAlAs QWs with respect to carrier confined band—band transitions. They concluded that in QWs e~decreases far below the corresponding bulk values at low photon energies. Cada et al. [4] showed that ~ in QWs stays above the bulk values in the same material system in the band gap region. Since the results of refs. [31and [4] are quite contrary to each other and since e~is related to important optical properties such as the refractive index and the is especially cused ongroup ~. Wevelocity, calculatethis theLetter dielectric function foin a lattice-matched InP—InGaAsP QW system. We indude all E 0 and E0 + i~ transitions, excitonic effects and broadening mechanisms in our calculation. The material parameters are adjusted for the application in integrated optical waveguides at 1.55 ~.tm.It is shown that both ~ and 2 exhibit an exciton-induced fine corresponding structure and that refractive index are exceeds the bulkthe values. The results discussed with respect to the articles cited above. The imaginary part of the dielectric function 2 can be derived from Fermi’s golden rule [5]. In semiconductor QWs the main contribution to 2 at optical frequencies arises from excitonic and band—band 0375-9601/90/S 03.50
~‘
transitions. Taking the sum over the conduction and valence subbands indexed c and v respectively, 2 is given in atomic units as [51 2w (~~4°(~ c,v ~2 (Q) = ,, cc~ ~
+ ~ I ePa. (k) I ~
—
k) £2)). —
(1)
Here c, e~ and Ware the velocity of light, the vacuum dielectric constant and the QW well width, respectively. Q is the photon energy, k the in-plane wave vector and £° a lineshape function which accounts for temperature and interface-roughness broadening as well as for compositional fluctuations in the well [6,7]. Q~and f~ are the excitonic transition energy 2and 4Q~(k) = and oscillator Ie~P~,,I Q~(k) —Q~(k)arestrength. the squared transition matrix element of polarisation e and the energy difference of the free electronic states. The real part of the dielectric function ~ is computed by a Kramers—Kronig transformation,
~
Q’ ~22(Q’)
(2) Q~ Eqs. (1) and (2) are the starting point of the numerical calculation, which is carried out in the framework of the envelope function approximation; the details can be found in ref. [51.In fig. 1 we show the calculated 2 spectrum of a W= 100 A InP—
1990— Elsevier Science Publishers B.V. (North-Holland)
2 P it
0
—
81
Volume 151, number 1,2
1.4
PHYSICS LETTERS A
Wavelength (pm)
1.3
,
BA2 el~(A)=A+A2C,
1.2
26 November 1990
(3)
A>Agap,
~
2n1 A
5
E1L1
\.
E111
1 Energy (eV)
0
‘
.
Fig. I. Complex part of the dielectric function ~2 of a 100 A InP— InGaAsP quantum well at room temperature. The polarisation vector is parallel to the heterostructure planes.
1 3
.
1 h Wave engt (pm) ~ 1•.
0
6 1 2
.
/
5 -
—
1 2
.
0
—
6
L—~.
ng
-/
—
-
.__I,
Energy
(eV)
.
9
with A = hc/Q in jim and the coefficients A, B and C equal to 8.38, 2.49 and 0.774, respectively [10]. It should be noted that below the band gap the refractive index n can be approximated by \/~ [11]. From fig. 2 it is seen that the real part of the dielectric function in the QW exceeds the bulk values in contrast to the results in ref. [3]. This is attributed to a background distribution ?° to e~,which is not included in the calculation of ref. [3]. Here we take into account all E0 and z~oband—band transitions [11] with transition energy higher than the barrier gap energy, of 1.35 eV, which give an important but nondispersive contribution ~° to e~.Since E1, E1 +~ and E2 transitions are not included in our calculation, more precise calculations should result in even higher values of ~ [11]. The downward bending of ~ below the band gap in fig. 2 results from excitonic transitions and the steplike rise ofthe band— band transitions. Additionally the maximum of e~is shifted to higher energies by AQ~~( 0) Q~due to —
carrier confinement in the QW. Our results are in qualitative agreement with those given in ref. [4] for the GaAs—A1GaAs system.
,.J~
Fig. 2. Solid curves: real part of the dielectric function ~. The upper (lower) curve is the ~ spectrum ofthe QW (bulk). Circles indicate the band gap energy. Dashed curve: group index flg as defined in the text.
The energy dependence of n = below the band gap in the QW structure discussed here can also be fitted to a Sellmeier formula (eq. (3)) with coefficientsA, B and C equal to 11.9, 0.092 and 1.70. Re-
InGaAsP QW at room temperature, whose quaternary well gap is taken to be 1.38 jim. The input band parameters of this calculation are taken from ref. [8] and the structure is designed to have the absorption edge,at 100 meV above the operation wavelength of 797 meV (& 1.55 jim), which is a typical case of practical application for waveguides [9]. The polarization vector is taken to be parallel to the heterostructure planes, thus both the first heavy- and lighthole exciton absorption (EIHI and E1L1 in fig. 1) is observed. At 1.04 eV the second conduction subband onset is seen. In fig. 2 the real part ~ of the dielectric function, which was calculated from eqs. (1) and (2) is displayed. For comparison we include the e~spectrum of the equivalent bulk structure in fig. 2, which is approximated by the modified Sellmeier formula [10],
placing n (Q) by its mean value ~lin the energy range between the band gap (900 meV) and 750 meV will lead to a maximum relative error of only 3%, thus for the computation of the absorption coefficient a =Q2/ndl in QWs [3], the refractive index can safely be regarded as constant with respect to energy. For the application of a QW as a laser we investigate the group index fl5, which is defined as n+Qôn/i9Q [12]. In a QW laser the excitons will be bleached by electrons, hence we calculate fl8 via eqs. (1) and (2), leaving out of account the excitonic transitions. In fig. 2 it is seen that fl5 is a strongly energy depending function. Since flg is related to the group velocity V5 by V5 = C/flg, which plays an important role in the rate equation model of semiconductor lasers [12], its explicit energy dependence should be essential in the theoretical description of QW lasers.
82
Volume 151, number 1,2
PHYSICS LET1’ERS A
The author is indebted to U. Wolff, G. MUller and L. Stoll of Siemens AG for helpful discussions regarding the group index and critical reading of the manuscript. This work was financially supported by the Siemens AG.
References (1] J.J. Coleman, ed., Special Issue on Quantum wells, heterostructures and superlattices, IEEE J. Quantum Electron. 24(1988) p. 1579. [2] L.W. Molenkamp, G.E.W. Bauer and R. Eppenga, Phys. Rev. B38(l988)6147. [3) H.N. Spector and H.H. Hassan, Phys. Lett. A 129 (1988) 121.
26 November 1990
[4] M. Cada, B.P. Keyworth, J.M. Glinski, C. Rolland, A.J. Spring Thrope, K.O. Hill and R.A. Soref, AppI. Phys. Lett. 54 (1989) 2509. ES] G.D. Sanders and Y.-C. Chang, Phys. Rev. B 35 (1987) 1300. [6) H.-S. Cho and P.R. Pruncal, IEEE J. Quantum Electron. 25 (1989) 1682. [7] S. Hong and J. Singh, J. AppI. Phys. 62 (1987) 1994. [8] I. Bar-Joseph, C. Klingshirn, D.A.B. Miller, D.S. Chemla, U. Koren and B.!. Miller, AppI. Phys. Lett. 50 (1987)1010. [9) K.J. Ebeling, Integrierte Optoelektronik (Springer, Berlin, 1989). [10) F. Fiedler and A. Schlachetzky, Solid State Electron. 30 (1987) 73. [11) S. Adachi, J. Appl. Phys. 53 (1982) 5863. [l2]A.Sudbø,IEEEJ.QuantumElectron.23(l987) 1127.
83
PHYSICS LETI’ERS A
26 November 1990
Complex dielectric function of InP—InGaAsP quantum wells B. Sauer Lehrstuhlfir Technische Elektrophysik, Technische Universiläz München, Arcisstrasse 21, 8000 Munich 2, FRG Received 2 May 1990; revised manuscript received 6 September 1990; accepted for publication 21 September 1990 Communicated by D. Bloch
The complex dielectricfunction
t=
+ i~
2of a lattice-matched InP—InGaAsP quantum well is calculated by the envelope function approximation and a Kramers—Kronig transformation. It is shown that both ~ and ~2 exhibit an exciton-induced fine structure at the band gap and that the refractive index, which is related to ~, exceeds the bulk values.
Recently both theoretical [1] and experimental [21 investigations have paid great attention to the optical absorption properties of semiconductor quantum wells (QWs). Less attention was given to the real part of the dielectric function. Spector and Hassan [3] calculated the complex dielectric function =~ +i 2 in GaAs—GaAlAs QWs with respect to carrier confined band—band transitions. They concluded that in QWs e~decreases far below the corresponding bulk values at low photon energies. Cada et al. [4] showed that ~ in QWs stays above the bulk values in the same material system in the band gap region. Since the results of refs. [31and [4] are quite contrary to each other and since e~is related to important optical properties such as the refractive index and the is especially cused ongroup ~. Wevelocity, calculatethis theLetter dielectric function foin a lattice-matched InP—InGaAsP QW system. We indude all E 0 and E0 + i~ transitions, excitonic effects and broadening mechanisms in our calculation. The material parameters are adjusted for the application in integrated optical waveguides at 1.55 ~.tm.It is shown that both ~ and 2 exhibit an exciton-induced fine corresponding structure and that refractive index are exceeds the bulkthe values. The results discussed with respect to the articles cited above. The imaginary part of the dielectric function 2 can be derived from Fermi’s golden rule [5]. In semiconductor QWs the main contribution to 2 at optical frequencies arises from excitonic and band—band 0375-9601/90/S 03.50
~‘
transitions. Taking the sum over the conduction and valence subbands indexed c and v respectively, 2 is given in atomic units as [51 2w (~~4°(~ c,v ~2 (Q) = ,, cc~ ~
+ ~ I ePa. (k) I ~
—
k) £2)). —
(1)
Here c, e~ and Ware the velocity of light, the vacuum dielectric constant and the QW well width, respectively. Q is the photon energy, k the in-plane wave vector and £° a lineshape function which accounts for temperature and interface-roughness broadening as well as for compositional fluctuations in the well [6,7]. Q~and f~ are the excitonic transition energy 2and 4Q~(k) = and oscillator Ie~P~,,I Q~(k) —Q~(k)arestrength. the squared transition matrix element of polarisation e and the energy difference of the free electronic states. The real part of the dielectric function ~ is computed by a Kramers—Kronig transformation,
~
Q’ ~22(Q’)
(2) Q~ Eqs. (1) and (2) are the starting point of the numerical calculation, which is carried out in the framework of the envelope function approximation; the details can be found in ref. [51.In fig. 1 we show the calculated 2 spectrum of a W= 100 A InP—
1990— Elsevier Science Publishers B.V. (North-Holland)
2 P it
0
—
81
Volume 151, number 1,2
1.4
PHYSICS LETTERS A
Wavelength (pm)
1.3
,
BA2 el~(A)=A+A2C,
1.2
26 November 1990
(3)
A>Agap,
~
2n1 A
5
E1L1
\.
E111
1 Energy (eV)
0
‘
.
Fig. I. Complex part of the dielectric function ~2 of a 100 A InP— InGaAsP quantum well at room temperature. The polarisation vector is parallel to the heterostructure planes.
1 3
.
1 h Wave engt (pm) ~ 1•.
0
6 1 2
.
/
5 -
—
1 2
.
0
—
6
L—~.
ng
-/
—
-
.__I,
Energy
(eV)
.
9
with A = hc/Q in jim and the coefficients A, B and C equal to 8.38, 2.49 and 0.774, respectively [10]. It should be noted that below the band gap the refractive index n can be approximated by \/~ [11]. From fig. 2 it is seen that the real part of the dielectric function in the QW exceeds the bulk values in contrast to the results in ref. [3]. This is attributed to a background distribution ?° to e~,which is not included in the calculation of ref. [3]. Here we take into account all E0 and z~oband—band transitions [11] with transition energy higher than the barrier gap energy, of 1.35 eV, which give an important but nondispersive contribution ~° to e~.Since E1, E1 +~ and E2 transitions are not included in our calculation, more precise calculations should result in even higher values of ~ [11]. The downward bending of ~ below the band gap in fig. 2 results from excitonic transitions and the steplike rise ofthe band— band transitions. Additionally the maximum of e~is shifted to higher energies by AQ~~( 0) Q~due to —
carrier confinement in the QW. Our results are in qualitative agreement with those given in ref. [4] for the GaAs—A1GaAs system.
,.J~
Fig. 2. Solid curves: real part of the dielectric function ~. The upper (lower) curve is the ~ spectrum ofthe QW (bulk). Circles indicate the band gap energy. Dashed curve: group index flg as defined in the text.
The energy dependence of n = below the band gap in the QW structure discussed here can also be fitted to a Sellmeier formula (eq. (3)) with coefficientsA, B and C equal to 11.9, 0.092 and 1.70. Re-
InGaAsP QW at room temperature, whose quaternary well gap is taken to be 1.38 jim. The input band parameters of this calculation are taken from ref. [8] and the structure is designed to have the absorption edge,at 100 meV above the operation wavelength of 797 meV (& 1.55 jim), which is a typical case of practical application for waveguides [9]. The polarization vector is taken to be parallel to the heterostructure planes, thus both the first heavy- and lighthole exciton absorption (EIHI and E1L1 in fig. 1) is observed. At 1.04 eV the second conduction subband onset is seen. In fig. 2 the real part ~ of the dielectric function, which was calculated from eqs. (1) and (2) is displayed. For comparison we include the e~spectrum of the equivalent bulk structure in fig. 2, which is approximated by the modified Sellmeier formula [10],
placing n (Q) by its mean value ~lin the energy range between the band gap (900 meV) and 750 meV will lead to a maximum relative error of only 3%, thus for the computation of the absorption coefficient a =Q2/ndl in QWs [3], the refractive index can safely be regarded as constant with respect to energy. For the application of a QW as a laser we investigate the group index fl5, which is defined as n+Qôn/i9Q [12]. In a QW laser the excitons will be bleached by electrons, hence we calculate fl8 via eqs. (1) and (2), leaving out of account the excitonic transitions. In fig. 2 it is seen that fl5 is a strongly energy depending function. Since flg is related to the group velocity V5 by V5 = C/flg, which plays an important role in the rate equation model of semiconductor lasers [12], its explicit energy dependence should be essential in the theoretical description of QW lasers.
82
Volume 151, number 1,2
PHYSICS LET1’ERS A
The author is indebted to U. Wolff, G. MUller and L. Stoll of Siemens AG for helpful discussions regarding the group index and critical reading of the manuscript. This work was financially supported by the Siemens AG.
References (1] J.J. Coleman, ed., Special Issue on Quantum wells, heterostructures and superlattices, IEEE J. Quantum Electron. 24(1988) p. 1579. [2] L.W. Molenkamp, G.E.W. Bauer and R. Eppenga, Phys. Rev. B38(l988)6147. [3) H.N. Spector and H.H. Hassan, Phys. Lett. A 129 (1988) 121.
26 November 1990
[4] M. Cada, B.P. Keyworth, J.M. Glinski, C. Rolland, A.J. Spring Thrope, K.O. Hill and R.A. Soref, AppI. Phys. Lett. 54 (1989) 2509. ES] G.D. Sanders and Y.-C. Chang, Phys. Rev. B 35 (1987) 1300. [6) H.-S. Cho and P.R. Pruncal, IEEE J. Quantum Electron. 25 (1989) 1682. [7] S. Hong and J. Singh, J. AppI. Phys. 62 (1987) 1994. [8] I. Bar-Joseph, C. Klingshirn, D.A.B. Miller, D.S. Chemla, U. Koren and B.!. Miller, AppI. Phys. Lett. 50 (1987)1010. [9) K.J. Ebeling, Integrierte Optoelektronik (Springer, Berlin, 1989). [10) F. Fiedler and A. Schlachetzky, Solid State Electron. 30 (1987) 73. [11) S. Adachi, J. Appl. Phys. 53 (1982) 5863. [l2]A.Sudbø,IEEEJ.QuantumElectron.23(l987) 1127.
83