Influence of nonparabolicity on electronic structure of quantum cascade laser

JID:PLA AID:22600 /SCO Doctopic: Condensed matter

[m5G; v 1.134; Prn:15/05/2014; 9:33] P.1 (1-4)

Physics Letters A ••• (••••) •••–•••

1

Contents lists available at ScienceDirect

2

67 68

3

Physics Letters A

4 5

69 70 71

6

72

www.elsevier.com/locate/pla

7

73

8

74

9

75

10

76

11 12 13

Influence of nonparabolicity on electronic structure of quantum cascade laser

14 15 16 17

81

School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia

83

24 25 26 27 28 29 30

82 84 85

a r t i c l e

i n f o

a b s t r a c t

86

21 23

79 80

19

22

78

´ Vitomir Milanovic, ´ Jelena Radovanovic´ ∗ Nikola Vukovic,

18 20

77

87

Article history: Received 9 December 2013 Received in revised form 28 April 2014 Accepted 29 April 2014 Available online xxxx Communicated by R. Wu Keywords: Conduction band nonparabolicity Quantum wells Quantum cascade laser

We analyze the influence of nonparabolicity on the bound electronic states in the conduction-band of quantum wells in external electric field. Numerical results, obtained by transfer matrix method are presented for active region of GaAs/Al0.3 Ga0.7 As quantum cascade laser. The structure was initially optimized by genetic algorithm, using Kane’s model of nonparabolicity, with emission wavelength set to λ ≈ 15.1 μm. However, our numerical results indicate the change in lasing wavelength to 14.04 μm when using a more comprehensive description of nonparabolicity. © 2014 Published by Elsevier B.V.

88 89 90 91 92 93 94 95 96

31

97

32

98

33 34

99

1. Introduction

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

When electrons occupy the energy levels on the order 100 meV above the bulk conduction-band edge of a semiconductor well ma terial, nonparabolicity effects become sign ficant. Several consider ations of this problem can be found in the literature, introducing energy-dependent electron effective mass. First models which in cluded nonparabolicity effects used a small number of bands, and treated the other bands as perturbations [1–4]. In [5,6] the authors used a more precise approach with higher conduction bands in volved, in order to provide better description of conduction-band when energies are greater than 50 meV above the band edge. Starting from the method presented in [6] which assumes 14-band k–p calculation, a more convenient method was developed, which used the bulk dispersion from [6] and from it determined the co efficients in an expansion of dispersion relation up to fourth order in wave vector [7–9]. It was stated in [9] that one possible bound ary condition candidate to be used at the well/barrier interface is the one satisfying the conservation of probability current along the structure. This above mentioned boundary condition was derived in detail in [10] by double integration of Schrödinger equation. An exact approach for solving the Schrödinger equation in case of a semiconductor quantum well (QW) under applied bias is rather cumbersome, especially with nonparabolicity involved. One of the numerous applications of a QW structure in the applied electric field is certainly quantum cascade laser (QCL).

60 61 62 63 64 65 66

*

Corresponding author. Tel.: +381 113370088. ´ E-mail address: [email protected] (J. Radovanovic).

http://dx.doi.org/10.1016/j.physleta.2014.04.069 0375-9601/© 2014 Published by Elsevier B.V.

QCLs are powerful light sources for mid- and far-infrared spec tral range which have turned out to be very fficient and reliable in free-space communications, medical diagnostics and chemical sensing [11–15]. Changing the design of the active region, one can obtain wide scope of operating wavelengths ranging from 3 μm up to 250 μm. Although the best performance in mid-infrared spectral range is achieved in GaInAs/AlInAs based QCLs [16,17], there are limitations concerning their chemical sensing applica tions [18]. On the other hand, GaAs/AlGaAs QCLs offer several benefits during fabrication since GaAs technology is well estab lished and does not require extremely demanding growth process and precise control [19,20]. Band nonparabolicity parameters rel evant for the effects considered in this work are well known, experimentally and theoretically, for GaAs/AlGaAs system, which is also lattice-matched to GaAs for any Al content. One needs to find optimal structure parameters for the desired wavelength us ing carefully selected optimization technique such as the genetic algorithm, which enables systematic search of the free parameters space and proves to be one of the best tools for global optimiza tion problems [21–23]. In this paper we describe our model of quantum well in applied electric field and utilize it on a suit able QCL active region designed for the mid-infrared spectral range (15.1 μm). This active region was previously obtained by structural parameters optimization using genetic algorithm which assumes a simpler relation for nonparabolicity effects. Our numerical results show that the bound states energies become shifted within the improved model, and that the lasing wavelength actually amounts to 14.04 μm.

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:PLA

AID:22600 /SCO Doctopic: Condensed matter

[m5G; v 1.134; Prn:15/05/2014; 9:33] P.2 (1-4)

N. Vukovi´c et al. / Physics Letters A ••• (••••) •••–•••

2

1

2. Theoretical considerations

67

2 3 4

68

In case of a nanostructure, the Schrödinger equation for the en velope wave function ψ( z) reads:

5 6 7

d2

ˆ ψ( z) = H

10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 26 27 28 29 30 31 32 33 34 35 36 37 38

−

h¯ 2 d

41 42 43 44 45 46 47 48 49 50 51 52 53



1

62 63 64 65 66

dz

74 75 76 77 78 79 80 81 82 83 84 85

Fig. 1. Dispersion relation for k > 0. Note that for k > kmax the branch does not have a physical meaning.

4

α0k +

2 2

h¯ k

2m∗

= E − U,

90

+ U ( z)ψ( z) = E ψ( z).

91

(2)

92 93 94 95

(3)

96 97

α0 < 0

(4)

 2 k

h¯

= 1/2





1± 4|α0 |m∗



1 − 16|α0 | E

m∗

2  .

h¯ 2

(5)

However, when E > 0 and parameter α0 approaches zero (the case of parabolic dispersion relation) it is obvious that a pair of solu tions with + sign diverges, and thus does not possess physical justification. The only two solutions remaining are from the − branch:

k2 =

h¯

2





1− 4|α0 |m∗

 1 − 16|α0 | E

m∗ h¯ 2

2  .

(6)

Similarly, when E < 0, it can be shown that solutions are in the following format:

κ2 =

h¯

2

4|α0

|m∗



 −1 +

87 89



where U is the potential energy at the bottom of the material’s conduction-band and we can set U = 0. Algebraically, Eq. (4) has four solutions: 2

86 88

where E g ( z) signifies the spatial dependence of the direct energy gap. The model presented in this paper uses the Schrödinger equa tion (1), and gives an improvement in the bound state energies compared to the second model which relies on Eq. (2). Since the solutions of the Schrödinger equation are ψ ∼ e ikz , the dispersion relation in a single material (i.e. GaAs) reads:

60 61

73

(1)

  E − U ( z) , M ( z, E ) = m ( z) 1 + E g ( z)

57 59

72

dz

∗

55

58

dψ( z)

Here M ( z, E ) represents the effective mass in form:

54 56

dψ( z)

2 dz M ( z, E )

39 40

1

where m∗ is the effective mass at the bottom of the conduction band, α0 is the nonparabolicity parameter and U ( z) is potential energy. If the concentration of ionized impurities is not very high, we may assume that U ( z) originates from the conduction-band discontinuity only. The formula (1) is valid when the in-plane wave vector is equal to zero. Otherwise, additional terms depending on the components of in-plane wave vector should appear, as given in [9]. This form of the Schrödinger equation arises from a model which includes interaction of the lowest conduction band with the light-hole band, split-off band and higher conduction bands, as stated in [9]. A simpler model for electrons in the conduction band, which considers only interaction between conduction and light-hole bands, yields Schrödinger equation valid at the bottom of the subbands, where the in-plane wave vector is zero [24]:

24

70 71

h¯ 2 d

α0 (z) − 2 dz m∗ ( z) dz2 dz2 + U ( z)ψ( z) = E ψ( z),

8 9

d2 ψ( z)

69

 1 + 16|α0 || E |

m∗ h¯

2

2  ,

(7)

where κ = ±ik. One can easily see from the first derivative of the dispersion relation that it has a minimum at k = 0 and maximum value of:

98 99 100 101 102 103 104

Fig. 2. In case when a moderate electric field is applied (i.e. K = 45 kV cm−1 ), it is acceptable to approximate the potential at structure’s ends with horizontal lines.

h¯ 4

h¯ 2

105 106 107

(8)

108

In case of GaAs, the maximum energy is E max = 384 meV (Fig. 1). Inflection point of − branch of the dispersion relation lies at the energy E π = (5E max )/9, which corresponds to 213 meV in GaAs. Starting from d finition of the effective mass:

110

E max =

1 m(k)

=

16|α0 |(m∗ )2

1 d2 E (k) h¯ 2

dk2

,

=

when k2max

12α0 k2 h¯ 2

+

1 m

=

, ∗

4|α0 |m∗

.

109 111 112 113 114 115

(9)

it is clear that for energies greater than E π , effective mass in conduction-band becomes negative. The application of electric field to the QW provides external ad justment of electronic structure and optical properties of the well, which can be utilized in numerous applications (i.e. realization of QCLs). The field alters the potential as shown in Fig. 2. If we assume that the field exists in the entire range of z-coordinate from −∞ to +∞, strictly speaking, bound states do not exist anymore and the energy spectrum became contin uous. However, in case of moderate external electric fields, the states that were previously bound, evolve into so-called quasi bound states after the application of bias, which can then still be treated as bound under additional assumption that the poten tial is constant far enough from the well [25]. This consideration is exploited in our model in order to solve Eq. (1) in a simpler

116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:PLA AID:22600 /SCO Doctopic: Condensed matter

[m5G; v 1.134; Prn:15/05/2014; 9:33] P.3 (1-4)

N. Vukovi´c et al. / Physics Letters A ••• (••••) •••–•••

matrix for the whole structure. In order to find the bound states, we calculate the transfer matrices for every bound-state energy candidate starting from the potential minimum. Having:

1 2 3



4 5 6 7



9 10 11 12 14 15

Fig. 3. Potential step which illustrates our approximation of linear potential between two points on the z-axes, when structure is placed in electric field. The cases in which the energy is below U 2 or above U 1 are analogous.

17 19 20 21 22 23 24 25 26 27 28 29 30 31 32

manner than within a rigorous approach which is rather arduous when nonparabolicity is involved. We numerically solve Eq. (1) us ing transfer matrix method (TMM), by dividing the potential profile into N equal segments, each having U ( z) = const. and two adjacent segments forming a potential step. Fig. 2 shows how our model (dotted line) approximates the real potential. We will illustrate our model on the example of a potential step shown in Fig. 3. In this example, for U 2 < E < U 1 in the vicinity of z = a, the wave function is given by:



A 1 e κ1 z + B 1 e −κ1 z ,

ψ( z) =

A 2 e ik2 z + B 2 e −ik2 z ,



34

κ12 = −

36

h¯

−1 +

2

2m∗1

37 38 39

k22 = −

42 43 44 45 46 47

h¯

1−

2

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

κ1 and k2 are real

h¯

(11)

2α01





1 + 4α02 ( E − U 2 )

2m∗

=

α1 β1 γ 1 δ1







h¯

2

(12)

Ak Bk



    h¯ 2 dψ(a− ) h¯ 2 dψ(a+ ) 2 −2α01 κ12 + = 2 α k + 02 2 ∗ ∗ dz

2m2

dz

(13)

at point z = z0



A2 B2



=

1 2



(1 + qa )e (κ1 −ik2 )a (1 − q )e (κ1 +ik2 )a a

  (1 − qa )e −(κ1 +ik2 )a A1 (14) −( κ − ik2 )a 1 B (1 + q )e a

1

72 74 75

at z = zk ,

76 77



h¯ 2 2m∗1

0 B N +1

=

N +1   i =1

αi βi γi δi



A0 0



 =

m11 m21

m12 m22





A0 . 0

+

81 82

(17)

obtains the form (i = j):



79 80

(16)

Theoretically, bound-states are obtained from the condition: m11 ( E ) = 0. In our model, we find the bound states energies at minima of moduli of matrix element m11 , because these quantities are in fact complex numbers. One of the key parameters in calculations related to quantum cascade lasers, for which it is necessary to solve the Schrödinger equation, is certainly the optical transition matrix element. At this point we will give a brief analysis of this quantity for the two cases of Schrödinger equation given by formulas (1) and (2). Eq. (2) contains the conventional energy-dependent mass de scription of nonparabolicity given by Eq. (3); since the effective mass depends on energy, corresponding wave functions are non orthogonal. However, as this model includes interaction of the conduction-band and the light-hole band, the wave function be comes a vector: ψ = [ψc , ψlh ], where ψc and ψlh stand for electron and light-hole wave functions respectfully and it was shown in [26] that the wave functions ψi and ψ j are orthogonal. The transi tion matrix element which is defined as:

z = z I ,

78

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

ψc∗,i z ψc, j dz h¯



2

2( E i − E j )

106

ψc∗,i

d dz

 z

1 mj

−

1 mi



dψc , j dz

107

dz.

108

(18)

110

A straightforward treatment shows that the model described by the Schrödinger equation (1) should use a different expression for transition matrix element:

di , j = ψi | z |ψ j  =

ψi∗ z ψ j dz,

109 111 112 113 114

(19)

115 116

where ψi and ψ j represent the wave functions of Eq. (1).

117 118 119 120



2α κ − κ1 i qa =  2  h ¯ 2α01 k22 + 2m ∗ k2 2 01 1



3. Numerical results

where expression for qa reads:

69 71

i =1



Now we can calculate A 2 and B 2 in terms of A 1 and B 1 as:

68 70

A0 , B0

  k   αi βi A0 = , B0 γi δi

di , j =

The first boundary condition for the solution is the continuity of the wave function at z = a, and the second boundary condition is obtained from the continuity of the probability current, derived in detail in Ref. [10], yielding:

2m1



67

73

di , j = ψi | z|ψ j ,

2m∗2 2

2α02

48 49

(10)

z>a

 2m∗ 2 1 − 4α01 (U 1 − E ) 21

2

40 41

z
A 1 , B 1 , A 2 , B 2 are complex constants, while quantities with positive values:

33 35



the co fficients B 0 and A N +1 must be zero in a bound state case:

16 18

A1 B1



and

8

13

3

(15)

2

Furthermore, in E < U 2 case, the quantity k22 should be replaced with −κ22 , where κ22 is calculated according to Eq. (11) but with U 2 instead of U 1 . On the other hand, in the case where U 1 < E, the quantity κ12 should be replaced with −k21 , and k21 is calculated according to Eq. (12) but with U 1 instead of U 2 . Similar calculation is carried out in different parts of the spectrum if U 1 > U 2 . Next, we compute the transfer matrices for each segment at the given energy, and by multiplying them together, we obtain a

Firstly, we numerically calculate the influence of nonparabolic ity on the ground-state energies in case of a rectangular GaAs/Al0.3 Ga0.7 As QW with finite barrier heights in applied electric field of magnitude K = 45 kV cm−1 . Table 1 shows two cases, first with the nonparabolicity included according to Eq. (1), and second, the parabolic case, as a function of the well width. The next example considers an active region of quantum cas cade laser. We carefully chose a QCL active region with barrier height of U b = 251 meV. The operation of the QCL in our exam ple is based on inter-subband transitions in the conduction-band. Structural parameters of the active region amount to 34, 55, 54, 19, 51 Å for well and barrier widths, respectively, going from left to

121 122 123 124 125 126 127 128 129 130 131 132

JID:PLA

AID:22600 /SCO Doctopic: Condensed matter

[m5G; v 1.134; Prn:15/05/2014; 9:33] P.4 (1-4)

N. Vukovi´c et al. / Physics Letters A ••• (••••) •••–•••

4

1 2 3 4

Table 1 Ground state confinement energies as a function of the well width L z in parabolic (PRB) and nonparabolic (NP) case from Eq. (1) when electric field of 45 kV cm−1 is applied.

5

L z [Å]

PRB [meV]

NP [meV]

6

30 40 50 60 70 80 90

132 105 87 74 65 59 55

146 116 95 80 70 63 58

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

4. Conclusion

68

In summary, the relevance of in-depth modeling of conduction band nonparabolicity in III–V compound semiconductor quantum

69

well structures in external electric field has been demonstrated on the example of active region of GaAs/Alx Ga1−x As based quantum cascade laser. The applied procedure can easily be modified for other material systems with band gaps large enough so that the nonparabolicity may be approximated by the fourth order expres sion in wave vector. Starting from a structure designed by using the genetic algorithm to operate at a specified wavelength, we find an increase in bound-state energies and a change in calculated lasing wavelength. The differences stem from the fact that our model utilizes a more complex description of nonparabolicity than the one implemented in the genetic algorithm driven optimization procedure. In addition, we have evaluated the energy range, mea sured from the bottom of the conduction band, where our model remains to provide valid results. Our method of choice in numeri cal calculations is the transfer matrix method, which allows us to approximate the linear potential resulting from the applied electric field with a step-wise constant profile.

71

22

Acknowledgements

24

This work was supported by the Ministry of Education, Sci ence and Technological Development (Republic of Serbia), ev. no. III 45010, NATO SfP Grant, ref. no. 984068.

26 27 28

References

30 32 33

Fig. 4. The active region of QCL, expected to lase at λ ≈ 14.04 μm. The layer widths (starting from the left well) are 34, 55, 54, 19, 51 Å, and the electric field is set to K = 45 kV cm−1 . The relevant energies and the wave functions squared are also displayed.

37 38 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

74 75 76 77 78 79 80 81 82 83 84 85 86 87 89 91 92 93 95 96

31

39

73

94

29

36

72

90

25

35

70

88

23

34

67

right and the barrier height is U b = 251 meV, which corresponds to aluminum mole fraction of 30%, in GaAs/Al0.3 Ga0.7 As struc ture. The material properties used in calculation are m∗ = 0.067m0 for GaAs and m∗ = 0.0919m0 for Al0.3 Ga0.7 As, the conduction band discontinuity between GaAs and AlAs is 835 meV. The ap plied electric field along z-axes is K = 45 kV cm−1 and non 4

4

parabolicity parameters are α0 = −1164 eV Å and −2107 eV Å for Al0.3 Ga0.7 As and GaAs, respectively, according to [9]. Layer widths and magnitude of K are results of genetic algorithm op timization, for emission at λ ≈ 15.1 μm, as described in [27]. Ex pected minima of energy subbands are at E 1 (k|| = 0) = 81.5 meV, E 2 (k|| = 0) = 117.3 meV and E 3 (k|| = 0) = 199.1 meV. However, in our simulations, bound energies amount to E 1 (k|| = 0) = 87.3 meV, E 2 (k|| = 0) = 124.5 meV and E 3 (k|| = 0) = 212.8 meV as presented in Fig. 4. The energy difference (E 3 − E 2 ) thus corresponds to a lasing wavelength of λ ≈ 14.04 μm. One can see that the changes in bound states energies and lasing wave length, due to a more comprehensive modeling of nonparabolicity, are not negligible. Correspondingly, when we set the nonparabol icity parameters to zero, the same results for quasi-bound energies are obtained as in previously used models which employed Eq. (2) [27], and as expected, when the bias is off (i.e. K = 0) our model matches analytically obtainable values.

[1] E.O. Kane, J. Phys. Chem. Solids 1 (1957) 249. [2] G. Bastard, Phys. Rev. B 25 (1982) 7584. [3] G. Bastard, in: L.L. Chang, K. Ploog (Eds.), Molecular Beam Epitaxy and Het erostructures, Martinus Nijhoff, Dordrecht, 1985, p. 381. [4] S. Yamada, A. Taguchi, A. Sugimura, Appl. Phys. Lett. 46 (1985) 675. [5] U. Rössler, Solid State Commun. 49 (1984) 943. [6] M. Braun, U. Rössler, J. Phys. C 18 (1985) 3365. [7] F. Malcher, G. Lommer, U. Rössler, Superlattices Microstruct. 2 (1986) 267. [8] U. Ekenberg, Phys. Rev. B 36 (1987) 6152. [9] U. Ekenberg, Phys. Rev. B 40 (1989) 7714. ´ J. Radovanovic, ´ S. Ramovic, ´ Phys. Lett. A 373 (2009) 3071. [10] V. Milanovic, [11] C. Gmachl, F. Capasso, D.L. Sivco, A.Y. Cho, Rep. Prog. Phys. 64 (2001) 1533. [12] J. Faist, F. Capasso, L.D. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Science 264 (1994) 553. [13] A. Kosterev, F. Tittel, IEEE J. Quantum Electron. 38 (2002) 582–591. [14] G. Wysocky, R. Lewicki, R.F. Curl, K.F. Tittel, L. Diehl, F. Capasso, M. Troccoli, G. Hofler, D. Bour, S. Corzine, R. Maulini, M. Giovannini, J. Faist, Appl. Phys. B 92 (2008) 305–311. [15] J.B. McManus, J.H. Shorter, D.D. Nelson, M.S. Zahnister, D.E. Glenn, R.M. McGov ern, Appl. Phys. B 92 (2008) 387–392. [16] A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, J. Faist, Appl. Phys. Lett. 89 (2006) 172120. [17] L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Höfler, M. Lonˇcar, M. Troccoli, F. Carpasso, Appl. Phys. Lett. 88 (2006) 201115. [18] I.T. Sorokina, K.L. Vodopyanov, Top. Appl. Phys. 89 (2003) 458–529. [19] M. Garcia, E. Normand, C.R. Stanley, C.N. Ironside, C.D. Farmer, G. Duxbury, N. Langford, Opt. Commun. 226 (2003) 39–43. [20] C. Sirtori, H. Page, C. Becker, Philos. Trans. R. Soc. Lond. A 359 (2001) 505–522. [21] D. Whitley, Inf. Softw. Technol. 43 (2001) 817–831. ´ V. Milanovic, ´ Z. Ikonic, ´ D. Indjin, J. Phys. D: Appl. Phys. 40 [22] J. Radovanovic, (2007) 5066–5070. ´ J. Radovanovic, ´ V. Milanovic, ´ D. Indjin, Z. Ikonic, ´ Nucl. Technol. Radiat. [23] A. Gajic, Prot. 29 (2014), in press. [24] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les edition de physique, 1998. ´ V. Milanovic, ´ in: Semiconductor Quantum Microstructures, University [25] Z. Ikonic, of Belgrade, 1997. [26] C. Sirtori, F. Capasso, J. Faist, S. Scandolo, Phys. Rev. B 50 (1994) 8663. ´ J. Radovanovic, ´ V. Milanovic, ´ D. Indjin, Z. Ikonic, ´ J. Phys. D 43 (2010) [27] A. Daniˇcic, 045101.

97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129

64

130

65

131

66

132