Spectra of 2D H-donors in magnetic field and their Zeeman splitting

16 Pages·0.11 MB·English

by  Maen Odeh

Spectra of 2D H-donors in magnetic field and
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0 their Zeeman splitting.
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2

n
a Maen Odeh
J

5
UGRU, UAEU,

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v Al-Ain, Abu Dhabi,
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3 UAE.
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1
email: [email protected]
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1
0
Omar Mustafa
/
h
p Department of Physics, Eastern Mediterranean University
-
h
t G. Magusa, North Cyprus, Mersin 10 - Turkey
a
m
email:[email protected]
:
v
i
X February 7, 2008
r
a
Abstract
The pseudoperturbative shifted - l expansion technique PSLET
[20-25] is employed to study the spectra of 2D H-donors in magnetic
field. The Zeeman effect is investigated and results are found to be
excellent compared to those in ref.[14].
1 Introduction
1
The most prominent featureof technology inthe last decades was the realiza-
tion of low dimensional structures. This achievement urged a huge body of
theoretical and experimental research to study different kinds of interactions
constrained by the new structures [1-14]. Perhaps, one of the most impor-
tant and simplest is that of hydrogenic impurity in bulk semiconductors in a
constant magnetic field. In such systems, the energy scales of bothcoulombic
and magnetic potentials are altered because of the dielectric function of the
∗
medium, ǫ = 1, and the effective mass of the electron m /m = 1. In semicon-
6 6
∗
ductors, typical valuesofthe effective mass m anddielectric constant ǫmake
∗
m /m
the effective Rydberg R∗(= 13.6 eV) about 103 104 times smaller and
ǫ2 −
∗
the dimensionless magnetic field γ = h¯ω /2R ( in a fixed magnetic field B
c
) about 104 105 times larger than those of the free hydrogenic atom [2,13].
−
As a result, the situation where γ 1 ( i.e., the intermediate magnetic
∼
field regime, where neither the Coulombic nore the confining potentials can
be considered small relative to each other ) is of special physical interest.
Therefore, many methods had been employed to resolve its significance [14].
However, all calculations have shown that the donor states strongly con-
fined by quantum wells can be correctly described by two dimensional (2D)
hydrongenlike atomswithproperlyvariationalparameters[9-12].Wecanthus
study 2D hydrogenic donors in a magnetic field to understand the behavior
of donors strongly confined by a quantum wells in different magnetic fields.
Although the Hamiltonian models in these settings can be separable, one is
still prevented from obtaining an analytically exact solution. Only for the
two extreme limits ofthemagnetic field (γ = 0andγ ) theproblemcan
→ ∞
be treated exactly [1,3]. Recently, Taut [15,16] has obtained ( conditionally)
exact solutions for specific values of γ, however no ground-state solutions
were obtained.
The adiabatic method [1,3] had been employed in γ 1 regime. How-
≫
2
ever, calculations based on this method have usually been restricted to the
groundstateandthefirstfewexcitedstates. Otherworkhasbeenestablished
using variational method. Makado and McGill [4] have based their approach
on that of Aldich and Greene [2] with different set of basis and they got quite
good results. Martin et al [11] have used a two-point quasifractional approx-
imation and found excellent interpolation between the weak and strong mag-
netic field perturbation expansions. MacDonald and Ritchie [13] have used
a two-point Pade’ approximation. The curves they obtained from different
Pade’ approximation are very different so that no regular pattern appears
and the results become unreliable [4]. Zhu et al [14] have used a power se-
ries expansion method. Villalbe and Pino [17] have used a finite-difference
scheme with a linear mesh of up to 2000 points and failed to provide a good
estimation of the ground state for the 2D hydrogen atom. Mustafa [18] and
Quigora et al [19] have used the shifted N-expansion technique and their
results were in good agreement with those of Martin et al [11].
Recently, we have introduced a pseudoperturbative shifted-l ( l is the an-
gular momentum quantum number ) expansion technique ( PSLET ) to solve
Schr¯odinger equation for states with arbitrary number of nodal zeros. It sim-
¯ ¯
ply consists of using 1/l as a pseudoperturbation parameter, where l = l β,
−
and β is a suitable shift. The shift β is vital for it removes the poles that
would emerge, at the lowest orbital states with l = 0, in our proposed expan-
sion below. Our new analytical, often semi-analytical, methodical proposal
PSLET has been successfully applied to quasi - relativistic harmonic oscil-
lator [20], spiked harmonic oscillator [21], anharmonic oscillators [22], two
dimensional hydrogenic atom in an arbitrary magnetic field [23], two elec-
trons in quantum parabolic dots [24], truncated coulombic potentials [25],
...etc [26].
Encouraged by its satisfactory performance, we extend, in this paper,
3
PSLETrecipe(insection2)andintroduceits2Dversiontotreattheproblem
of2Dhydrogenicdonorsinmagneticfields. Insection3,wediscusstheresults
of PSLET. We conclude our work in section 4.
2 Hamiltonian model and PSLET recipe
2.1 Hamiltonian model
The problem of a 2D hydrogenic donor in the presence of a magnetic field
that is perpendicular to the 2D plane, in the effective mass approximation,
can be described by the Hamiltonian [14]
h¯2 → 2 e2 → → h¯e → → we2
Hˆ = + A . A +i A . (1)
2 2
−2µ ∇ 2µc2 µc ∇ − ǫρ
→
where A is the vector potential, µ is the electron reduced mass, ǫ is the static
→ ∂ ∂
dielectric constant, w is equal to 0 or 1 and represents (ˆı +ˆ ). If we
2
∇ ∂x ∂y
choose the cylindrical gauge such that
→ 1 → →
A= B r (2)
2 ×
and the wavefunction Ψ(ρ,ϕ) to be in the form
u(ρ)
Ψ(ρ,ϕ) = eimϕ (3)
√ρ
then the Schr˝odinger like equation that describes the system reads
d2 m2 1 γ2 2w
+ − 4 +( ρ2 ) u(ρ) = [E(m) mγ] u(ρ) (4)
"−dρ2 ρ2 4 − ρ # −
4
where m is a well-defined magnetic quantum number of the electron in the
cylindrically symmetric magnetic and 2D Coulomb potentials, E(m) is the
eigenenergy in units of R∗ = µe4/2h¯2ǫ2, the effective Rydberg, γ = h¯ω /2R∗,
c
ω (= eB/µc) is the cyclotron frequency and ρ is in units of the effective Bohr
c
radius (a∗ = ǫh¯2/e2µ).
2.2 PSLET recipe
The 2D version of PSLET starts with shifting the magnetic quantum number
through l = l β, where l = m , and β is a suitable shift introduced
− | |
technically to avoid the trivial case l = 0, and to be determined below.
e
Eq.(4) thus becomes
d2 (l+β +1/2)(l+β 1/2) l2
+ − + V(ρ) u(ρ) = εu(ρ) (5)
"−dρ2 ρ2 Q #
e e e
1 2w
where ε = E(m) mγ, V(ρ) = γ2ρ2 , and Q is a scaling factor that
− 4 − ρ
is set equal to l2 at the end of the calculations .
We employ Taylor’s theorem in expanding Eq.(5) about ρ , an arbitrary
e o
point on the ρ axis. It is convenient then to transform the coordinates in
Eq.(5) via x = l1/2(ρ ρ )/ρ . Expansions about x = 0 ( i.e., ρ = ρ ) yield
o o o
−
e
d2 ρ2
+V(x(ρ)) u(x) = o εu(x), (6)
"−dx2 # l
e
where
e
1 V(ρ ) V′(ρ )ρ3
V(x(ρ)) = ρ2l + o +l1/2 2+ o o x
o "ρ2 Q # "− Q #
o
e e V′′(ρ )ρ4 e ∞
+ 3+ o o x2 +2β ( 1)n(n+1)xnl−n/2
" 2Q # −
n=1
X
e
5
∞ dnV(ρ ) ρ2(ρ x)n
+ ( 1)n(n+1)xn + o o o l−(n−2)/2
" − dρn ! n!Q #
n=3 o
X
1 ∞ e
+ (β2 ) ( 1)n(n+1)xnl−(n+2)/2 +2β, (7)
− 4 −
n=0
X
e
and the prime on V(ρ ) denotes the derivative with respect to ρ . It is
o o
convenient to expand ε as
∞
ε = E(n)l−n.. (8)
nρ,l
n=−2
X
e
Equation (6) is exactly of the type of Schr¨odinger equation for one - dimen-
sional anharmonic oscillator. This leads to the identification of the leading
correction in the energy series, Eq.(8), namely
1 V(ρ )
(−2) o
E = + (9)
nρ,l ρ2 Q
o
(−2)
Here ρ is chosen to minimize E , i. e.
o nρ,l
dE(−2) d2E(−2)
nρ,l = 0 and nρ,l > 0, (10)
dρ dρ2
o o
which in turn gives, with l = √Q,
e
l β = ρ3V′(ρ )/2. (11)
− o o
q
Eq.(11) is an explicit equation in ρ . However, closed form solutions are
o
usually hard to be found, if not impossible. Thus we use numerical solu-
tions to solve for ρ ( hence, the technique is often called semi-analytical
o
6
(−1)
). The shifting parameter β is determined by choosing lE = 0. This
nρ,l
choice is physically motivated. It requires not only the agreements between
e
PSLET eigenvalues andthe exact known ones for the harmonic oscillator and
Coulomb potentials but also between the eigenfunctions as well. Hence
1 1
β = (n + )Ω, (12)
ρ
−2 2
where
ρ V′′(ρ )
o o
Ω = 2 3+ . (13)
v V′(ρ )
u o
u
t
Equation (7) thus becomes
∞
1 V(ρ )
V(x(ρ)) = ρ2l + o + v(n)(x)l−n/2, (14)
o "ρ2 Q #
o n=0
X
e e e
where
1
v(0)(x) = Ω2x2 +2β, (15)
4
ρ5V′′′(ρ )
v(1)(x) = 4βx 4x3 + o o x3, (16)
− − 6Q
and for n 2
≥
v(n)(x) = ( 1)n2β(n+1)xn +( 1)n(β2 1/4)(n 1)x(n−2)
− − − −
+B xn+2, (17)
n
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ρ(n+4) dn+2V(ρ )
B = ( 1)n(n+3)+ o o (18)
n − Q(n+2)! dρn+2
o
Equation (6) thus becomes
d2 + ∞ v(n)(x)l−n/2 u(x) = ρ2 ∞ E(n−1)l−n u(x)
"−dx2 # o" nρ,l #
n=0 n=1
X X
e e (19)
Up to this point, one would conclude that the above procedure is nothing
but an animation of the eminent shifted large-N expansion ( SLNT ). How-
ever, because of the limited capabilities of SLNT in handling large -order
corrections via the standard Rayleigh-Schr˝odinger perturbation theory, only
low-order corrections have been reported, sacrificing in effect its preciseness.
Therefore, one should seek for analternative and proceed by setting the wave
functions with any number of nodes as
u(x) = F (x)exp(U (x)) (20)
nρ,l nρ,l
Eq.(19) readily transforms into the following Riccati type equation
F′′ (x)+2F′ (x)U′ (x) +F (x) [U′′ (x)+(U′ (x))2]
− nρ,l nρ,l nρ,l nρ,l {− nρ,l nρ,l
(cid:20) (cid:21)
(21)
+2β + 1Ω2x2 + ∞ v(n)(x)l−n/2 = ρ2F (x) ∞ E(n−1)l−n
4 n=1 } o nρ,l n=1 nρ,l
P P
e e
where primes denotes derivatives with respect to x.It is evident that (21)
admits solutions of the form
∞ nρ−1
F (x) = xnρ + a(n) xpl−n/2 (22)
nρ,l p,nρ
n=0 p=0
X X
e
8
∞ ∞
U′ (x) = U(n)(x)l−n/2 + G(n)(x)l−(n+1)/2, (23)
nρ,l nρ nρ
n=0 n=0
X X
e e
where
n+1
U(n)(x) = D x2m−1 ; D = 0, (24)
nρ m,n,nρ 0,n,nρ
m=0
X
n+1
G(n)(x) = C x2m. (25)
nρ m,n,nρ
m=0
X
Substituting equations (22)-(25) into equation (21) implies
∞ ′ ′ ∞ n
F (x)[ (U(n) (x)l−n/2 +G(n) (x)l−(n+1)/2) (U(m)(x)U(n−m)(x)l−n/2
nρ,l nρ nρ − nρ nρ
n=0 n=0m=0
P P P
e e e
∞
+G(m)(x)G(n−m)(x)l−(n+2)/2 +2U(m)(x)G(n−m)(x)l−(n+1)/2)+ v(n)l−n/2
nρ nρ nρ nρ
n=0
P
e e e
∞ ∞
ρ2 E(n−1)l−n] 2F′ (x)[ (U(n)(x)l−n/2
− 0 nρ,l − nρ,l nρ
n=1 n=0
P P
e e
+G(n)(x)l−(n+1)/2)] F′′ (x) = 0
nρ − nρ,l
(26)
e
Obviously, the solution of equation (26) follows from the uniqueness of power
series representation. Therefore, for a given n we equate the coefficients of
ρ
the same powers of l and n, respectively. One can then calculate the energy
eigenvalue and eigenfunctions from the knowledge of C ,D and
e m,n,nρ m,n,nρ
a(n) in a hierarchical manner. Nevertheless, the procedure just described
p,nρ
is suitable for a software package such as MAPLE to determine the energy
9
eigenvalue and eigenfunction up to any order of the pseudoperturbation se-
ries.
Although the energy series Eq.(8), could appear divergent, or at best,
asymptotic for small l one can still calculate the eigen series to a very good
accuracy by performing the sophisticated [N,M] Pade’ approximation [25],
e
P +P /l+ +P /lN
0 1 M
PM(1/l) = ··· (27)
N (cid:16) (cid:17)
1+q /l+ +q /lM
1 e N e
···
e (cid:16) (cid:17)
to the energy series, Eq(8). The energey series, Eq(e8), is calculated up to
E(8) /l8 by
nρ,l
e ε = l2E(−2) +E(0) + +E(8) /l8 +O(1/l9), (28)
nρ,l nρ,l nρ,l ··· nρ,l
e e e
and with the P4(1/l) Pade’ approximant it becomes
4
e ε [4,4] = l2E(−2) +P4(1/l). (29)
nρ,l nρ,l 4
e e
Our method is therefore well prescribed.
3 Results and Discussion
For the potential in hand, Eq.(13) reads
2γ2ρ3 +2w
Ω = 2√2 o (30)
v γ2ρ3 +4w
u o
u
t
Eq(11) in turn yields
1 2γ2ρ3 +2w 1
l+√2(n + ) o = γ2ρ4 +wρ (31)
ρ 2 v γ2ρ3 +4w s4 o o
u o
u
t
10