Impact of polarization on quasi-two-dimensional exciton and barrier-width dependence of the exciton associated transition in wurtzite III–V nitride quantum wells

PERGAMON

Solid State Communications 122 (2002) 287±292

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Impact of polarization on quasi-two-dimensional exciton and barrier-width dependence of the exciton associated transition in wurtzite III±V nitride quantum wells Shou-Pu Wan*, Jian-Bai Xia National Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, People's Republic of China Received 25 May 2001; accepted 10 November 2001 by Z. Gan

Abstract Excitonic states in AlxGa12xN/GaN quantum wells (QWs) are studied within the framework of effective-mass theory. Spontaneous and piezoelectric polarizations are included and their impact on the excitonic states and optical properties are studied. We witnessed a signi®cant blue shift in transition energy when the barrier width decreases and we attributed this to the redistribution of the built-in electric ®eld between well layers and barrier layers. For the exciton the binding energies, we found in narrow QWs that there exists a critical value for barrier width, which demarcates the borderline for quantum con®nement effect and the quantum con®ned Stark effect. Exciton and free carrier radiative lifetimes are estimated by simple argumentation. The calculated results suggest that there are ef®cient non-radiative mechanisms in narrow barrier QWs. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 73.21.Fg; 71.35. 2 y; 77.84.Bw; 81.05.Ea Keywords: A. Quantum wells; A. Semiconductors; D. Piezoelectricity; D. Optical properties

1. Introduction Group III nitride alloys, especially AlN/GaN/InN systems, have widely adjustable bandgaps (from visible red to ultraviolet); thus are considered as promising material systems for the fabrication of optoelectronic devices working at high power and high frequency. One of the most distinguished properties of QWs and epi®lms based on wurtzite AlN/GaN/InN systems is their enormous built-in electric ®eld (BEF) generated by macro-polarization with intensity as high as several 10 mV/nm. A great number of evidences have con®rmed the existence of BEF, and thereby, macro-polarization [1±5]. According to previous reports [6], polarization in wurtzite III±V nitrides includes both spontaneous (SP) which is bred by the wurtzite structure and piezoelectric (PZ) which is induced by the lattice mismatch. The strong BEF in these materials leads to a giant quantum con®ned Stark effect [7]. At the same time, due to * Corresponding author. Tel.: 186-10-82304291. E-mail address: [email protected] (S.-p. Wan).

their large bandgaps, large effective masses, and small dielectric constants, exciton holds a very important position in III±V nitrides as contrasted to III±V arsenides. The near band-edge luminescence spectra observed from many GaN samples, especially under low temperature, are dominated by strong, sharp emission lines of excitons, both bound and free [8,9]. In this article, we study the quasi-two-dimensional (quasi2D) exciton as well as exciton associated transition (EAT) in AlxGa12xN/GaN QWs within the framework of effectivemass theory. We ®rstly provide the background theory of polarization in nitride semiconductors then we calculate the electronic structure and interband transition matrix elements. Upon all these, we study the excitonic state and EAT and calculate the dependence of transition energy on barrier thickness and thereby theoretically testify the blue shift effect observed by Leroux et al. [10]. We show that for AlxGa12xN/GaN QWs with a certain well width, for exciton binding energies, there exists a critical value for barrier width. Below this value, the quantum con®ned Stark effect overwhelms the quantum con®nement effect, and vice versa. At last, we estimate exciton radiative lifetime and employ

0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(02)00037-6

288

S.-P. Wan, J.-B. Xia / Solid State Communications 122 (2002) 287±292

where e33 and e31 are the piezoelectric constants. The total polarization difference across the heterojunction is DP ˆ PbSP 2 Pw SP 1 PPZ :

…2†

It is DP that causes the electric charge deposit s e on the heterojunction, which in turn, results in the BEF. For QWs with well width lw and barrier width lb, the BEF for well layers and barrier layers is

Fig. 1. Schematic map for AlN/GaN lattices and heterojunction. As shown in the ®gure, the Ga±N and Al±N are both polarized bonds but with different magnitude, and this difference eventually causes the charge deposit s e on the heterojunction.

our results to corroborate the prediction that there is increasing effective non-radiative recombination as barrier width diminishes [11]. 2. Polarization and the built-in electric ®eld Bulk wurtzite GaN and AlN tend to develop SP. It is so for the following two facts: one is that the N±Ga (N±Al) bonds are polarized; the other is that in wurtzite structure, the crystal lattice lacks inverse symmetry due to the asymmetric arrangement of child lattices of nitrogen and gallium (aluminum) atoms [1]. Fig. 1 shows the crystal lattices of AlN and GaN and the formation of polarization and charge deposit on the AlN/GaN heterojunction. Many researchers have conducted theoretical calculation and designed experiments to reveal and determine the SP in nitride ®lms or QWs materials [6,7]. According to the calculation of Bernardini et al. [6], SP can be as high as 0.029 C/m 2 in GaN and 0.081 C/ m 2 in AlN. Lattice mismatch in epitaxial materials is commonly present, especially in wurtzite III±V nitrides. For instance, AlxGa12xN/GaN QWs is usually epitaxially grown on buffer layers GaN along the crystal axis (0001), taken sapphire or 6H±SiC as the substrate. The AlxGa12xN layers, which have smaller lattice parameters, are biaxially strained while the GaN layers remain unstrained. PZ thus is introduced into the barrier layers. Following Bernardini et al. [6], PZ can be expressed as PPZ ˆ e33 1zz 1 e31 …1xx 1 1yy †;

…1†

Eb ˆ

lw DP ; lb 1w 1 lw 1b

…3†

Ew ˆ

lb DP ; lw 1b 1 lb 1w

…4†

respectively, where 1 w and 1 b are the dielectric constants. Clearly, we can effectively alter the electric ®eld intensity in the wells by changing the barrier width based on Eqs. (3) and (4).

3. Quasi-2D exciton It is expedient to review the feature of GaN electronic structure near the Brillion zone center. Wurtzite GaN is a kind of direct-band semiconductor material. The conduction band minimum (CBM) is almost immune from the outer in¯uences and remains parabolic. The valence band maximum (VBM), however, is distorted and somewhat complex. The originally degenerate three p-bands, which originate from px, py, pz states of the nitrogen atoms, split into three bands, namely, HH, LH, and HH 0 due to the combined effects of crystal ®eld and spin±orbit interaction. In the effective-mass theory, the Hamiltonian for electron is "2 2 7 1 Eg ; 2mpc

…5†

where mpc is the effective mass of electron, Eg is the bandgap. The Hamiltonian for hole is [16] 2

Heff

F 6 p 6K 6 6 p 6H 6 ˆ6 6 6 0 6 6 6 0 4 0

K

H

0

0

G

Hp

0

H

W

0

0 p 2l3

0

0 p 2l3

F

Kp

K

G

0

H

Hp

0 p 2 2l3

3 0 p 7 2 2l3 7 7 7 0 7 7 7; …6† 7 p 7 H 7 7 H 7 5 W

S.-P. Wan, J.-B. Xia / Solid State Communications 122 (2002) 287±292

289

Table 1 Ê ), dielectric constants (1 0), free carrier Crystal lattice parameters (A and A-exciton bandgaps, and bowing parameters (eV) [20]

a0

c0

1 [1]

Eg a

Egex ; [21±23]

b [20]

GaN AlN

3.1893 3.1330

5.1851 4.9816

10.0 8.5

3.503 6.28

3.485 6.20

1.3

a

Free carrier bandgap at 1.6 K.

(hole) the expressions can be expressed as   8 lb 1 lw > > ; 2 eF z 2 DE > c;…v† b > 2 > > > < V 0c;…v† …z† ˆ eFw z; > > >   > > l 1l > > : DEc;…v† 2 eFb z 1 b w ; 2

Fig. 2. Sketch for band pro®les, and electron and hole wavefunctions in AlxGa12xN/GaN QWs. The bottoms of the wells are tilted severely by the strong BEF, and as a consequence, the electron and hole tend to aggregate at the corners of the well. The broken line portrays the situation of unpolarized QWs (¯at QWs).

where Fˆ

   1 L1M 2 Px 1 P2y 1 NP2z 2 l2 2 …C1 1 C3 †1zz 2 2

2 …C2 1 C4 †1' ; Gˆ

   1 L1M 2 Px 1 P2y 1 NP2z 1 l2 2 …C1 1 C3 †1zz 2 2 2 …C2 1 C4 †1' ;

 i 1h  2 S Px 1 P2y 1 TP2z 1 Dc 2 C1 1zz 2 C2 1' ; 2    1 L2M 2 Px 2 P2y 2 iRPx Py ; Kˆ 2 2   1 A Q p P0 …Px 2 iPy † 1 p …Px 2 iPy †Pz ; Hˆ 2 2 2 Wˆ

in which, L, M, N, R, S, T, Q, A are the Luttinger-like parameters; C1, C2, C3, C4, the deformation potentials (DPs); D c, the crystal-®eld splitting (CFS) energy; and l 1, l 2 are the spin±orbit coupling (SOC) energies. This 6 £ 6 effectivemass Hamiltonian is based on the following six basic functions u11l "; u1 2 1l "; u10l "; u1 2 1l #; u11l #; u10l # which describe the six hole states (spin state included) at the VBM. In polarized AlxGa12xN/GaN QWs, perturbation comes from two sources, namely, the band offset and the electrostatic potential. They are treated as a whole, and for electron

lw l 1l #z# b w; 2 2 lw ; 2 l 1l l 2 b w , z,2 w ; 2 2

uzu ,

…7† respectively, where DEc;…v† is conduction (valence) band offset (We assume DEc =DEv ˆ 75 : 25) and Fw;…b† is the BEF intensity in well (barrier) layers. Perturbation and well pro®le are plotted in Fig. 2. General discussions about the eigen energies and interband transition in nitride QWs can be seen in Refs. [12,13]. Within the framework of effective-mass theory, the envelop function of free exciton is the solution of the following SchroÈdinger equation " "2 2 "2 22 "2 " 22 2 7k 2 2 7 k2 2 p p p 2 2mpzv 2z2v 2mc 2mc 2zc 2mkv # e2 0 0 1V c …zc † 1 V v …zv † 2 1u~r c 2 ~r v u £ F ex …~rc ; ~rv † ˆ …E 2 Eg †F ex …~r c ; ~r v †;

(8)

where we have made the so-called `decouple approximation' for the hole. The subscript k denotes quantities related to the directions in the x±y plane, while z denotes those related to z direction. For E1±HH1 exciton, the hole effective mass in Eq. (8) can be deduced from the Luttinger parameters as, mpkv ˆ 2=…L 1 M†; and mpzv ˆ 1=N: Parameters and constants used in this article are listed in Tables 1±6. Although both electron and hole can move freely along directions within the x±y plane, their movements are modulated by the well-barrier pattern and the sawteeth of BEF in the z direction. According to the ready theory of two-particle system, the movement of exciton in QWs can be separated into two independent ones: the centroid movement within the x±y plane and the relative movement between electron and hole. The exciton envelope functions can be written as 1 ~ ~ F ex ˆ p eiK k ´Rk F 0ex ; NVk

…9†

~ k is the exciton wave vector in x±y plane, V k is the where K

290

S.-P. Wan, J.-B. Xia / Solid State Communications 122 (2002) 287±292 Table 3 Deformation potentials (eV) [24]

C1 1 Dc a

C2 1 Dc b

C3

C4

GaN AlN

13.87 12.34

13.71 9.36

22.92 24.76

1.56 2.08

a

A-exciton emission line. Dc is the DP of conduction band. It is conventionally as well as expediently ®gured in the corresponding DPs of valence band. b

where Eex ˆ E 2 Ec 2 Eh is the binding energy of exciton, and

r† ˆ I…~

Fig. 3. Binding energies of exciton as functions of barrier width in AlxGa12xN/GaN QWs with x ˆ 0.03, 0.10, 0.17, 0.24, 0.31. Two well widths, 16 and 4 ML, are employed.

area covered by the projection of the basic crystal cell, and F 0ex abides by " "2 2 "2 22 "2 22 7r 2 2 2 2mk 2mpzc 2z2c 2mpzv 2z2v # e2 1Vc …zc † 1 Vv …zv † 2 F 0ex …zc ; zv ; r~ † ˆ EF 0ex …zc ; zv ; r~ †; 1r …10† in which m k is the reduced effective mass of the two particle system in the x±y plane 1 1 1 ˆ p 1 p ; mkc mkv mk ~r is the relative position vector between electron and hole ~r ˆ ~r c 2 ~r v ; and r~ is its x±y component. We assume

F 0ex …zc ; zv ; r~ † ˆ c c …zc †c v …zv †fn …~ r †;

…11†

where c c …zc † and c v …zv † are electron and hole envelop functions. On substituting Eq. (11) into Eq. (10) and carrying out the integral over the whole QWs along the z direction, we get a more concise formula " # "2 2 2 p 7 r 1 I…~ r † fn …~ r † ˆ Eex fn …~ r †; …12† 2mk

lb lb Z lw 1 Z lw 1 2 2 lw 1 lb 22

lw 1 lb 22

dzc dzv uc c …zc †u2 uc v …zv †u2

2e2    1 r2 1 …zc 1 zv † 1=2 is the z direction average of Coulomb interaction between electron and hole. To solve Eq. (12), we use variational approach with trial functions to be the linear combination of the non-orthogonal Gaussian functions. This practice proves to be pretty accurate [17,18]. Fig. 3 shows the free exciton binding energies as functions of barrier width for AlxGa12xN/GaN QWs, with x ˆ 0.03, 0.10, 0.17, 0.24, 0.31, respectively. Two well widths are Ê ). The most evident charused: 4 and 16 ML (1 ML < 2.6 A acteristic of the value is that the binding energies of all the QWs with 4 ML-well width converge at the critical barrier width of 7.5 nm. We explain this by the competition of two opposite effects, i.e. the quantum con®nement effect and quantum con®ned Stark effect. As previously mentioned, the intensity of electric ®eld in the well layers increases as the barrier width increases and vice versa. Simultaneously, the quantum con®nement effect is enhanced as the barrier width increase and vice versa. So, if the barrier width is decreased, the quantum con®ned Stark effect will be weakened and, at the same time, the con®nement effect will be impaired. However, there is a signi®cant difference between their speeds. So when the barrier width is below a certain value, 7.5 nm for the case of 4 ML QWs, the quantum con®nement effect fails away more than quantum con®ned Stark effect does, that is, the tunneling of electron and hole is so strong that they over¯ow to an extensive area into the barriers. Thus, we observed a reversed order of the excitonic binding energy as plotted against the fraction of aluminum. The barrier width goes up well above this critical value, however, the dependence of the excitonic

Table 2 Conduction and valence band parameters in the framework of effective-mass theory (m0)

GaN AlN

mpe ; [24]

L [16]

M

N

R

S

T

Q

A

0.20 0.27

6.3055 2.784

0.1956 0.174

0.3813 0.281

6.1227 2.614

0.4355 0.467

7.3308 3.668

4.0200 0.850

0.6751 1.688

S.-P. Wan, J.-B. Xia / Solid State Communications 122 (2002) 287±292 Table 4 Stiffness coef®cients (GPa)

291

Table 6 CFS energy, and SOC energies of GaN (eV)

[25]

C11

C12

C13

C33

C44

C66

[26]

Dc

l2

l3

GaN AlN

396 398

144 140

100 127

392 383

91 96

126 129

GaN

0.01

0.0062

0.0055

can be obtained as: binding energy on the fraction of aluminum is set back to its ordinary order. 4. Exciton associated transition and exciton radiative lifetime in AlxGa12xN/GaN QWs EAT occurs between the so-called `vacuum state' u0l and an exciton state uC ex l: In nitride QWs, exciton becomes quasi-2D as a consequence of quantum con®nement effect, and excitonic binding energy is much larger than that in bulk materials. However, due to the presence of macropolarization and thereby the BEF, great changes take place in quasi-2D exciton as well as EAT. Ground state EAT energy is the algebraic sum Egf 1 Ec 2 Ev 2 Eb ;

…13†

where Egf is the free carrier bandgap; Ec and Ev are electron and hole ground state energies; Eb is the binding energy of the ground state exciton. As mentioned earlier, the energy levels of electron and hole fall into the forbidden band because of the tilting and lowering effect of the BEF. The binding energy of exciton is also reduced by quantum con®ned Stark effect. Fig. 4 shows the calculated EAT energies vs barrier thickness for Al0.17Ga0.83N/GaN QWs. Except the 4 ML QWs, all others undergo signi®cant blue shift in the transition energies with the decreasing of barrier width. In the 4 ML QWs, quantum con®nement effect and quantum con®ned Stark effect neutralize each other and the total effect is that the transition energy is almost constant. Also shown in Fig. 4 are experimental data taken from Leroux et al. [10]. However, comparing the experimental and theoretical results, we ®nd that there are some discrepancies between experimental and theoretical results. A possible explanation is that the magnitude of polarization in wurtzite III±V nitrides epilayers is sensitive to the subtle growth conditions which frequently fail to be identical. Besides, there is possible error that comes from the ¯uctuation of epilayer thickness. After all, transition energies are very sensitive to the well width as pointed out in many previous works [14,15]. EAT transition matrix element in AlxGa12xN/GaN QWs

 2 2  ~ 2 ~^ C v uMex ˆ ukC ex uH^ eF u0lu2 < F…zc ; zv ; r~ †~rˆ0 ´ C c ueik´~r a~ ´Pu 2 l 1l Z b w 2 ´uMcv u2 ˆ u f …0†u ´ c …z† c …z† dz c v lb 1 lw 2 2 2

(14)

Expression (14) indicates that the EAT transition matrix element is the product of the interband transition matrix element and the value of exciton envelop function with the relative position vector ~r set zero. Exciton radiative lifetime is proportional to the reciprocal of the EAT transition matrix element, namely 1= tr , ; …15† uMex u2 thus we can take the reciprocal of EAT matrix element as an analog of exciton radiative lifetime. Fig. 5 (a) displays the reciprocal of the EAT matrix elements; (b) displays that of the intersubband transition matrix elements. We observe two opposite trends in (a) and (b): the carrier lifetime of intersubband transition increases when increase the barrier width, while the exciton radiative lifetime decreases, especially for narrow QWs. For example, in 4 ML-well width QWs, the exciton radiative lifetime decreases more than nine-tenth

Table 5 SP values and PZ coef®cients (C/m 2) [6]

Psp

e33

e31

GaN AlN

20.029 20.081

0.73 1.46

20.49 20.60

Fig. 4. EAT energies are plotted as functions of barrier width for Al0.17Ga0.83N/GaN QWs with several well widths. The squares, up triangles, diamonds, and circles mark the experimental values of the samples with well widths of 4, 8, 12, and 16 ML, respectively, taken from recent experiment of Leroux et al. [10].

292

S.-P. Wan, J.-B. Xia / Solid State Communications 122 (2002) 287±292

Fig. 5. As an estimate for free carrier lifetime and exciton radiative in III±V nitride QWs, (a) the reciprocals of exciton associated and (b) intersubband transition matrix elements are plotted as functions of barrier thickness, respectively. Samples used in the calculation are Al0.17Ga0.83N/GaN QWs with well width ˆ 4, 8, 12, 16, 20 ML.

when barrier width changes from 5 to 50 nm, while free carrier lifetime increases more than 2 times. This phenomenon indicates that EAT will dominate radiative recombination in wide barrier QWs. We thus get results consistent with these made by Gallart et al. [11,19]. As barrier width decreases, the interband radiative lifetime declines, but the exciton radiative lifetime increases. In addition, experiments [11] show that the total carrier lifetime decreases abruptly. (See Fig. 6, where the PL experiment decay time of the 4 ML Al0.17Ga0.83N/GaN QWs are shown as functions of barrier width.) Therefore, there may be a non-radiative mechanism that accounts for the decrease of carrier lifetime in narrower barrier QWs. And one of the possible mode is the speci®c percolative transportation of hole between aluminum atoms [19]. 5. Conclusion To sum up, we have calculated quasi-2D exciton states in AlxGa12xN/GaN QWs employing the effective-mass approximation theory. The EAT has been considered in detail. As a result, we ®nd that the EAT energy undergoes a signi®cant blue shift when the barrier width is decreased and that in narrow QWs, for the exciton binding energies there exists a critical value for the barrier width that characterizes the competition between the two effects taking place in QWs: quantum con®nement effect and quantum con®ned Stark effect. We have also studied the EAT transition matrix elements and the exciton radiative lifetime. As a result, we ®nd that as the barrier width increases, the exciton radiative lifetime decreases, and the EAT dominates the radiative recombination gradually in AlxGa12xN/GaN QWs.

Fig. 6. The PL decay times of the 4 ML Al0.17Ga0.83N/GaN QWs vs barrier width (taken from recent experiment of Gallart et al. [11]).

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Acknowledgements

[19] [20] [21] [22] [23] [24]

This work is supported by the National Natural Science Foundation of China.

[25] [26]

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