Two-mode squeezed magnetopolarons in two-dimensional quantum dot

Physics Letters A 360 (2007) 632–637 www.elsevier.com/locate/pla

Two-mode squeezed magnetopolarons in two-dimensional quantum dot Yanmin Zhang ∗ , Ze Cheng, Zixia Wu, Yunxia Ping Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Received 22 May 2006; received in revised form 17 August 2006; accepted 25 August 2006 Available online 5 September 2006 Communicated by A.R. Bishop

Abstract In this Letter, some properties of magnetopolarons in two-dimensional quantum dot are investigated by two-mode squeezed states transformation. This method considers linear functions, bilinear functions of the phonon operators and the correlation between two longitudinal optical (LO) phonon modes, which is based on the Lee–Low–Pines and Huybrechts (LLP–H) canonical transformations. So it can provide results not only for the ground state energy but also for the excited states energies, furthermore, it can be applied to the entire range of the electron–phonon coupling strength. Using two-mode squeezed states transformation, we have obtained more accurate results for the ground state energy, excited states energies and renormalized cyclotron masses for some possible transitions. © 2006 Elsevier B.V. All rights reserved. PACS: 63.20.Kr; 71.38.Fp; 71.38.Cn Keywords: Squeezed magnetopolarons; Electron–phonon interaction; Renormalized mass

1. Introduction In recent years, there has been a great deal of interest in the investigation of low-dimensional nanostructures [1–12]. Of particular interest is quantum dot for the electrons confined in all three dimensions in these systems [13–21], which brings in quantum effects when the electron wavelength is of the same order as the confinement length. Because of the potential device applications and the interesting physical effects in such structures, understanding the electronic properties of these systems is of particular importance [22,23]. The electron–phonon interaction plays an important role in low-dimensional systems, particularly, in quantum dots [24,25]. Study of the squeezed states is another interesting problem both in quantum optics and condensed matter physics. In 1988, Hang Zheng introduced the squeezed states of phonons to condensed matter physics according to the squeezed states of photons in quantum optics [26,27]. Later, squeezed polarons attracted much attention. For example, the squeezed states of * Corresponding author. Tel.: +86 2787545361.

E-mail address: [email protected] (Y. Zhang). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.066

phonons are successfully applied to small polarons [28] and Fröhlich optical polarons[29–33]. Recently, the polaronic effect has been calculated by the Feynman–Haken path integral [34] and the Lee–Low–Pines– Huybrechts (LLP–H) canonical transformations [35] for a parabolic QD in the entire range of the electron–phonon coupling strength. The Feynman–Haken path integral approach yields more accurate results than those of the LLP–H transformations, but it can only be applied to the ground state. Comparatively, the LLP–H transformations can give results for both the ground state and the excited states with fairly reasonable accuracy. In addition, it can be improved by choosing more suitable trial wave function [33]. The aim of the present Letter is to improve the LLP–H results for the polaronic effects in quantum dots obtained from Ref. [35]. We consider magnetopolaronic effects in a QD embedded in a two-dimensional material, where the dot electron is confined in a parabolic potential. In this Letter, bilinear functions of the phonon operators that arise from the LLP–H transformations are considered by two-mode squeezed states transformation. Although these have been studied in Ref. [32] by single-mode squeezed states transformation, unfortunately it ignores the correlation between two LO phonon modes. In or-

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der to improve the results, we now consider two-mode squeezed states transformation of magnetopolarons in which the phonon subsystem is in a correlated state. The results of two-mode squeezed states transformation in this Letter are compared with those of single-mode squeezed states transformation [32] and the LLP–H transformations [35]. It shows that more judicious and accurate results are given in this Letter using two-mode squeezed states transformation in the following sections.

Here, the terms which do not depend on the phonon operators are

2. Model and theory

The linear terms of the phonon operators are

The Hamiltonian for a 2D electron interacting with LO phonons in an isotropic harmonic potential and a uniform magnetic field along the Z direction is given by  2  1 e 1 P + A + μ 2 r 2 + h¯ ω0 bq† bq H= 2μ c 2 q   Vq bq eiq·r + h.c. , + (1)

H1 =

q

where P = hk ¯ is the electron momentum with a band mass μ, P and r are electronic momentum and position operators, respectively. The vector potential A is taken in the symmetrical Coulomb gauge: A = B(−y, x, 0)/2. The second term in Eq. (1) is a parabolic potential that confines the electron to zero dimension and so produces the QD with confinement frequency  . bq† (bq ) is the creation (annihilation) operator of an optical phonon with wave vector q and energy hω ¯ 0 . The last term is the electron–phonon interaction, |Vq |2 = (h¯ ω0 )2 (2παr0 /V q) is the 2D electron–phonon interaction amplitude, α is the electron–phonon coupling constant, r0 = (h¯ /2μω0 )1/2 is the polaron radius, and V is the surface area of the 2D crystal in which the LO phonons are confined. First, we use the Lee–Low–Pines and Huybrechts (LLP–H) transformations [36,37]:    † qbq bq , U1 = exp −iλr · (2)

 2 

1 1 e Vq fq ei(1−λ)q·r + h.c. P + A + μω2 r 2 + H0 = 2μ c 2 q +

 h¯ 2 λ2 q 2 2u

q

   λh¯ e + h¯ ω0 − P + A · q |fq |2 . u c

(5)



Vq bq ei(1−λ)q·r + h.c.

q

    λh¯ e λ2 h¯ 2 q 2 q· P+ A + hω + ¯ 0− μ c 2μ q  ∗  × fq bq + h.c. .

(6)

The bilinear and higher order terms of the phonon operators are given by H2 =

 h¯ 2 λ2 q 2 q

+

2u

+ h¯ ω0 −

   λh¯ e P + A · q bq† bq u c

h¯ 2 λ2   q · kFq† Fk† Fq Fk , 2u q

(7)

k

where Fq = bq + fq . Next, in order to deal with bilinear terms of the phonon creation and annihilation operators arising from the above canonical transformations, and take into account of the correlation between two relevant LO phonon modes, we introduce twomode squeezed states transformation in this work [29]  

ϕqk † † b q b k − bq b k , U3 = exp N

(8)

q=k

q

   †  ∗ U2 = exp b q f q − bq f q ,

(3)

q

where λ is a variational parameter and fq is a variational function to be determined by minimizing the energy function. The first transformation U1 makes our method valid for the entire range of the electron–phonon coupling strength, and the second transformation U2 generates coherent state for phonon fields. When λ = 1, the LLP–H method reduces to the LLP method and provides a good description in the extended state limit, whereas for λ = 0, this approach is equivalent to the Landau– Pekar method [38] and is valid in the localized state limit. Thus treating λ as a variational parameter (0 < λ < 1) one can have a consistent theory for the entire parameter space [39]. Under these transformations, the Fröhlich Hamiltonian becomes [30] H  = U2−1 U1−1 H U1 U2 =

2  i=0

Hi .

(4)

where q and k are the wave vectors of two different LO phonon modes, respectively. N is the total number of LO phonon modes. ϕqk is the squeezing angle and it will be used as an additional parameter to minimize the energy. In order to keep the unitarity of this transformation, the squeezing angle is taken as a real number. Then the squeezed vacuum state |ψph  is formed as |ψph  = U3 |0, where |0 is the zero phonon state. If squeezing angle ϕqk = 0, |ψph  returns to the vacuum state |0; as long as squeezing angle ϕqk = 0, |ψph  is the squeezed state of phonon. Then the quadratic terms in Eq. (4) can be diagonalized by performing two-mode squeezed states transformation. In order to reflect the feature that two LO phonons are involved in the trial wave function, we rewrite the Hamiltonian symmetrically and transform it by U3 , then the Fröhlich Hamiltonian H  becomes [33] H¯ = U3−1 H  U3 =

 1 Hi (q) + Hi (k) , 2 2

2   1 i=0

634

H0 (q) =

Y. Zhang et al. / Physics Letters A 360 (2007) 632–637

  h¯ λ  e + P + A · q |fq |2 u h ω c ¯ 0 q    2   + r02 λ2 q|fq |2 + V¯q fq ei(1−λ)q·r

 2 1 1 e P + A + μω2 r 2 2μ c 2 

i(1−λ)q·r Vq fq e + + h.c. q

+

 h¯ 2 λ2 q 2

 + h¯ ω0 |fq |2

q

2u q    hλ  h¯ 2 λ2  2 e ¯ − P + A · q|fq |2 + q|fq |2 , u c 2u q q      ϕqk Vq bq cosh H1 (q) = N q    ϕqk + i(1−λ)q·r + bk sinh + h.c. e N     λh¯ e λ2 h¯ 2 q 2 + q· P+ A + h¯ ω0 − μ c 2μ q        ϕqk ϕqk × fq∗ bq cosh + bk+ sinh + h.c. , N N     2 2 2   h¯ λ q  2 ϕqk + h¯ ω0 sinh H2 (q) = 2u N q k     λ2 h¯ 2

2ϕqk q · k fq fk + fq∗ fk∗ sinh + 4u N q k     h¯ 2 λ2 2  ϕqk q · k |fk | sinh2 + u N q k   + 0 bq , bq† , (9) where 0(bq , bq† ) represents the terms that contain bq or bq† and vanish on taking expectation value for the zero phonon state. In order to solve Eq. (1) within the scheme of variational approach, we use a variational state vector |ψ = U1 U2 U3 |0ph ⊗ |n, ∓m, which is the direct product between the wave function of phonon part and that of electronic part. For phonon part, it is generated from the vacuum state by successive canonical transformations. For electronic part, its coordinate representation is given by [32] m+1 

1/2 n! r|n, ∓m = (−1) √ π (n + m)! 

2 2 × exp −β x + y 2 /2 2 2 

2 . × (x ∓ iy)m Lm n β x +y nβ

(10)

Averaging H over the variational state vector |ψ, and divided by h¯ ω0 on both sides of the equation, we get the expectation value of the energy in dimensionless form for the squeezed magnetopolarons as   1 1 2 2 ω¯ ¯ ¯ (2n + m + 1) ∓ m c En,∓m = + ω ¯ β 2 ¯ 8 2 2β    1 + 1 + r02 λ2 q 2 |fq |2 2 q



  + V¯q∗ fq∗ e−i(1−λ)q·r

  

ϕqk 1 + r02 λ2 q 2 sinh2 N q    ϕqk 2r02 λ2 q · k|fq |2 sinh2 + N q

+

 

k

+

 

2ϕqk q · k fq fk + fq∗ fk∗ sinh 2 N

  r 2 λ2 0

q

k

1 + {q ↔ k}, 2

(11)

where E¯ n,∓m = En,∓m /h¯ ω0 , ω¯ c = ωc /ω0 , ω¯ 2 = 4 ¯ 2 + ω¯ c2 , 1/2 ¯ ¯ and β = (μω0 /h¯ )1/2 /β. In Vq = Vq /h¯ ω0 , r0 = (h¯ /2μω 0 ) Eq. (11), it is assumed that q q|fq |2 = 0 due to the symmetry of QD. With the help of Ref. [32], we can obtain

 i(1−λ)q·r  = σnm (q) = n, ∓m| exp ∓i(1 − λ)q · r |n, ∓m e   ∞ 1  (m + p)! (1 − λ)2 q 2 p = nm (p), − n!m! [(p)!]2 4β 2 p=0 (12) where  1+m+p m+p A2  d n F ( 2 , 1 + 2 , 1 + m, B 2 ) nm (p) = n , dh (1 − h)1+m B 1+m+p h=0 and F is the hypergeometric function with A2 = 4h/(1 − h)2 , B = (1 + h)/(1 − h). Now minimizing the energy with respect to fq∗ and ϕqk , we obtain two coupled equations:   fq 1 + r02 q 2 λ2 + V¯q∗ σnm (q)    2ϕqk 2 2 ∗ r0 λ q · kfk sinh = 0, + (13) N q k     2ϕqk 1 + r02 q 2 λ2 + 1 + r02 k 2 λ2 sinh N     2ϕqk = 0. + 2λ2 r02 q · k fq fk + fq∗ fk∗ cosh (14) N Inserting these equations into Eq. (11), we can rewrite the expression of the energy as   1 1 2 2 ω¯ ¯ (2n + m + 1) ∓ m c + ω ¯ E¯ n,∓m = β 2 2β¯ 2 8  + V¯q fq σnm (q) q

+

  (1 + r 2 λ2 q 2 + 1 + r 2 λ2 k 2 ) 0

q

k

0

2

 sinh2

 ϕqk . N (15)

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In order to calculate this energy, we require the solutions of fq and ϕqk in Eqs. (13)–(14). It is difficult to solve these equations exactly, so we try to get results by making first order approximation [29]. Firstly, making use of the first term in the expansion of the hyperbolic function in powers of the small quantity ϕqk /N , we can obtain  2ϕ −V¯q∗ σnm (q) − k λ2 r02 qk cos γfk∗ Nqk fq = (16) , 1 + r02 q 2 λ2 −2λ2 r02 qk cos γfq fk ϕqk = , N (1 + r02 q 2 λ2 + 1 + r02 k 2 λ2 )

(17)

where cos γ = cos(θ1 − θ2 ), γ is the angle between two different wave vectors q and k, θ1 and θ2 are the angles between corresponding wave vectors and the polar axis. Then, we can choose fq as

−V¯q∗ σnm (q) 1+r02 q 2 λ2

corresponding to the unsqueezed case,

and put it into Eqs. (16)–(17) to obtain new fq and ϕqk . By making use of these new values of fq and ϕqk , the energy expression (15) becomes E¯ n,∓m   1 1 2 2 ω¯ c ¯ = + ω¯ β (2n + m + 1) ∓ m 2 ¯ 8 2 2β 2 (q)  |V¯q |2 σnm − 2 2 2 q 1 + r0 λ q 2 (q)σ 2 (k)  2λ4 r04 q 2 k 2 cos2 γ |V¯q |2 |V¯k |2 σnm nm − 2 2 2 2 2 2 2 2 2 2 2 q

k

(1 + r0 λ q + 1 + r0 λ k )(1 + r0 λ q ) (1 + r0 λ2 k 2 )2

Fig. 1. The polaronic correction for the ground state energy as a function of the electron–phonon coupling constant α for u0 = 1.0, ω¯ c = 1.0, where the dotted line, dashed line and straight line represent the results of two-mode squeezed states transformation, single-mode squeezed states transformation and the LLP–H transformations, respectively.

.

(18) We change sums into integrals and minimize the energy with respect to β¯ and λ, then the energy of the squeezed magnetopolarons can be obtained. It is also interesting to study the renormalized cyclotron mass m ¯ ∗ , which can be defined as the ratio of the cyclotron mass m∗ to the electron band mass μ and given by [40,41] m ¯ ∗n,∓m = ω/ ¯ ω¯ c∗ ,

(19)

where ω¯ c∗ = E¯ n,∓m (ω) ¯ − E¯ 0,0 (ω) ¯ is the cyclotron resonance frequency for the transition (0, 0) → (n, ∓m). 3. Results and discussions From Eq. (15), we calculate the polaronic correction for the ground state energy E¯ 00 = E¯ 00 (α) − E¯ 00 (α = 0) with the optimum value of λ and β, and plot the polaronic correction for the ground state energy as a function of the electron–phonon coupling constant α for u0 = 1.0, ω¯ c = 1.0 in Fig. 1, where the dotted line, dashed line and straight line represent the results of two-mode squeezed states transformation (TMSS), singlemode squeezed states transformation (SMSS) and the LLP–H transformations, respectively. It shows that the polaronic correction becomes large as the electron–phonon coupling constant α increases, and the results are qualitatively same for these three methods. Quantitatively however the polaronic enhancements are much larger for two-mode squeezed states transformation than single-mode squeezed states transformation and

Fig. 2. The polaronic correction for the ground state energy versus the confinement length u0 for α = 4.0, ω¯ c = 1.0, the dotted line, dashed line and straight line represent for the results of two-mode squeezed states transformation, single-mode squeezed states transformation and the LLP–H transformations, respectively.

the LLP–H transformation. This means that energy levels of magnetopolarons have been lowered and more stable states have been obtained from two-mode squeezed states transformation. It is not enough for us to only consider linear forms and bilinear ones of phonons. In addition, the correlation between two LO phonon modes should also be taken into account, especially for large values of the electron–phonon coupling constant α. Fig. 2 shows the polaronic correction for the ground state energy versus the confinement length u0 . We can see that the polaronic correction decreases deeply with the increase of the confinement length and tends asymptotically to a constant value at large values of the confinement length, which is the same fea-

636

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Fig. 3. The ground state energy E¯ 00 (solid line), the two components of the internal relaxed excited state energy E¯ 0−1 (dashed line), E¯ 0+1 (dashed-dotted) and the first excited Landau state energy E¯ 10 (dotted line) as a function of confinement length u0 for α = 4.0, ω¯ c = 1.0.

(a)

(b) Fig. 5. The renormalized cyclotron masses m ¯ ∗nm as a function of confinement length u0 for (a) α = 1.0, ω¯ c = 1.0, (b) α = 4.0, ω¯ c = 1.0, where the dotted line, dashed line and straight line represent m ¯ ∗0−1 , m ¯ ∗0+1 and m ¯ ∗10 , respectively.

Fig. 4. The ground state energy E¯ 00 (solid line), the two components of the internal relaxed excited state energy E¯ 0−1 (dashed line), E¯ 0+1 (dashed-dotted) and the first excited Landau state energy E¯ 10 (dotted line) as a function of the dimensionless magnetic ω¯ c for α = 4.0, u0 = 1.0.

ture in these three lines. The two-mode squeezed states transformation gives more stable state and this effect increases as the confinement length u0 increases, and gives rise to a substantial shift in the polaronic correction for large values of u0 . In Figs. 3 and 4 we plot energy E¯ nm versus confinement length u0 , and E¯ nm versus dimensionless magnetic field ω¯ c , respectively. The solid line, dashed line, dashed-dotted line and dotted line represent the ground state energy E¯ 00 , the two components of the internal relaxed excited state energy E¯ 0−1 , E¯ 0+1 and the first excited Landau state energy E¯ 10 , respectively. Both figures show that the two components of the internal relaxed excited state energy E¯ 0−1 and E¯ 0+1 are split because of the ex-

istence of the magnetic field. One of them E¯ 0−1 is very close to the ground state level E¯ 00 , while the other E¯ 0+1 is very close to the first excited Landau state level E¯ 10 . From Fig. 3, we can see that all these energy levels increase deeply as the decreasing confinement length and tend asymptotically to a constant value at large value of the confinement length. From Fig. 4, it can be seen that all these energy levels increase with increasing magnetic field strength. The two components of the internal relaxed excited state energy level E¯ 0−1 and E¯ 0+1 overlap when the magnetic field does not exist, and they were split more as the magnetic field strength increases. Using Eqs. (18)–(19), we can obtain the renormalized cyclotron masses m ¯ ∗nm for squeezed magnetopolarons in twodimensional quantum dot. In Fig. 5, we plot the renormalized cyclotron masses m ¯ ∗nm as a function of confinement length u0 for (a) α = 1.0, ω¯ c = 1.0, (b) α = 4.0, ω¯ c = 1.0. The dot¯ ∗0+1 ted line, dashed line and straight line represent m ¯ ∗0−1 , m

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account of linear terms, bilinear ones of phonons and the correlation between two LO phonon modes. Therefore, this method improves the LLP–H canonical transformations [35] and singlemode squeezed states transformation [32], so it gives more exact results in this Letter. We have shown that the correlation between two LO phonon modes is very important, and it should not be ignored in the study of the properties of magnetopolarons. Acknowledgements This research was supported by the National Natural Science Foundation of China under Grants Nos. 10174024 and 10474025. References Fig. 6. The renormalized cyclotron masses m ¯ ∗nm as a function of the dimensionless magnetic field ω¯ c for α = 4.0, u0 = 1.0. The dotted line, dashed line and ¯ ∗0+1 and m ¯ ∗10 , respectively. straight line represent m ¯ ∗0−1 , m

and m ¯ ∗10 , respectively. It gives the following features: (1) The renormalized cyclotron masses m ¯ ∗nm increase with increasing confinement length and tend asymptotically to a constant value at large confinement length values. (2) The renormalized cyclotron masses m ¯ ∗nm decrease as the coupling constant α increases. (3) For large confinement length values, the renormalized cyclotron mass m ¯ ∗0−1 for the transition (0, 0) → (0, −1) is larger than the band mass, while m ¯ ∗0+1 for the transition ∗ (0, 0) → (0, +1) and m ¯ 10 for the transition (0, 0) → (1, 0) are smaller than the band mass. Fig. 6 shows the renormalized cyclotron masses m ¯ ∗nm as a function of the dimensionless magnetic field ω¯ c for α = 4.0, ¯ ∗0+1 and m ¯ ∗10 are u0 = 1.0. It gives that the cyclotron masses m smaller than the band mass and tend asymptotically to a constant value at high magnetic field limit; while cyclotron mass m ¯ ∗0−1 is larger than the band mass. 4. Conclusions In this Letter, we have investigated the properties of squeezed magnetopolarons in two-dimensional quantum dot using twomode squeezed states transformation. Some particular conclusions have been obtained as follows: (1) Energy levels of magnetopolarons have been lowered due to introducing two-mode squeezed states into the phonon subsystem. (2) The squeezing corrections to the energy of magnetopolarons become quite large as the strength of the electron–phonon interaction increases. (3) The squeezing effects are enhanced with the increase of the confinement length. Furthermore, we have studied three excited states energies of the squeezed magnetopolarons and the renormalized cyclotron masses for some possible transitions, and obtained some interesting results. In conclusion, two-mode squeezed states transformation which based on the Lee–Low–Pines and Huybrechts (LLP–H) canonical transformations has been introduced, and it takes into

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