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The Ground State Energy of a Bound Magneto-Polaron in Ternary Mixed Crystals
I. Zorkani, R. Belhissi
To cite this version:
I. Zorkani, R. Belhissi. The Ground State Energy of a Bound Magneto-Polaron in Ternary Mixed Crystals. Journal de Physique I, EDP Sciences, 1997, 7 (2), pp.385-392. �10.1051/jp1:1997151�. �jpa- 00247334�
The Ground State Energy of a Bound Magneto-Polaron
in Ternary Mixed Crystals
1. Zorkani (*) and R. Belh188i
Laboratoire de physique des solides, Ddpartement de Phy8ique, Facultd des Sciences, BP 1796 Atlas, Fds, Morocco
(Received 15 April 1996, revised 10 July 1996, accepted 21 October 1996)
PACS.75.30.-m Intrinsic properties of magnetically ordered materials
PACS.78.40.-q Absorption and reflection spectra: visible and ultraviolet
Abstract. Theoretical calculations for the ground state energy of a bound magneto-polarou
in the ternary mixed crystal A~BI-zC are presented. The theory is based on a variational approach using a trial wave function in the adiabatic limit. The electronic part of the wave
function is the one used in the harmonic approximation. Numerical results are obtained for Ga~ Ah-z As and show that the effect of phonons decreases on increasing the magnetic field and
on decreasing the fraction x of the compounds.
1. Introduction
At present, the application of ternary mixed crystals in the commercial production of quantum- well lasers, high electron mobility transistors, fast field-effect transistors, etc has prompted the necessity for a more detailed study of the impurities in these compounds. An electron bound to a Coulomb center and interacting with the longitudinal optical (LO) phonon, is called a bound polaron [I-4]. Many properties of ternary mixed crystals are different from those of binary crystals and can be modulated by changing the fraction of the compounds.
However, some physical parameters, such as frequency (~J), Fr6hlich coupling parameter (o),
dielectric constant (e) and band mass (m), are dependent on composition z [5-7]. There is
only one LO-phonon branch in binary crystals, while there are two (one is GaAs-like and the other is AlAs-like optical phonon) in ternary crystals. The behaviour of phonons in ternary mixed crystals has been of considerable interest [8-12]. Adachi [5] pointed out that the ground
state energy of the shallow bound polaron for A[Gai-~As varies linearly and increases with
increasing composition z. The energy levels of the shallow impurity state in a ternary mixed crystal has been discussed in the presence of a magnetic field [14-17j. The authors [13-17]
did not fully consider the interaction between an electron and LO-phonon in ternary mixed crystals.
In this paper, we consider the effect of phonons on the energy of the shallow impurity state in the presence of a magnetic field and derive the ground state energy of the bound magneto- polaron. In Section 2, we present the general formalism and deduce the expression for the shallow bound polaron ground state energy as a function of the magnetic field strength for (*) Author for correspondence
© Les #ditions de Physique 1997
386 JOURNAL DE PHYSIQUE I N°2
all coupling parameters al and a2 for the two LO-phonon branches in ternary mixed crystals.
The numerical results and discussion are presented in Section 3.
2. General Formalism
The Hamiltonian describing an electron moving in the Coulomb field of a donor and interacting
with the longitudinal optical phonons in the presence of an external magnetic field is given by [18,20,21]:
H=He+HLO+Hi (I)
where He is the electronic part, HLO is the LO-phonon contribution, and Hi is the e-phonon
interaction Hamiltonian. These are given respectively by:
He =
£ (P
+ ~A)~ ~~ (2a)
m c eon
where A
= B A r is the vector potential of the field, c is the velocity of light in vacuum, (-e)
2
is the electronic charge, eo and m are the static dielectric constant and the effective mass, in ternary mixed crystal A~BI-~C, respectively, which are given by [5-7]:
60 " Z610 + (1 Z)620 (~~)
m = z mAc + (I z)mBc. (2c)
eio, e20 and mAc, mBc are the static dielectric constants and band masses for corresponding binary crystals AC and BC respectively.
~~~ ~~'~~~~~~~ ~ ~ 2'f02b~bq
~
(3)
where rfoi l'fo2) is the ratio between the LO-phonon energy for the crystal AC (BC) and the Coulomb one. a) (ak) and bj (bq) are the creation (annihilation operators for the crystals AC and BC respectively. The electron-phonon interaction:
Hi "
~j ~~ at exp(-ikr) + H
C) + £ ~~ b+ exp(-iqr) + H
C) (4a)
~
kW
~
qW ~
where H.C is the Hermitian Conjugate and:
gi "
-I(2rfoi)~~~fi$ (4b)
g2 " -I(2rfo2)~~~fi$ (4c)
with V being the crystal volume and al and 02 the coupling parameters for the crystals AC and BC respectively.
Choosing the symmetric gauge and using the cylindrical polar coordinate system (p,#,z)
with the magnetic field oriented along the z axis, equation (2a) becomes:
~ ~2
He=-V)--+~p~+rf.Lz (5)
there we have introduced the effective radius a* as the unit of length and the effective Rydberg
R* as the unit of energy; these are functions of the composition z and are given, in the ca8e of
Ga~Ali-~As, by:
z + ~~° (l z)
a*(z)
= 100.7 £)) 1 (6a)
z + -(I z)
IIIAC
z + ~°~~ (l z)
R*(z)
= 5.6 ~~~
~
mev. (6b)
z + ~~° (l z)
610
rf is a dimensionless unit of the magnetic field strength given by (rf = eh/2mcR*)B, Lz is the
z component of the angular momentum operator in units of h.
We first apply the unitary transformation [22]:
Ui " exp I-ia £ k r a)ak ib £ q r b)bq (7)
q
where a and b are variational parameters. The Hamiltonian in equation (I) becomes:
Hi = Up~HUI
= H( + £(2rfoi + a~k~)a)ak + ~j(2rfo2 + b~q~)b)bq
+a~£kk k'a)a( akak, + b~~jq q'b)b)bqbq, + 2ab~jk qa)akb)bq
kk' Qq' kQ (8)
+1 (Sat exPiiii a)k r)
+
-C)
+1 lsbi exPiiii b)~ r)
+
-C)
in which
H( ~
=
-V) ~ + '~ p~ + rf h(z(aky bqy) y(ak~ bq~)). (9)
r 4
We then apply the Lee-Low-Pines transformation [23]:
U2 " exp l~(fka) flak + £(hqb) h(bq) (10)
k
where fk and hq are variational parameters. The ground state wave function in the adiabatic approximation can be written as [24]
lfl~) = 14ell°1 (ii)
where )#e) is the wave function of electron and )0) is the phonon vacuum state wave function.
Using this approximation, the ground state energy of a bound magnetopolaron is given by:
l~ " (~e)li~)~e)+~j(~'f01+a~k~))fk)~+~(2'f02+b~q~))~IQ)~
+~~£h' h')fk)~)fk')~k + b~~q'q')~IQ)~)~IQ')~ + 2~~~h'q)fk)~)hq)~
(~~)
kk, qq, kq
+1 lsf<Pk + -C) + j (§hiPq +
-C)
388 JOURNAL DE PHYSIQUE I N°2
But, by symmetry, we have [25]
£ fk)~k
= 0 (13a)
k
and ~j jhqj~q
= 0 (13b)
~
thus the ground state energy becomes:
l~ " (~e)li~)~e) + ~(2'f01 + ~~k~))fk)~ + ~(2'f02 + b~q~))~lq)~
+i IA lPk
+ H-Cl + j
Ii
hiP~ +
-C) ~~~~
where
Pk " (<ei ~XP(I(1 ~)k '~ii<ei (15~)
P~ " (<ei ~XP(I(1 b)~l ~ii<ei (lsbi
Minimising equation (14) with respect to fk and hq, we obtain:
~~ =
91Pl (16aj
kli(2rfoi + a2k2)
~~
W(/~~~+
b2q2) ~~~~~
and get the following energy:
l~ " (~e)lf~)~e) + ~(2'f01 + a~k~))fk)~ + ~(2'f02 + b~q~))/l~)~
k ~
~k2V)~)~~~~~2k2)
~
°q2V~j~~~~~2q21' ~~~~
~
We use, in the harmonic approximation, the following ground state wave function of the elec- tron:
j#e) = (~i/~)~@exp(-~(r~/2). (18)
when ii is a variational parameters. This wave function is similar to the one used in references
[18,19] (with I(
= 1/2a2 in the isotropic case). We find for:
pk " exp(-(I a)~k~/41j) (19a)
pq = exp(-(I b)~q~/41jj (19bj
that the ground state energy is given by:
~~~~~
~(2~'f~)~~~j ~ (2~~l'f~)~~~
'~~~ 2~~~) ~~ ~'~~
and erfc(z)
= 1
~ exp(-z~)dz is the complementary error function.
V~~
By minimising the expectation values of the energy equation (20) with respect to the vari- ational parameters a, b and ii, we obtain the ground state energy of the shallow bound
magnetopolaron.
ic,oo
-
9.oo x Wf
uJ
c,oo
-i,oo
ODD ma
7
Fig. 1. The electronic ground state energy as a function of the magnetic field strength in
Ga~Ali-zAs in units of R*(z).
Table I. The parameters of binary crystals used in calculation.
eo eco h~JLo m a
AlAs (1) 10.06 [26] 8.16 [26] 50.09 [11] 0.15 [5] 0.126 [5]
GaAs (2) 13.18 [27] 10.89 [27] 36.25 [11] 0.067 [5] 0.068 [5]
3. Numerical Results and Discussion
In Figure I we present the ground state energy as a function of the magnetic field strength
without the effect of phonons (Eq. (20) with al
" 02 " 0). The parameters of binary crystals
used in the calculation are shown in Table I. Because the ternary mixed crystal is defined by
its composition z, the curve in Figure I is universal. It is valid for all fraction z in the region
0 < z < I and it increases with increasing magnetic field.
In Figure 2 we present the ground state energy with and without the polaron effect in the
case of Ga~Ali-~As for z = 0.2 and 0.8 as a function of the magnetic field strength in units of the R*(z
= I) (R*(1) = 5.6 mev). We notice that, the difference /hE, between the two curves
(with and without the polaron effect) decreases with increasing the magnetic field. This can be
explained physically as follows: the presence of the magnetic field makes the electron motion
in the transverse plane ix, y-plane) harmonic oscillator like. The electron-phonon coupling is
effective only when the electron moves slowly so that it effectively clothed by the phonons.
390 JOURNAL DE PHYSIQUE I N°2
with the pclaron effect withoLl the polaron effect
' '
/
- '
C / ,
~Q$ / /
~ / ,'
Lu '
Zoo ,'
, x_o~
,$- ,-'
---_-
-3.00
0.00 5.00
Fig. 2. Ground state energy as a function of the magnetic field strength with (solid line) and without the polaronic effect (dashed line) in Ga~Ali-zAs mixed crystal for x
= 0.2 and x
= 0.8 in
units of R*(1).
'f= O-C
i,~i
1.51
o,oo o-co o.80
AlAs GaAs
Fig. 3. The energy shift between the bound polaron energy and the electronic one as a function of mole fraction x for ~
= 0.4, in Ga~Ali-zAs.
7
7=05
- '
y=1 7
~ff y=ai
cr w
5.60
0.00 loo
x
Fig. 4. Ground state energy of a shallow bound polaron as a function of the mole fraction
x in units of R*(z
= 1) for magnetic strengths of ~ = 0.1, 0.5 and 1 for Ga~Ali-zAs.
In Figure 3, we present the energy shift between the bound polaron energy and the electronic
one as a function of mole fraction z in the case of Ga~Ali-zAs case for a fixed magnetic field strength at rf = 0.4: this energy shift decreases from the more polar compound (AlAs where
a = 0.126) towards the less polar one (GaAs with a
= 0.068).
Figure 4 shows the ground state energy of the shallow bound polaron as a function of the mole fraction z in units of RI (z
= I) for a magnetic field strength of rf
= 0.1, 0.5 and I. For
the low field limit the effect of a magnetic field is weaker and is important when the magnetic field strength tends to the intermediate (rf < 0.5) and high limit region (rf > I).
For z = 0 and rf = 0 our results (E
= -1.62) are comparable with those presented by Shi et al. [28] (E[28] = -1.550), Adachi [5] (E[5] = -2.3) and Zorkani et al. [18] (E[18]
= -1.5516) in units of the Rydberg effective constant (R*
= 5.6 mev). The authors [18,28] have calculated and discussed the magnetopolaron effect on the shallow donor states in GaAs. For
z = 0,
where the magnetic field is switched on, our result (E
= -8.91 mev) is very close, for example
for a magnetic field fixed at rf = 0.2, to the one in reference [28] (E[28]
= 8.512 mev). For nf = I, our result is E
= -7.366 mev and the one of Shi et al. [28] is E = -6.664 mev
so a disagreement begins to be established where the magnetic field tends to stronger values (nf > I). In effect, in a strong magnetic field, the symmetry of the wave function is axial and not spherical. Therefore the spherical approximation of the trial wave function will lead to
unphysical results, especially for stronger magnetic field (nf > 1).
392 JOURNAL DE PHYSIQUE I N°2
Acknowledgments
One of the authors (I.Z.) would like to thank the ICTP, Trieste (Italy) for hospitality.
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