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Dependence of the photoluminescence energy and carrier lifetime of the carrier density in nitride quantum well
ARTICLE IN PRESS
Microelectronics Journal 39 (2008) 447–449 www.elsevier.com/locate/mejo
Dependence of the photoluminescence energy and carrier lifetime of the carrier density in nitride quantum well M.R. Lo´pez, G. Gonza´lez de la Cruz Departamento de Fisica, Centro de Investigacio´n y de estudios Avanzados del IPN, Apartado Postal 14–740, 07000 Me´xico D.F., Mexico Available online 27 August 2007
Abstract We investigated the high-carrier screening of macroscopic polarization fields in GaN quantum wells (QWs) using a variational wave function for electrons and holes. In particular, we studied of the influence of free-carrier screening on the photoluminescence (PL) energy emission and carrier lifetime in the QW. We show that the energy transition between electrons and holes are explained well by the freecarrier screening effect that compensates for the built-in electric field in the QW. r 2007 Elsevier Ltd. All rights reserved. PACS: 78.20.Bh; 78.55.m; 78.67.De Keywords: III–V nitrides; Quantum wells; Screening
1. Introduction GaN-based quantum wells (QW’s) attracted much interest in the last few years since they have become the key material in commercial fabrication of long-time light emitting diodes (LED’s) and laser diodes for the blue to ultraviolet spectral range. Up to now InGaN/GaN is used as an active region in these devices. However, the lightemitting mechanism is not yet fully understood because this material exhibits some peculiarities. A peculiarity results from the polar axis of the wurtzite crystal structure and the strong polarity of III-N bindings. All group-III nitrides in the wurtzite phase have a strong spontaneous macroscopic polarization and large piezoelectric coefficients. This has been found from ab initio calculations [1,2]. The abrupt variation of the polarization at the surfaces and interfaces gives rise to large polarization sheet charges that in turn create internal electric fields of the order of MV/cm. The field-induced linear bending of the band edges causes a spatial separation of confined electrons and holes within the active layers of the devices and has, therefore, important consequences on the optical Corresponding author.
E-mail address: mrlopez@fis.cinvestav.mx (M.R. Lo´pez). 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.07.044
properties of the nitride-based LED’s or lasers. It is worth noting that the piezoelectric field present in the III–V nitrides appears in the presence of strain, due to, e.g., epitaxy, while the spontaneous polarization is a property of low-symmetry materials in their ground state, independent of strain, and it is absent in zinc-blende materials e.g., GaAs. In this work, we focus on the carrier density dependence of the photoluminescence (PL) energy. In QW’s the freecarrier screening effect built-in electric field is affected by the carrier density. Sala et al. [3] reported that the freecarrier screening effect explains some puzzling experimental data on nitride lasers, such as the unusually high lasing excitation thresholds and emission blue shifts that accompany increasing excitation levels. Recently Kuroda and Tackeuchi [4] performed a systematic time-resolved photoluminescence measurements of QW’s for various carrier densities and different well widths, and showed that the PL energy shift and the variation on the carrier lifetime can be explained by the free-carrier screening effect, which effectively screens the built-in spontaneous and piezoelectric polarization fields inducing a field-free band profile. We show approximate results on the electron (hole) energy in the QW in the presence of the internal electric field as a function of the carrier density. The Schrodinger
ARTICLE IN PRESS 448
M.R. Lo´pez, G. Gonza´lez de la Cruz / Microelectronics Journal 39 (2008) 447–449
equation is solved in the envelope approximation using a variational wave function for electrons and holes in the QW and the energy potential for the electron is assumed to consist of the confined potential, the potential produced by the built-in electric field and the contribution to the potential energy from the charge distribution of the electrons and holes in the QW. Our theoretical results are in qualitative agreement with the experimental results reported in the literature. 2. Theoretical model Since our study is based on the comparison of the PL spectra with the calculated transition energies, the model to calculate the well and barrier profiles to include in the Schro¨dinger equation is presented here. Consider first the Schrodinger equation of our system disregarding the Coulomb interaction. Following the usual choice for the electron three-dimensional wave function, i.e., the product of a plane wave and an envelope function cn;k ¼ eikr xn where r ¼ (x,y), k is the in-plane wave vector and n is the subband index, it is easy to put into the 3D Schrodinger equation to generate two equations, one for the 2D free motion and other for the bound state. The sum of the energy eigenvalues forms the total energy n;k ¼ _2 k2 =2m þ E n and it reduces the task to the resolution of a one dimensional Scrodinger equation. In the presence of the piezoelectric field in the QW in the z-axis, it reads _2 d 2 xn þ ½V F ðzÞ þ V H ðzÞxn ¼ E n xn , 2m dz2 where V 0 ðzÞ þ eFz jzjoa , V F ðzÞ ¼ 0 otherwise
(1)
(2)
and V0(z) ¼ V0[y(za)+y(za)] is the potential of the square QW, V0 is the band offset, y(z) is the Heaviside unit step function and 2a the QW width. This potential well and the electron and hole wave functions are schematically presented in Fig. 1. VH(z), the Hartree potential, is the
contribution to the potential energy from the charge distribution of the electrons and holes in the conduction and valence bands in the QW, respectively, and is given by the solution of Poisson’s equation with the charge density in the ground state as source term. The result is Z z e2 ðz0 zÞðx2h ðz0 Þ x2e ðz0 ÞÞ dz0 , (3) V H ¼ ns z þ 1 where ns is the 2D electron (hole) concentration in the lowest subband and e is the static dielectric constant of the QW, xe and xh are solutions of Eq. (1) for the electrons in the conduction band and holes in the valence band in the QW, respectively. It is the VH(z) in the effective one-electron potential of Eq. (1) that makes the Schrodinger equation a nonlinear eigenvalue problem, because the potential depends on the wave functions and besides that, Eq. (1) has to be solved simultaneously with Poisson‘s equation in a self-consistent approximation. Simple analytic wave functions make calculations of properties of QW’s much more convenient than they would if numerical self-consistent solutions or cumbersome analytic solutions like Airy functions had to be used. For that reason approximate solutions have been widely used in QW’s calculations [5]. One of the simplest of these is dz xðzÞ ¼ N 1 þ (4) j ðzÞ, 2a 0 1=2 is a normalization factor, NR ¼ 1 þ ðd=2aÞ2 z2 where 1 z2 ¼ 1 z2 j20 dz, j0 is the ground state particle wave function in the absence of the built-in electric field in the QW, and the parameter d in Eq. (4) is determined by minimizing the total energy of the system for given values of the QW density charges of electrons and holes. Because of the simplicity of the wave function, it is easy to evaluate the expectation values of all the terms in the Hamiltonian, Eq. (1). We find 2 2 eF d 2 2 _ d h H ix ¼ E 0 þ N z þ a 8ma2 e 2 ns 4 d d2 d3 N A1 þ A2 2 þ A3 3 , ð5Þ a 0 a a where Ai depending of the QW parameters, and are well determinated. Then the value of d that minimizes the total energy satisfies the following polynomial equation of fourth order: d4 þ
Fig. 1. Schematic picture of energies and wave functions of electrons and holes in a strained quantum well with piezoelectric field.
awþ 3 12a2 b 2 4a3 w 16a4 g ¼ 0. d þ 2 d þ 2 d þ a ma m z a z a
(6)
Here a ¼ 4CA3nseF/z2S2, b ¼ A1/z2S;4A3, g ¼ eF /z2SCA1ns, w ¼ _2 8mCA2 ns and C ¼ e2/e. It is worth to note that in the limit of low density carrier such that ns 5_2 =8maCA2 the solution of Eq. (6) reduces to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " aw 16m2 ag d¼ 1 1 . (7) w2 2ma
ARTICLE IN PRESS M.R. Lo´pez, G. Gonza´lez de la Cruz / Microelectronics Journal 39 (2008) 447–449
PL peak energy. [eV]
3.5
in Fig. 2, upon increasing the sheet carrier density we observe a blue shift in the transition energy. This is due to the progressive recovery of flat band conditions upon increasing the sheet carrier density. As can be seen our theory is in agreement with the self-consistent calculation developed in Ref. [3]. We obtain the recombination rate from Z Rr / xe ðzÞxhh ðzÞ dz,
3.0
Variational calculations. 50A 100A
2.5
and we found that the recombination rate increase nonlinearly for low densities and become constant for high- carrier densities in agree with the Refs. [3,5].
Selfconsistent approach. 50A 100A
2.0
449
3. Conclusions 0
5
10
15
20
ηs [ 1013 cm-2] Fig. 2. Electron–hole transition energy versus sheet carrier density for several well widths.
Once we know the variational parameter that minimizes Eq. (5), the PL energy peak can be calculated by the simple relationship as a function of the carrier density in the QW and the internal electric field as a fitting parameter E PL ¼ E g þ E e þ de þ E h þ dh ,
(8)
where Eg is the gap of the semiconductor QW, Ee(h) is the electron (hole) energy without electric field in the QW and deðhÞ ¼ hH ieðhÞ E eðhÞ is the induced energy shift for electrons (holes) which depends on the carrier density and the built-in electric field. In Fig. 2 we present the calculated transition energy between the highest hole level and the lowest electron level for an internal piezoelectric field F ¼ 3.25 MV/cm. As seen
In conclusion, we have shown that free carriers can effectively screen macroscopic polarization fields in nitride quantum wells, resulting in a non vanishing recombination rate for large wells (contrary to previous claims) in normal laser operation. A rather high sheet density is needed to achieve these conditions. We also explained the red shifts versus well width and blue shifts versus sheet density as resulting from the interplay of free-carrier screening and polarization fields. References [1] F. Bernardini, V. Fiorentini, D. Vanderbilt, Phys. Rev. B 56 (1997) R10024. [2] F. Bernardini, V. Fiorentini, Phys. Rev. B 57 (1998) R9427. [3] F.D. Sala, A. Di Carlo, P. Lugli, F. Bernardini, V. Fiorentini, R. Scholz, J.M. Jancu, Appl. Phys. Lett. 74 (1999) 2002. [4] T. Kuroda, A. Tackeuchi, T. Sota, Appl. Phys. Lett. 76 (2000) 3753. [5] G. bastard, E.E. Mendez, L.L. Chang, L. Esaki, Phys. Rev. B 28 (1983) 3241.
Microelectronics Journal 39 (2008) 447–449 www.elsevier.com/locate/mejo
Dependence of the photoluminescence energy and carrier lifetime of the carrier density in nitride quantum well M.R. Lo´pez, G. Gonza´lez de la Cruz Departamento de Fisica, Centro de Investigacio´n y de estudios Avanzados del IPN, Apartado Postal 14–740, 07000 Me´xico D.F., Mexico Available online 27 August 2007
Abstract We investigated the high-carrier screening of macroscopic polarization fields in GaN quantum wells (QWs) using a variational wave function for electrons and holes. In particular, we studied of the influence of free-carrier screening on the photoluminescence (PL) energy emission and carrier lifetime in the QW. We show that the energy transition between electrons and holes are explained well by the freecarrier screening effect that compensates for the built-in electric field in the QW. r 2007 Elsevier Ltd. All rights reserved. PACS: 78.20.Bh; 78.55.m; 78.67.De Keywords: III–V nitrides; Quantum wells; Screening
1. Introduction GaN-based quantum wells (QW’s) attracted much interest in the last few years since they have become the key material in commercial fabrication of long-time light emitting diodes (LED’s) and laser diodes for the blue to ultraviolet spectral range. Up to now InGaN/GaN is used as an active region in these devices. However, the lightemitting mechanism is not yet fully understood because this material exhibits some peculiarities. A peculiarity results from the polar axis of the wurtzite crystal structure and the strong polarity of III-N bindings. All group-III nitrides in the wurtzite phase have a strong spontaneous macroscopic polarization and large piezoelectric coefficients. This has been found from ab initio calculations [1,2]. The abrupt variation of the polarization at the surfaces and interfaces gives rise to large polarization sheet charges that in turn create internal electric fields of the order of MV/cm. The field-induced linear bending of the band edges causes a spatial separation of confined electrons and holes within the active layers of the devices and has, therefore, important consequences on the optical Corresponding author.
E-mail address: mrlopez@fis.cinvestav.mx (M.R. Lo´pez). 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.07.044
properties of the nitride-based LED’s or lasers. It is worth noting that the piezoelectric field present in the III–V nitrides appears in the presence of strain, due to, e.g., epitaxy, while the spontaneous polarization is a property of low-symmetry materials in their ground state, independent of strain, and it is absent in zinc-blende materials e.g., GaAs. In this work, we focus on the carrier density dependence of the photoluminescence (PL) energy. In QW’s the freecarrier screening effect built-in electric field is affected by the carrier density. Sala et al. [3] reported that the freecarrier screening effect explains some puzzling experimental data on nitride lasers, such as the unusually high lasing excitation thresholds and emission blue shifts that accompany increasing excitation levels. Recently Kuroda and Tackeuchi [4] performed a systematic time-resolved photoluminescence measurements of QW’s for various carrier densities and different well widths, and showed that the PL energy shift and the variation on the carrier lifetime can be explained by the free-carrier screening effect, which effectively screens the built-in spontaneous and piezoelectric polarization fields inducing a field-free band profile. We show approximate results on the electron (hole) energy in the QW in the presence of the internal electric field as a function of the carrier density. The Schrodinger
ARTICLE IN PRESS 448
M.R. Lo´pez, G. Gonza´lez de la Cruz / Microelectronics Journal 39 (2008) 447–449
equation is solved in the envelope approximation using a variational wave function for electrons and holes in the QW and the energy potential for the electron is assumed to consist of the confined potential, the potential produced by the built-in electric field and the contribution to the potential energy from the charge distribution of the electrons and holes in the QW. Our theoretical results are in qualitative agreement with the experimental results reported in the literature. 2. Theoretical model Since our study is based on the comparison of the PL spectra with the calculated transition energies, the model to calculate the well and barrier profiles to include in the Schro¨dinger equation is presented here. Consider first the Schrodinger equation of our system disregarding the Coulomb interaction. Following the usual choice for the electron three-dimensional wave function, i.e., the product of a plane wave and an envelope function cn;k ¼ eikr xn where r ¼ (x,y), k is the in-plane wave vector and n is the subband index, it is easy to put into the 3D Schrodinger equation to generate two equations, one for the 2D free motion and other for the bound state. The sum of the energy eigenvalues forms the total energy n;k ¼ _2 k2 =2m þ E n and it reduces the task to the resolution of a one dimensional Scrodinger equation. In the presence of the piezoelectric field in the QW in the z-axis, it reads _2 d 2 xn þ ½V F ðzÞ þ V H ðzÞxn ¼ E n xn , 2m dz2 where V 0 ðzÞ þ eFz jzjoa , V F ðzÞ ¼ 0 otherwise
(1)
(2)
and V0(z) ¼ V0[y(za)+y(za)] is the potential of the square QW, V0 is the band offset, y(z) is the Heaviside unit step function and 2a the QW width. This potential well and the electron and hole wave functions are schematically presented in Fig. 1. VH(z), the Hartree potential, is the
contribution to the potential energy from the charge distribution of the electrons and holes in the conduction and valence bands in the QW, respectively, and is given by the solution of Poisson’s equation with the charge density in the ground state as source term. The result is Z z e2 ðz0 zÞðx2h ðz0 Þ x2e ðz0 ÞÞ dz0 , (3) V H ¼ ns z þ 1 where ns is the 2D electron (hole) concentration in the lowest subband and e is the static dielectric constant of the QW, xe and xh are solutions of Eq. (1) for the electrons in the conduction band and holes in the valence band in the QW, respectively. It is the VH(z) in the effective one-electron potential of Eq. (1) that makes the Schrodinger equation a nonlinear eigenvalue problem, because the potential depends on the wave functions and besides that, Eq. (1) has to be solved simultaneously with Poisson‘s equation in a self-consistent approximation. Simple analytic wave functions make calculations of properties of QW’s much more convenient than they would if numerical self-consistent solutions or cumbersome analytic solutions like Airy functions had to be used. For that reason approximate solutions have been widely used in QW’s calculations [5]. One of the simplest of these is dz xðzÞ ¼ N 1 þ (4) j ðzÞ, 2a 0 1=2 is a normalization factor, NR ¼ 1 þ ðd=2aÞ2 z2 where 1 z2 ¼ 1 z2 j20 dz, j0 is the ground state particle wave function in the absence of the built-in electric field in the QW, and the parameter d in Eq. (4) is determined by minimizing the total energy of the system for given values of the QW density charges of electrons and holes. Because of the simplicity of the wave function, it is easy to evaluate the expectation values of all the terms in the Hamiltonian, Eq. (1). We find 2 2 eF d 2 2 _ d h H ix ¼ E 0 þ N z þ a 8ma2 e 2 ns 4 d d2 d3 N A1 þ A2 2 þ A3 3 , ð5Þ a 0 a a where Ai depending of the QW parameters, and are well determinated. Then the value of d that minimizes the total energy satisfies the following polynomial equation of fourth order: d4 þ
Fig. 1. Schematic picture of energies and wave functions of electrons and holes in a strained quantum well with piezoelectric field.
awþ 3 12a2 b 2 4a3 w 16a4 g ¼ 0. d þ 2 d þ 2 d þ a ma m z a z a
(6)
Here a ¼ 4CA3nseF/z2S2, b ¼ A1/z2S;4A3, g ¼ eF /z2SCA1ns, w ¼ _2 8mCA2 ns and C ¼ e2/e. It is worth to note that in the limit of low density carrier such that ns 5_2 =8maCA2 the solution of Eq. (6) reduces to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " aw 16m2 ag d¼ 1 1 . (7) w2 2ma
ARTICLE IN PRESS M.R. Lo´pez, G. Gonza´lez de la Cruz / Microelectronics Journal 39 (2008) 447–449
PL peak energy. [eV]
3.5
in Fig. 2, upon increasing the sheet carrier density we observe a blue shift in the transition energy. This is due to the progressive recovery of flat band conditions upon increasing the sheet carrier density. As can be seen our theory is in agreement with the self-consistent calculation developed in Ref. [3]. We obtain the recombination rate from Z Rr / xe ðzÞxhh ðzÞ dz,
3.0
Variational calculations. 50A 100A
2.5
and we found that the recombination rate increase nonlinearly for low densities and become constant for high- carrier densities in agree with the Refs. [3,5].
Selfconsistent approach. 50A 100A
2.0
449
3. Conclusions 0
5
10
15
20
ηs [ 1013 cm-2] Fig. 2. Electron–hole transition energy versus sheet carrier density for several well widths.
Once we know the variational parameter that minimizes Eq. (5), the PL energy peak can be calculated by the simple relationship as a function of the carrier density in the QW and the internal electric field as a fitting parameter E PL ¼ E g þ E e þ de þ E h þ dh ,
(8)
where Eg is the gap of the semiconductor QW, Ee(h) is the electron (hole) energy without electric field in the QW and deðhÞ ¼ hH ieðhÞ E eðhÞ is the induced energy shift for electrons (holes) which depends on the carrier density and the built-in electric field. In Fig. 2 we present the calculated transition energy between the highest hole level and the lowest electron level for an internal piezoelectric field F ¼ 3.25 MV/cm. As seen
In conclusion, we have shown that free carriers can effectively screen macroscopic polarization fields in nitride quantum wells, resulting in a non vanishing recombination rate for large wells (contrary to previous claims) in normal laser operation. A rather high sheet density is needed to achieve these conditions. We also explained the red shifts versus well width and blue shifts versus sheet density as resulting from the interplay of free-carrier screening and polarization fields. References [1] F. Bernardini, V. Fiorentini, D. Vanderbilt, Phys. Rev. B 56 (1997) R10024. [2] F. Bernardini, V. Fiorentini, Phys. Rev. B 57 (1998) R9427. [3] F.D. Sala, A. Di Carlo, P. Lugli, F. Bernardini, V. Fiorentini, R. Scholz, J.M. Jancu, Appl. Phys. Lett. 74 (1999) 2002. [4] T. Kuroda, A. Tackeuchi, T. Sota, Appl. Phys. Lett. 76 (2000) 3753. [5] G. bastard, E.E. Mendez, L.L. Chang, L. Esaki, Phys. Rev. B 28 (1983) 3241.