The effect of confined Longitudinal Optical Phonons on the Binding Energy of an impurity in CdSe Quantum Dot
B. El Amrani1, N. Essbai2, M. Fliyou1, H.Chikhaoui1 and M. Chaouch3
1 Equipe de Physique du Solide E.N.S BP 5206 Bensouda Fès Morocco.
2 Faculté des Sciences & Techniques, Rte d’Imouzzer , BP 2202 Fès Atlas Morocco.
3 Faculté Des Sciences Dhar el Mahrez ,Fes Atlas-Morocco
Using a variational approach, the effect of the confined LO-phonon on the binding energy in CdSe Quantum Dot have been calculated. The charge-carrier-phonon coupling is treated within the adiabatic approximation. Our results show that the effect of the confined LO-phonon on the binding energy decreases with the dot size.
However the correction of the confined LO-phonon to bound state energy increases with dot size.
PACS number: 03
I. INTRODUCTION
With the development of semiconductor growth techniques, the study of electron state in semiconductor microstructures has advanced rapidly. The reduction to quasi-zero dimensional semiconductor is becomes possible. In Quantum Dot (QD) the charge carrier confinement is in all directions and the impurity plays a fundamental role in some physical properties such as optical and transport phenomena at low temperature.
Many extensive works in different low dimensional systems [1-9], using the effective mass approximation, the first principle, the plane wave semiempirical pseudo- potential method [10-21] have been done. The binding energies and the density of states of shallow impurities in cubic [6] and in spherical quantum dot [3-5,14] have been calculated as function of dot size. The results show that the binding energy in quasi-zero dimensional systems is the higher one compared with the other systems (quasi-one and quasi-two dimensional systems) [6,10,17,18]. For dot size larger than the impurity Bohr radius, the results obtained by the semiempirical pseudo- potential method and by the effective mass approximation are comparable [21].
These microstructures are based on polar and semi- polar semiconductors; the charge carrier LO-phonon interaction is imposed. The previous calculations have not taken into account of the ion - confined LO-phonon interaction. The effect of LO-phonon on shallow donor impurity in cubic quantum dot has been studied just for the subband state by evaluating the effective potential induced by the electron-confined LO-phonon [12] and by using the Feynman-Haken method [22].
In this paper we propose to complete the calculation of the polaron binding energy by using an adequate wave function which describing the system in large interval of dot size. We use the Fröhlich-Hamiltonian, in order to describe the interaction between the charge carrier and confined LO-phonon modes.
We have been interested on the effect of confined LO-phonons in CdSe quantum dot in the region where the dot size is larger than the impurity Bohr radius, so the contribution of the Surface Optical (SO)-phonon is
neglected. We organize the paper as follows : In Section II the Hamiltonian of the system is given using a modified Lee-Low-Pines transformation, the polaron binding energy of the impurity is derived as a function of the size of the quantum dot. In Section III we give the discussion of our results.
II. MODEL
In the effective mass approximation, the Hamiltonian of a bounded polaron located at the center in quantum dot (QD) can be written as :
int ph 2
e 2
H + H + z) , y , x ( r V e m 2
= P
H +
−ε
∞
(1)
where me is the effective mass of the electron in the conduction band, P is the momentum of the electron, r= x2+y2+z2 and V(x,y,z) is the infinitely deep potential well of the quantum confinement, and that the barrier potential V is taken as zero inside the QD and infinite otherwise.
Hph is the Hamiltonian of the field of the confined LO-phonons contribution [24]
∑ ∑ ω +
∑
=
2
2 3 3
1
1 m ,p (mi)
pi A i) m i( Ap p ,
m 0
p , ph m
H η (2)
Where (mi))
pi A i)(
m i(
Ap+ is the operator of creation (annihilation) of LO-phonons with frequency ω0, wave vector (m1π/L1, m2π/L2, m3π/L3) and parities pi
(i=1,2,3). The parities pi refer to the mirror symmetry with respect to the planes x = 0, y = 0 and z = 0 and the positive integers mi refer to the discrete values of x, y and z components of the LO-phonon wave vector. For even parities (pi takes +), mi is odd and for odd parities (pi takes -), mi is even. It should be noted that the LO- phonon wave vector is limited by the Brillouin boundary: m1π/L1 ≤ π/2a, m2π/L2 ≤ π/2a and m3π/L3≤ π/2a (a is the lattice constant). By consequent, m1, m2, M.J. CONDENSED MATTER VOLUME 3, NUMBER 1 1 JULY 2000
3 37 © 2000 The Moroccan Statistical Physical Society
m3 can be any integer within the range 1<m1 ≤ N1/2, 1<m2 ≤ N2/2 and 1<m3 ≤ N3/2, (Li=Ni a ).
Hint is the Hamiltonian of the charge carrier-LO- phonon interaction [25]:
p1 1, m
C . H ) m p( A m3 m2 m1 D
p , Sm B p1 1, 1 m p 1, int m
H = ∑ ∑ ∑ +
(3)
where
L ) z (m csn L )
y (m csn L )
x (m csn
L ) z (m csn L )
y (m csn L )
x (m csn
S S S
3 0 3 2
0 2 1
0 1
3 3 2
2 1
1 ion
p , m e
p , m p , m
π π
− π
π π
= π
−
=
(4)
m = (m1 , m2 , m3 ) , p = (p1 , p2 , p3 ) and x0 = y0 = z0 =0
csn(m t L
m t L m
m t L m
i i i
i i i i
i i j i
π
π π /
cos( /
sin( / )
) , , ...
) , , ...
=
=
=
1 3 5
2 4 6
(5)
i= 1 , 2 , 3 t1 = x , t2 = y , t3 = z
( )
Dm m m1 2 3 = (m1π/L1)2 + (m2π/L2)2+(m3π/L3)2 1/2 (6)
2 / 1
0 2
0 1 1)
e ( 6 i 1 -
=
B
−ε ε Ω ω π
∞
η (7) Ω is the quantum dot volume and ε∞ ( ε0 ) are the optic (static ) dielectric constant respectively.
In this work, we have used a modified unitary transformation in Lee-Low-Pines scheme. This transformation takes into account the eigenfunctions of the phonon modes [14,22] :
∑ −
= m,p * p
p , m +
p p ,
m
)
(
F (r)A (m) F (r)A (m)exp
U (8)
with
) r ( S ) r ( F
) r (
Fm,p = m1p1,m2p2,m3p3 = fm,p *m,p
p3 m3 2, 2p m 1, 1p m p ,
m f
f = is a variational parameter and can be determined by minimizing the energy of the system.
The wave function describing the system, in the adiabatic approximation, can be written as the product of the impurity wave function ψ(r) and the vacuum state of phonons 0 :
0 ) r ( U ) r
( = ψ
Φ (9)
The only effect of the charge carrier-LO-phonon coupling, within the adiabatic approximation, is to displace the ion equilibrium positions. This displacement is performed by means the canonical transformation.
Then the Hamiltonian into the expectation value in such state given by :
ψ ψ
= Φ Φ
= H Q
E (10)
Where
∑ +
+
∑ ω + ∇
+
=
= −
p ,
m m,p
p , m
2 p , m p , m
2 p , m e 2 2 p , m 0 2 p , m e
1
) c . D H
BS (
) m S
S 2 ( H
0 U H U 0 Q
f
f η η (11)
Finally the corresponding eigenenergy is :
∑
+
∑
ω + +
=
p ,
m m,p
p , m 1
2 p , p m
,
m 2
e 2 1 0 0
H.c D +
B T
m T T 2 E
E
f
η f η
(12)
with :
H
E0= ψ e ψ ,T1= ψ Sm,p2ψ and ψ
∇ ψ
= m,p2
2 S
T
In order to determine the form of
f
m p, we use the variational condition E E 0* p , p m , m
∂ =
= ∂
∂
∂
f f , then
f
m p, is obtained as:m T T 2
T D
B
2 e 2 01 0
01 p
, m
* p ,
m ηω + η
= −
f (13)
where T01=T1− ψ Sionm,p2ψ =T1−Sionm,p2 the last term is the same in the bound and in subband state that will be omitted in the next step of work.
By substituting the
f
m p, value ( equation (13)) into (12), we obtain the total energy E as:LO
0 E
E
=
E − (14) In which ELO is the contribution of the confined LO- phonon in the impurity state.
∑
+ ω
=m,p
2 e 2 01 0
2 01 2
p , m 2 LO
m T T 2
T D
E B
η η
(15)
The trial wave function is chosen as the non separable form because this wave function recovers a large domain of the system :
r) exp(
L ) cos( z L ) cos( y L ) cos( x A
3 2
1 π π −λ
= π
ψ (16)
where A is the normalisation constant and λ is a variational parameter. The minimization of the total energy, )E(λ)=E0(λ)−ELO(λ obtained by substituting 38 B. EL AMRANI, N. ESSBAI, M. FLIYOU, H.CHIKHAOUI AND M. CHAOUCH 3
Eq.(16) into eq.(14), gives the ground-state energy of the impurity. Then, we define the polaron binding energy as:
) ( E sub min b E
E = − λ
(17) where the subband energy is corrected by the interaction with phonon taking into account the presence of ion in center[13]:
) ( 2
3 ) ( L 2 2 ) ( L 2 1 ) ( L
2 2 − → ∞
+ +
= π π π E LO λ
m e E sub η
III. RESULTS AND DISCUSSIONS
For our numerical calculations we have chosen a CdSe quantum dot as a typical example. All parameters used (ε∞ = 6.23 ; ε0 = 9.56 ; ηω0 = 26.46 meV ; a = 4.30 Å , c = 7.011 Å , me = 0.13 m0 ) are taken from a literature where m0 is the free electron mass [26].
We use the effective atomic units so that all energies are expressed in the effective Rydberg units
2 2 0 4 ee 2 m
=
Ry ε η and all distances are expressed in the effective Bohr radius units
a * = η
2ε
0m e
e 2 ( here Ry =19.345 meV and a* = 38.98 Å ).In our study we have taken the trial wave function as the non separable function (eq. 16) because this wave function describes and recovers a large domain of the dot especially the limit cases (small and large dots ) [1,6]. For small dots the polaron binding energies are strongly affected by the electron correlation, the confinement is strong then the impurity wave function is compressed in a little volume. For large dots the confinement becomes very weak, the bulk-like approach takes place. For CdSe material the coupling electron-LO-phonon value is α = 0,43 . The polaron binding energy increases as this constant increases [22]. At high temperature the wave function becomes
Φ = U ψ n
k , wheren
k is the phonon excited state, these works are in progress.0 2 4 6 8 10 12
0 5 10 15 20 25
Fig. 1 Variations of the binding energy with (curv A) and without (curve B) confined LO-Phonon. The correction of confined LO-phonon (curve C).
C B
A
EB (Ry)
Dot Size (a*3)
Our work consists to calculate the binding energy of an impurity in CdSe quantum dot, placed at the center, with the absence and the presence of the confined LO- phonon interaction as a function of dot size taking into account the ion-LO-phonon interaction and the phonon effect on the subband energy. In fig.1 we have plotted the variations of the polaron binding energy (curve A) and in the absence of the confined LO-phonon interaction (curve B). In our model, we are always interesting only a region where dot size is greater than a*3. In this region the binding energies corresponding to the finite or infinite barrier potential are approach and the approximation of the effective mass is in agreement with the others models of the first principle [19]. Here we have just used the assumption of an infinite barrier potential, which gives an approach results as the finite barrier potential. We can see that all binding energies curves decrease monotonically with dot size. The difference between the binding energies with the presence and the absence of the confined LO-phonon interaction is shown in (curve C) . It is apparent that this energy shift decreases progressively with the dot size.
In fig2. We present the variation of the polaronic contribution due to the effective potential induced by the charge carrier - confined LO-phonon interaction (eq.15) in the impurity (curve D) and in the subbed ( curve E) state. For small dots this polaronic correction remains weak because only a few number of confined LO- phonon modes contributes 1<mi≤Li 2a=Ni 2 with ( i = 1,2,3 ) , the surfacic optical(SO)-phonon is neglected because we are in the region (VD ≥ a*3 ). For a large dots the approach of confined phonons is not valid and the bulk- like approach take place, the polaronic correction is more pronounced because the number of LO-phonon is increased.
0 2 4 6 8 10 12
0.2 0.4 0.6 0.8
Fig. 2 The contribution variation due to the effective potential induced by confined LO-Phonon interaction (Eq. 15) in the impurtity (curve D) and in the subband (curve E) state.
E D
ELO (Ry)
Dot Size (a*3)
3 BOGOLIUBOV TRANSFORMATION FOR DISTINGUISHABLE PARTICLES 39
FIG. 1 : Variations of the binding energy with (curve A) and without (curve B) confined LO-Phonon. The correction of confined LO-phonon (curveC).
FIG. 2 : The contribution variation due to the effective potential induced by confined LO-Phonon interaction (Eq. 15) in the impurtity (curve D) and in the subband (curve E) state.
As expected, the values of the impurity polaron binding energies are higher in the case of quasi-zero- dimensional QDs than the quasi-one and two-dimension systems (QWWs and QWs) [4-11,17]. The comparison with others works shows that our results are in good agreement with that obtained by Ribeiro et al [6] for binding energy in the absence of the confined LO- phonon interaction and present the same behaviour than that obtained by Mukhopadhyay et al [22] for polaronic correction to the ground state using the Feynman- Haken method with parabolic confinement in quantum dot.
In conclusion, we have studied the LO-phonon effects on the polaron binding energy in a quantum dot
structure. The results obtained in this paper will be extended to impurity position and SO-phonon effect when the impurity is placed near the surface of the dot.
ACKNOWLEDGEMENTS
This work was effected in « Programme d’Appui à la Recherche Scientifique P.A.R.S » Physique 03.
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40 B. EL AMRANI, N. ESSBAI, M. FLIYOU, H.CHIKHAOUI AND M. CHAOUCH 3
