Double bound polaron in polar semiconductor heterostructures

Superlattices and Microstructures 33 (2003) 53–62 www.elsevier.com/locate/jnlabr/yspmi

Double bound polaron in polar semiconductor heterostructures Zi-xin Liu∗, Ping Liu, Xin-li Chu, Zhi-jie Qin, Ya Liu Department of Physics, Henan Normal University, Henan 453002, Xinxiang, China Received 19 August 2002; received in revised form 3 June 2003; accepted 5 June 2003

Abstract In this paper we discuss the single as well as the double polaron bound to a helium-type donor impurity in a magnetic field in a general step quantum well. We calculate their binding energy and investigate the influence of the well step. We find that the polaronic effects are strengthened due to the well step. We give a reasonable explanation for this phenomenon. © 2003 Elsevier Ltd. All rights reserved. PACS: 71.38.+i; 63.20.Kr Keywords: Double bound polaron; General step well

1. Introduction As we know, it is necessary to study the behaviour of a bound polaron in polar semiconductor QWs and other low-dimensional structures [1–10]. Although much attention is focused on hydrogenic impurities, the study of the state of a two-electron impurity centre (double donor state) in QWs is of considerable interest [11–13]. On the other hand, some asymmetric polar semiconductor heterostructures have aroused widespread interest in their applications to some special devices [14–16]. The present work deals with the polaronic effects on the binding energy of a helium-type impurity in general step quantum wells (GSQWs) in the presence of a magnetic field. Although the experimental results are not yet available, this work is motivated by similar previous works on hydrogenic impurities [1] in which the theoretic values considering polaronic effects are in excellent agreement with the experimental data and are better than other theoretic results. We found that the polaronic effects were strengthened due to the existence of the well step, ∗ Corresponding author.

E-mail address: [email protected] (Zi-xin Liu). 0749-6036/03/$ - see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0749-6036(03)00044-2

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and a reasonable explanation for this phenomenon is also given. The non-parabolicity of the conduction band is taken into account in our calculations. 2. Theory Consider a GSQW composed of four different polar crystals. Layers of the materials l, w, s and r are located at z < −L, −L ≤ z ≤ 0, 0 < z ≤ b and z > b, respectively. Let the z axis be perpendicular to the interfaces which are located at z = 0, −L and b. After some suitable canonical transformations, the total Hamiltonian for a double donor impurity can be written as H=

2
e− ph e c B (n) + H(n) (H(n) + H(n) + H(n) + H ) + Hee + H ph ,

(1)

n=1

with e H(n)


2 =− 2m 



1 ∂ ρn ∂ρn

    
2 d d 1 ∂2 1 ∂ ρn + 2 2 − + V (z n ), ∂ρn 2 dz n m(z n ) dz n ρn ∂ϕn (2)

where is the energy operator of a bare electron (n) which is confined to the potential well V (z). V (z) = Vl , for z < −L; V (z) = Vw = 0, for −L ≤ z ≤ 0; V (z) = Vs , for 0 < z ≤ b; V (z) = Vr , for z > b; and e H(n)

2e2 . 4πε f ∞ rn

c =− H(n)

(3)

c H(n) corresponds to the Coulomb term between the electron (n) and the impurity centre which is localized at (0, 0, z d ); ε f is the permittivity of free space, and ∞ is the highfrequency dielectric constant and  (4) rn = ρn2 + (z n − z d )2 ,

e− ph H(n) = [Γν j k (z n )eik ·ρn aν j k + h.c.] + [Γik (z n )eik ·ρn aik + h.c.], (5) ν j k

(n) = H


j

+

e2 πε f Tµ


ik



1 ∞µ

−

1 0µ



ik

K 0 (ρn qµj ) sin(qµj z n ) sin(qµj z d )

1 Γik (z n )Γik(+)∗ (z d ) cos(k · ρn ), 
ωik 

(6) e− ph

where K 0 is the Bessel function of the imaginary argument, µ = w or s. H(n) and (n) stands for the electron–phonon interaction as well as the interaction of the positive H point charge of the impurity with the phonon field, in which the first series corresponds to the confined and half-space longitudinal optical (LO) phonons, and the second to the interface phonons. The definition of Γν j k (z) is given in [18]. It describes the intensity

Zi-xin Liu et al. / Superlattices and Microstructures 33 (2003) 53–62

55

of the interaction between an electron and the confined phonons or the half-spaced LO phonons. Γik (z) describes the intensity of the interaction between an electron and the interface phonons in GSQWs and has been given in [17]. Γ (+) (z d ) describes the intensity of the interaction between the impurity and the phonon fields; it can be obtained from Γ by replacing e in Γ with 2e. e2 2 e B ·L + A (7) 2m 2m (n) where the vector potential A is defined as A = B × r/2, B and L are the magnetic induction intensity and the angular momentum respectively. Hee is the electron–electron interaction term, B H(n) =

Hee =

e2 . 4πε f ∞ |r1 − r2 |

(8)

The free-phonon Hamiltonian is

+ H ph =
ω Lν aν+j k aν j k +
ωik aik aik ,  ν j k

(9)

ik

which includes the confined bulk-like LO, the half-space LO and the interface phonons;
ω Lν is the bulk LO phonon energy in material ν(ν = l, w, s, r );
ωik is the interface phonon energy, and m(z) is the electron effective mass [19]. Considering the subband nonparabolicity, m(z) is given by  m l (E) = m l [1 − (Vl − E)/E gl ], z < −L,    m w (E) = m w [1 − (Vw − E)/E gw ], −L ≤ z < 0, m(z) = 0 ≤ z ≤ b, m s (E) = m s [1 − (Vs − E)/E gs ],    m r (E) = m r [1 − (Vr − E)/E gr ], z > b, where E gν (ν = l, w, s, r ) is the energy gap between the conduction and light-hole valence bands in the material ν, E is the electron energy level which can be obtained by solving the following subband energy Eq. (14), m ν is the mass constant in the material ν. Within the range of error of variational calculation, following the usual method we assumed that m  = m(z) for simplicity. If the double donor centre binds a single electron, it is positively charged; for this ionized donor the only modifications of Hamiltonians in the above equations are the removal of the summation sign 2n=1 and the subscript ‘n’, and then multiplying the coefficients in front of Γ (+) by 2 in Eq. (6). We introduce the following variational trial wavefunction ψ in the case of an ionized donor: |ψ = U Φ(ρ, z)|0, √ 2 2 Φ(ρ, z) = Ne−µ ρ +z φ(z), where N is the normalization constant.

(10) (11)

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Zi-xin Liu et al. / Superlattices and Microstructures 33 (2003) 53–62

 A1 ekl z ,    A2 sin(kw z) + A3 cos(kw z), φ(z) = A4 eks z + A5 e−ks z ,    −kr z , e

z < −L, −L ≤ z < 0, 0 ≤ z ≤ b, z > b,

(12)

with kν ≡ [2m ν (E 1 )(Vν − E 1 )]1/2/
, (ν = l, s, r ), and kw ≡ [2m w (E 1 )E 1 ]1/2/
, where E 1 is the ground state eigenvalue of the electron Hamiltonian H e . Am (m = 1, 2, 3, 4, 5) is defined as A1 A2 A3 A4 A5

= = = = =

flw A3 ekl L /[ flw cos(kw L) + sin(kw L)], A3 [cos(kw L) − flw sin(kw L)]/[ flw cos(kw L) + sin(kw L)], [cosh(ks b) + f sr sinh(ks b)]e−kr b , (1 − f sr )e−(ks +kr )b /2, (1 + f sr )e(ks −kr )b /2.

(13)

The subband energy equation in our QW potential V (z) is obtained by [cos(kw L) − flw sin(kw L)][cosh(ks b) + fsr sinh(ks b)] + fws [sinh(ks b) + fsr cosh(ks b)][ f lw cos(kw L) + sin(kw L)] = 0,

(14)

where flw ≡

m l (E 1 )kw , kl m w (E 1 )

f sr ≡

m s (E 1 )kr , ks m r (E 1 )

f ws ≡

m w (E 1 )ks . kw m s (E 1 )

(15)

In Eq. (10), |0 is the phonon vacuum state, and the canonical transformation U is given by U = U1U2,  
 U1 = exp [γν j k (ρ, z)aν+j k − γν∗j k (ρ, z)aν j k ] ,   ν j k   
 + ∗ [γik (ρ)aik − γ (ρ)a ] U2 = exp , ik  ik    

(16)

ik

with γν j k (ρ, z) = αν j k u(qνj , z)e−ik ·ρ , γik (ρ) = βik e−ik ·ρ .

(17)

In Eq. (10), there are three real variational parameters µ > 0, αν j k and βik which will subsequently be determined by minimizing the energy of the system. The binding energy in the case of an ionized donor is obtained from E B (ionized) = E sub + E L − ψ|Hion |ψmin , where E sub is the lowest energy solution of Eq. (14); EL =

eB
2m

(18)

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57

is the ground-state Landau level; Hionmin is the minimum of Hion, the expectation value of the Hamiltonian of an ionized donor system. In the case of a neutral donor centre, the trial wavefunction of the double bound polaron is   2  U(n) Φ(ρn , z n ) |0, (19) |Ψ  = n=1

where U(n) and Φ are similar to those in the case of the single bound polaron. The binding energy in the neutral donor case is obtained by E B (neutral) = E sub + E L − ( Ψ |Hneu|Ψ min − ψ|Hion |ψmin ),

(20)

where the expectation value of the electron–electron interaction term is approximately evaluated in [11]. We also treat the electron–electron interaction term as the only correlation term between these two electrons, i.e. we will disregard any other correlations of these two electrons via a phonon field in our further approximate numerical evaluations. 3. Numerical results and discussion As an example of the application of our theory, we have calculated the binding energy of a single bound polaron and a double bound polaron for a helium-type impurity in Alx1 Ga1−x1 As/GaAs/Alx3 Ga1−x3 As/Alx4 Ga1−x4 As asymmetric general step wells. The parameter E gν is given as E gν = 1424 + 1266x ν + 260x ν2 meV (ν = l, w, s, r ) and the other parameters are the same as those in [17]. Fig. 1 shows the binding energy E B as a function of the well width L for Al0.35 Ga0.65 As/GaAs/Al0.2 Ga0.8 As/Al0.4 Ga0.6 As GSQWs. In Fig. 1, the donor impurity centre is localized at the centre of the well (z d = −L/2) and the polaronic effect is obvious only for some special cases: 12 nm < L < 22 nm in the case of a magnetic field B = 0, near L = 20 nm in the case of B = 20 T, (we also found that it is obvious for small well width (L < 6 nm) in both cases). The magnetic field significantly modifies the binding energy of the donor. When the magnetic field intensity and well width are large enough, the binding energy of the double donor will be low. In this case the double donor easily loses one of its electrons and becomes an ionized donor. We can observe from Fig. 2 that the polaronic effect is strengthened for binding energy due to the donor impurity localized at the centre of the well step (z d = b/2), and the polaronic effect can contribute appreciably to the binding energy in both cases of B = 0 and 20 T for a wide range of well widths. The polaronic effect due to the well step can also be found from Fig. 3 in which the relationship between the binding energy and the position of the donor impurity centre are indicated. When the donor impurity centre is localized at the well step whose potential height is higher than that of the well, the interaction between the donor impurity centre and phonon fields are strengthened, and the wavefunction of the electron bounded by the impurity centre can easily cross over the interface and spread into the well due to the lower well height. The action of the interface phonon mode on the electron is strengthened,

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Zi-xin Liu et al. / Superlattices and Microstructures 33 (2003) 53–62

45 40

(a)

B C D E

B=0T

35

EB(meV)

30 25 20 15 10 5 8

10

12

14

16

18

20

L(nm) 55 50

(b)

B C D E

B=20 T

45 40

30 25

B

E (meV)

35

20 15 10 5 0 8

10

12

14

16

18

20

L (nm) Fig. 1. Binding energy as a function of the well width L for Al0.35 Ga0.65 As/GaAs/Al0.2 Ga0.8 As/Al0.4 Ga0.6 As GSQWs with fixed step width b = 5 nm, the donor impurity localized at the centre of the well (z d = −L/2). The lines B and C correspond to the single bound polaron in the case of an ionized donor; the lines D and E correspond to the double bound polaron in the case of a neutral donor. For lines B and D the total binding energy includes the interaction of the electron and the impurity with phonons, i.e. the lines C and E are obtained by neglecting the presence of phonons. The meaning of curve letters B, C, D, E in other figures are the same as in this figure.

Zi-xin Liu et al. / Superlattices and Microstructures 33 (2003) 53–62

59

35

B C D E

B=0T

(a)

30

B

E (meV)

25

20

15

10

5

0 7.5

10.0

12.5

15.0

17.5

20.0

L (nm) 40

(b)

B C D E

B=20T

20

B

E (meV)

30

10

0 7.5

10.0

12.5

15.0

17.5

20.0

L (nm) Fig. 2. As Fig. 1, but the donor impurity is localized at the centre of the well step (z d = b/2).

and the action of the constrained bulk phonon mode in the step is supplemented by the constrained bulk phonon mode in the well. The cumulative effects of the impurity– phonon coupling and the electron–phonon coupling are increased in this case. Although this increase is in a reasonable range (≤3 meV), its ratio with total binding energy is

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Zi-xin Liu et al. / Superlattices and Microstructures 33 (2003) 53–62 40

(a)

35

B C D E

30

EB (meV)

25 20 15 10 5 0 -5 -10

-8

-6

-4

-2

0

2

zd (nm) 50

(b)

45

B C D E

40 35

EB (meV)

30 25 20 15 10 5 0 -5 -10

-8

-6

-4

-2

0

2

zd (nm) Fig. 3. Binding energy as a function of the position coordinates of an impurity centre in Al0.35 Ga0.65 As/ GaAs/Al0.2 Ga0.8 As/Al0.4 Ga0.6 As GSQWs, the well width L = 20 nm, the step width b = 5 nm, and the magnetic field B = 0 T in (a), B = 20 T in (b).

quite large. This is due to the fact that the step well can decrease the binding energy if the width of step well is small. The polaronic effects are more important in this case, and we can say we found an interesting enhancement of the polaronic effect due to the well step.

Zi-xin Liu et al. / Superlattices and Microstructures 33 (2003) 53–62

61

40

B C D E

35 30

EB (meV)

25 20 15 10 5 0 10

20

30

40

B (T) Fig. 4. Binding energy as a function of the magnetic field B for Al0.35 Ga0.65 As/GaAs/Al0.2 Ga0.8 As/ Al0.4 Ga0.6 As GSQWs with fixed step width b = 5 nm, the donor impurity is localized at the centre of the well (z d = −L/2, L = 20 nm).

Fig. 4 shows that the magnetic field significantly modifies the binding energy. It also shows that for a large magnetic field the binding energy of a double donor can be small and the neutral donor is easily ionized. 4. Summary For a helium-type impurity in a GSQW in the presence of a magnetic field, the Hamiltonian of single and double bound polarons is proposed, in which the interactions of the electron and the impurity with the phonon field are included. We have calculated the binding energy of the single and double bound polarons in GSQWs including all the polaronic contributions due to (i) electron-confined bulk-like LO phonon coupling, (ii) electron-half-space LO phonon coupling, (iii) electron-interfacial phonon coupling, as well as the coupling between the positive charge of the impurity ion and the above various phonon modes. The impurity centre can be located at an arbitrary position in the well or at the well step. In addition, the effects of the subband non-parabolicity and the finiteness of barrier height are all taken into account at the same time. In numerical calculation we found that the interaction of the electrons and the positive point charge of impurity with phonons has a significant contribution to the binding energy of the single or double bound polarons in GSQWs in some cases. We found that the polaronic effects are strengthened due to the well step. This may be important for the design of certain optoelectronic devices in

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the future. When the magnetic field intensity is large enough and the well width is beyond a certain value, the neutral donor is easily ionized, by losing one of its electrons. Acknowledgements This work was supported by the Natural Science Foundation of Henan Province and the National Natural Science Foundation of China under Grant No. 60276004 and 60290083. References [1] Zi-xin Liu, Jun-jie Shi, Guo-xin Ju, Shao-hua Pan, Superlatt. Microstruct. 22 (1997) 273. [2] R. Chen, K.K. Bajaj, J.P. Cheng, B.D. McCombe, Phys. Rev. B 51 (1995) 9825. [3] J.M. Shi, F.M. Peeters, G.A. Farias, J.A.K. Freire, J.T. Devreese, S. Bednarek, J. Adamowski, Phys. Rev. B 57 (1998) 3900. [4] A. Ercelebi, M. Tomak, Solid State Commun. 54 (1985) 883. [5] B.A. Mason, S.D. Sarma, Phys. Rev. B 33 (1986) 8379. [6] Y.C. Li, S.W. Gu, Phys. Rev. B 45 (1992) 12102. [7] A. Elangovan, K. Navaneethakrishnan, J. Phys.: Condens. Matter 5 (1993) 4021. [8] Hai-Yang Zhou, Shi-Wei Gu, Solid State Commun. 89 (1994) 4113. [9] G.A. Farias, M.H. Degani, O. Hipolito, Phys. Rev. B 43 (1991) 4113. [10] Hong-Jing Xie et al., Phys. Rev. B 61 (2000) 4827. [11] E.C. Niculescu, Phys. Lett. A 197 (1995) 330. [12] E.C. Niculescu, Phys. Lett. A 202 (1995) 305. [13] Zi-xin Liu, Xing-yi Li, Xing-li Chu, Yong-chang Huang, Ya Liu, Superlatt. Microstruct. 27 (2000) 235. [14] D.D. Coon, R.P.G. Karunasiri, Appl. Phys. Lett 45 (1994) 649. [15] Xi-xia Liang, J. Phys.: Condens. Matter 4 (1992) 9769. [16] Jun-jie Shi, Shao-hua Pan, Zi-xin Liu, Z. Phys. B 100 (1996) 353. [17] Jun-jie Shi, Shao-hua Pan, Phys. Rev. B 51 (1995) 17681. [18] Jun-jie Shi, Xiu-qin Zhu, Zi-xin Liu, Shao-hua Pan, Xing-yi Li, Phys. Rev. B 55 (1997) 4670. [19] D.F. Nelson, R.C. Miller, D.A. Kleinman, Phys. Rev. B 35 (1987) 7770.