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Metastable impurity perturbed states of 2D single carrier quantum dots
Chemical Physics Letters 468 (2009) 216–221
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Metastable impurity perturbed states of 2D single carrier quantum dots Ram Kuntal Hazra a, Manas Ghosh b, S.P. Bhattacharyya a,* a b
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India Department of Chemistry, Physical Chemistry Section, Visva-Bharati, Santiniketan 731 235, India
a r t i c l e
i n f o
Article history: Received 5 November 2008 In final form 1 December 2008 Available online 7 December 2008
a b s t r a c t We investigate the characteristics of a few metastable states in 2D quantum dots in the presence of complex impurity potentials. The interplay of electrical confinement potential, transverse magnetic field strength, position of the impurity centre, real and imaginary amplitudes of the complex dopant potential and their role in stabilizing or destabilizing the metastable states are analyzed. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Low dimensional semiconductors are objects of great scientific curiosity and practical utility [1,2]. Practical realization of quasitwo dimensional quantum wires and quasi-zero dimensional quantum dot atoms (superatoms) are real break-throughs in electronics and optics [3–7]. In a model 2D quantum dot external electric field (harmonic) may be used to confine carrier electron(s). Shape of such a quantum dot can be controlled by the nature and the geometry of electrostatic gates. Occurrence of impurity is natural in the fabrication of nanoscale objects, but such impurities (dopants) can play profound role in determining the properties of the objects of interest. For example, impurity plays an important role in transistor technology [1]. Researchers are naturally expected to focus their attention on the controllability of the impurity induced effects. For single carrier 2D dots with harmonic confinement, the Schrödinger equation is exactly solvable [8,9]. One can therefore find the impurity perturbed states by diagonalizing the hamiltonian including impurity interactions, in the basis of eigenstates of the unperturbed hamiltonian. Alternatively, one may make use of the idea of quantum adiabatic switching [10– 20] and solve the appropriate evolution equations for the impurity modulated states, starting from the corresponding unperturbed eigenstate of the impurity free dot hamiltonian. A switching function is then used to interpolate continuously between the initial and the final hamiltonians. For adiabaticity of the transition, the switching function must fulfill Kato conditions [13,14]. In reality, it is a challenging task to construct such a function that is computationally efficient without sacrificing adiabaticity of the evolution. The impurity can be a hole or an electron localized in a trap. The impurity may have stable (bound) states or metastable states (autoionizing states, for example). If the impurity supports only metastable states, they could interact with bound dot states making all the states of the composite (dot + impurity) system metasta-
ble. One way to model the effects of metastable impurity states could be to introduce a phenomenological complex impurity-dot interaction potential. Alternatively, we could use a real interaction potential and use the complex coordinate rotation method to generate a non-hermitian hamiltonian for describing the metastable states of the composite system. In the present work, we have investigated the signatures of a complex dopant interaction with the carrier electron and the role of electrical confinement, transverse magnetic field, the location of the impurity centre and the strength of complex dopant potential on the life times of different states of the metastable impurity perturbed dot. To the best of knowledge, no such studies have been so far reported in literature. 2. Model The stationary states of a carrier electron with effective mass m in 2D parabolic dots confined by potential v ðx; yÞ ¼ 12 m x20 ðx2 þ y2 Þ in the presence of transverse magnetic field B (¼ r A) of symmetric gauge f¼ 12 ðB rÞ ¼ 12 ðBy; Bx; 0Þg, correspond to the eigenstates of the hamiltonian [21,22] given by the equation
Hðx; y; x0 ; xc Þ ¼
1 e 2 1 p þ A0 þ m x20 x2 þ y2 ; 2m 0 c 2
ð1Þ
where xc ¼ ðmeB cÞ. The energy levels are obtained by solving timeindependent Schrödinger equation
Hðx; y; x0 ; xc ÞWn ðx; yÞ ¼ En Wn ðx; yÞ
ð2Þ
and elaboration of the hamiltonian leads to the equation
Hðx; y; xc ; x0 ÞUnm ðx; yÞ i o o xc y x Unm ðx; yÞ ¼ Hðx; x0 ; xc Þ þ Hðy; x0 ; xc Þ þ h 2 ox oy ð3Þ
* Corresponding author. Fax: +91 33 2473 2805. E-mail address: [email protected] (S.P. Bhattacharyya). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.12.005
) Hðx; y; xc ; x0 ÞUnm ðx; yÞ ¼ Enm Unm ðx; yÞ;
ð4Þ
217
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
where 2
2
h o 1 þ m X2 x2 ox2 2 2m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Hðx; x0 ; xc Þ ¼
xc 2 and X ¼ fx20 þ 14 ðm Þ g. The system can be viewed as a composite of two harmonic oscillators interacting with each other by the coub int ¼ i hxc ðy oxo x oyo Þ. At very low magnetic field it pling potential V 2 reduces to two separable harmonic oscillators Hðx; x0 ; 0Þ and Hðy; x0 ; 0Þ, respectively oscillating along x- and y-axes having eigenfunctions
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 1 X Un ðx; x0 ; xc ! 0Þ ¼ expðz2 =2ÞHn ðzÞ 2n n! p
ð5Þ
and
z¼
We may expand the wave function w in a Fourier integral over real energy eigenfunctions of H0 , with their time-dependence 0 determined by the exponentials eiE t=h and try to find out the order of magnitude of the energy interval over which the amplitudes of the real energy functions are non-vanishing. Now, the Fourier expansion of the wave function with complex energy Ec is, R
I
eiEc t=h ¼ eCk t=2h eiEk t=h ¼
Xx
ð6Þ
For a non-zero magnetic field strength, an exact solution of the wave equation for doped parabolic 2D superatom is not possible. We have therefore adopted a variational approach for approximately solving the problem by taking,
Wðx; yÞ ¼
X
C n;m /n ðX; xÞ/m ðX; yÞ ¼
ð11Þ
where the amplitude aðEÞ is given by
aðEÞ ¼
Z
1 2ph
1
R
eCk t=2h eiðEk EÞ dt ¼ I
0
X
ck jki
jaðEÞj2 ¼ 4p
2
1 ERk
1.0
where eigenstate jki corresponds to product basis elements f/n ðX; xÞ/m ðX; yÞg. Interaction among these two oscillators can be easily evaluated exactly with the help of standard recurrence relation
0.8
hn; mjV int ðx; y; xc Þjn0 ; m0 i ! rffiffiffiffiffiffi " rffiffiffiffiffiffiffiffiffiffiffiffiffiffi hxc m0 þ 1 m0 dm;m0 þ1 þ ¼i dm;m0 1 2 2 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi n0 n0 þ 1 dn;n0 þ1 dn;n0 1 2 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi n0 þ 1 n0 dn;n0 þ1 þ dn;n0 1 2 2 !# rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi 0 0 m m þ1 dm;m0 þ1 dm;m0 1 2 2
0.4
h
2p
CIk 2
1 i : þ i ERk E
ð12Þ
2
þ
CI2 k
:
ð13Þ
4
þ V p ðx; yÞ 2
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2
1.0
ð8Þ
Norm
0.8
Dynamic Non-dynamic
0.6 0.4 0.2
ð9Þ
CIk
0.0 1.0 Dynamic Non-dynamic
0.8 0.6 0.4 0.2
2
and the corresponding energy eigenfunctions are decaying wave functions. The probability amplitude is then proportional to I R eCk t=2h eiCk t=h . The second factor accounts for the exponential falloff of the amplitude, the first factor being the dynamical phase factor of the amplitude of the initial state which is now damped. The mean decay time (Dt k ) is given by I
0.6
0.0
where V p ðx; yÞ ¼ V 0 ecððxax0 Þ þðyay0 Þ Þ or V 0 ecð ðxax0 Þ þðyay0 Þ Þ with c > 0, V 0 > 0 (electron doping), V 0 < 0 (hole doping) and ðax0 ; ay0 Þ is the centre of impurity. The matrix elements of V p ðx; yÞ may be computed numerically. Let us assume that the amplitude of the impurity potential is I complex V 0 ¼ ðV R0 iV 0 Þ. In that case, H is no longer hermitian. The eigenvalues of H are now generally complex
Ec ¼ Ek ¼ ERk i
Dynamic Non-dynamic
0.2
1 Hp ðx0 ; xc Þ ¼ Hðx; x0 ; xc Þ þ Hðy; x0 ; xc Þ þ V int ðx; y; xc Þ m
1
E
k
In the presence of impurity perturbations, the hamiltonian becomes
0
aðEÞeiEt=h dE;
1
ð7Þ
n;m
Z
þ1
The square of the amplitude, i.e., the probability is
pffiffiffiffi
Dt k ¼
Z
eCk t=h dt ¼
h
CIk
:
ð10Þ
0.0 0
1
2
3
4
5
Time x 10 -6 (a.u.) Fig. 1. A plot of norm versus time of (solid circle (SC)) the ground state with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me , (solid triangle (ST)) the ground state with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me and (solid diamond (SD)) the first excited state with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me .
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R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
Table 1 Half-lives of different states (V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u., m ¼ 1:5me ) in impurity doped dots. Quantum states
j0i j1i j8i
s1=2 for hole doping (V R0 ¼ 0:8 a.u.)
s1=2 for electron doping (V R0 ¼ 0:8 a.u.)
Dynamic (a.u.)
Nondynamic (a.u.)
Dynamic (a.u.)
Nondynamic (a.u.)
5:616 104 3:464 107 6:931 106
5:62 104 3:45 107 6:92 106
5:691 105 7:719 106 9:418 106
5:70 105 7:72 106 9:42 106
Clearly,
1 jaðEÞj2 2 CI as ERk E ! k : 2
jaðEÞj2 !
The energy width is therefore CIk and its reciprocal is linked to the mean decay time. From definition, it is evident that the dimension of the amplitude aðEÞ is (energy)1. Therefore, if we integrate the square of the amplitude over energy we obtain a quantity whose dimension is also (energy)1. But it is also the total area of jaðEÞj2 versus E curve in the interval in which amplitudes of the expansion are non-zero. If we denote this interval as DE, we obtain, by definition
Z þ1 1 ¼ jaðEÞj2 dE DE 1 Z þ1 1 1 dE ) ¼ 2 2 CI2 DE 4p 1 R Ek E þ 4k
ð14Þ ð15Þ
1 1 ¼ DE 2pCIk
Comparing Eqs. (10) and (16), we have,
DEDt ¼ 2ph
ð17Þ
The relation Eq. (17) should be taken to imply the energy of a state existing for a limited time ‘Dt’ is determinate to within the accuracy h=DtÞ. We have therefore two options at our of the order ð2p disposal: (a) we diagonalize the impurity perturbed complex non-hermitian hamiltonian (H) in the basis of eigenstates of H0 , find the imaginary parts of the eigenvalues and from that, the level width (C), the inverse (s) of which gives the decay rate (K C ). We may call it the static way of computing lifetime or decay rate; (b) we solve the time-dependent Schrödinger equation for H numerically in the basis of eigenstates of H0 , by an accurate Adams–Moulton Predictor-corrector algorithm or 6th order Runge–Kutta–Fehlberg method with small enough time steps and numerically compute the rate at which jwj2 decay, yielding directly the decay rate ðdtd jwj2 ¼ K C Þ. We may term it the dynamic way of computing decay rate or the lifetime. The questions that we wish to probe are: (i) if the two ways of computing decay rate (lifetime) produce same results, (ii) if the strength of the transverse magnetic field affects the computed decay rates or lifetimes, and if so, in what ways. (iii) if the location, the type and strength of the impurity potential has significant effects.
ð16Þ Probability Density X 10 10
1.6
c
1.2
(x 10-8 a.u.)
8
0.8 0.4
6
0.0 0 2 4 6 8 10
1.2 0.9 0.6 0.3
4 2
-100 -50
0
8
50 100
0 50 100 X
-100 -50
0 Y
0 45 90 X
-45
0 Y
45
-90
0 -60 -30 Y
30
b
10 8 6 4 2 0 2 4 6 8 10
10 (x 10-7 a.u.)
Half-Life Period
4
2.5 2.0 1.5 1.0 0.5
6 4
-90-45
2 12
6.0
90
-4
(x 10 a.u.)
5.6
10
5.2
a
4.8
0 2 4 6 8 10
8
5 4 3 2 1
6 4
-60 0
2
4
6
8
10
ωc (x 10 a.u.) 4
Fig. 2. A plot of half-life (in a.u.) versus xc with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), x0 ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states.
-30
0 30 60 X
60
Fig. 3. Probability density map of (a) the ground, (b) the first excited and (c) the eighth excited states with xc ¼ 0:000001 a.u., x0 ¼ 0:0005 a.u., m ¼ 1:5me , V R0 ¼ 0:8 a.u., V I0 ¼ 0:0 a.u and impurity centre R = (3, 0).
219
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
3. Results and discussion We have investigated the time-profiles of the norms for the quasi-stable states in the presence of both electron and hole doping impurities in 2D single quantum dots (Fig. 1). The impurity is located at the point (3, 0). Fig. 1 (SC and ST) shows how the norm decays in the metastable ground states of both the hole and electron impurity perturbed quantum dots (V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me Þ and Fig. 1 (SD) clearly highlights the decay phenomenon of the first excited state in the case of a electron impurity perturbed dot. The half-lives estimated from the exponential decay of the norms turn out to be 5:616 104 a.u., 3:464 107 a.u. and 6:93 106 a.u. of time for the ground, first, and eighth excited states, respectively. On the other hand, the imaginary part of the corresponding complex eigenvalue (obtained by diagonalizing H in 36-dimensional basis) yields the half-lives of 5:62 104 a.u., 3:47 107 a.u. and 6:94 106 a.u. of time. The agreement between the two sets is quite close. Table 1 reports the dynamically and non-dynamically computed half-lives (s1=2 ) of several states of the same system. The agreement between the two sets of values is satisfactory. Since half-lives of the first and the eighth excited states in hole-doped single dots are rather high, the time scale of final decay becomes very large and we refrain from presenting the corresponding figures. Fig. 2 displays how the computed half-lives of hole-doped states vary with the cyclotron frequency or the strength of the transverse magnetic field. For the ground state, half-life is high when xc is small and it registers a moderate decrease as xc increases for the hole-doped impurity with the amplitudes of the real and the imag-
inary parts of the impurity potential V R0 ¼ 0:8 a.u. and V I0 ¼ 0:004 a.u. (Fig. 2 (SC)). The half-life of the first excited state mildly oscillates with increase in xc (Fig. 2 (ST)). The half-life of the eighth excited state decreases to a rather low value at xc ¼ 4:0 1044 a.u., then rises abruptly around xc ¼ ½6:0 104 7:0 10 a.u. (Fig. 2 (SD)) and saturates thereafter. The unperturbed dot states have infinite life-times either at low or at high cyclotron frequencies. At low xc , the ground state is described by a rather diffuse wave function which overlaps and interacts only weakly with the metastable hole impurity states. The weak mixing causes significant decrease in the life-time relative to the impurity free ground state. At higher xc , the ground state wave function is more compact around the centre of the potential and interacts more strongly with the metastable impurity states. The density plots of the relevant states of the impurity free dot are shown in Fig. 3a–c at low xc (=0.000001 a.u.). The impurity perturbed states are generally seen to have much lower life-times than the corresponding free states. The density plots (Fig. 3a–c) show that the impurity free excited states have considerable accumulation of electron density in the neighborhood of the impurity centre (3, 0). The mixing with the metastable impurity states is thus more pronounced in the excited states causing decrease in the life-times. The variation of s1=2 with xc reflects the changes in the electron density distribution in the free dot as xc varies. The behaviour of the ‘half-life period versus xc ’ plot of the ground state in an electron doped system with the amplitude of V R0 ¼ 0:8 a.u. and V I0 ¼ 0:004 a.u. (Fig. 4a (SC)) is quite different from what was observed in the previous case (Fig. 2 (SC)). Thus for a 2D single quantum dot with complex impurity potential, hole doping progressively destabilizes the ground state as xc increases although the extent of destabilization is small. The repulsive elec-
1.6
-7
(x 10 a.u.)
(x 10-8 a.u.)
3 1.2 0.8
1
0.4
0 0
2
4
6
8
10
6
0.6
Half-Life Period
-9
(x 10 a.u.)
0.8
0.4 0.2 0.0 1.0
4
2
0 1.2
0.8 (x 10-6 a.u.)
-7
(x 10 a.u.)
Half-Life Period
1.0
(x 10-7 a.u.)
0.0 1.2
2
0.6 0.4 0.2 0.0
0.9 0.6 0.3 0.0
0
2
4
6
ωc(x 10 a.u.) 4
8
10
-0.8
-0.4
0.0
0.4
0.8
R
Amplitude ( V0 a.u. )
Fig. 4. (a) A plot of half-life (in a.u.) versus xc with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., impurity centre R = (3,0), x0 ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states. (b) A plot of half-life (in a.u.) versus V R0 with V I0 ¼ 0:008jV R0 j a.u., impurity centre R = (3, 0), x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states.
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
tron doping potential scatters the carrier electron, reducing its lifetime which remains virtually constant over a range of xc . For much larger xc , the carrier electron remains strongly localized around the centre of the confinement potential, reducing the scattering induced effect and the life-time increases. For the first excited state, the half-lives computed by the diagonalization method nicely agree with the respective dynamically computed values. The excited state life-times are generally two orders of magnitudes lower than the life-times of the metastable ground state. The density plot for the first excited state of the impurity free dot shows (Fig. 3b) build up of carrier electron density near the impurity centre which is strongly scattered by the repulsive impurity potential leading to a general decrease in life-time. When xc increases substantially the carrier electron density is drawn closer to the centre of the potential, reducing the effects of scattering from the repulsive metastable impurity potentials. These values decrease initially as xc increases and finally saturate (Fig. 4a (ST)). For the eighth excited state, the plot of ‘half-life versus xc ’ reveals oscillations which get damped reducing s1=2 to near zero value (Fig. 4a (SD)) as xc increases. This feature can be similarly explained from the nature of variations in the density distribution in the unperturbed eighth excited state of the dot as xc changes. The details will be presented elsewhere. When the magnitude of the real part of the (repulsive) electron impurity potential is gradually enhanced, with a proportional increase the imaginary part, the computed half-life of the ground state passes through a clear minimum at V R0 ¼ 0:45 a.u.; but for the hole-doped system it diminishes to an almost vanishing value making the ground state appear very much like a transient state (Fig. 4b (SC)). The minimum appears presumably due to a delicate balance between the effects of repulsive scatterer at (3, 0) (tending
to push the carrier electron towards the centre of the confining potential) and the confinement potential which also tends to draw the carrier electron to itself. There is a clear discontinuity in the computed half-lives at V R0 . Since in this case V R0 ¼ 0 implies V I0 ¼ 0, the life-time of any state for V R0 ¼ 0 is infinite explaining the discontinuity in the ‘s1=2 versus V R0 ’ plots. The computed halflife (s1=2 ) remains almost constant for the first excited state when V R0 < 0. For V R > 0, s1=2 decreases with increasing amplitude of the real part of the perturbation potential (Fig. 4b (ST)) and there is a discontinuity at V R ¼ 0. In the case of the eighth excited state (chosen randomly) the pattern of the ‘s1=2 versus V R0 ’ plot is similar to what was found for the ground state and s1=2 reaches a minimum at V R0 = 0.42 a.u. (Fig. 4b (SD)). One can explain the observed features using arguments already provided for the ground state. If the imaginary part of the impurity potential is held fixed at a small value, V I0 ¼ 0:004 a.u., s1=2 -V R0 profiles of the quasi-stable states in a 2D single dot do not reveal any discontinuous jump near V R0 ¼ 0 (Fig. 5) as expected since V I0 –0 even when V R0 ¼ 0, so that the life-time remains finite even at V R0 ¼ 0. The life-time of the
a
6 5
(x 10-7 a.u.)
220
4 3 2 1 0
1.4
-7
(x 10 a.u.)
1.3 1.2 1.1 1.0
(x 10-8 a.u.)
Half-Life Period
1.6
0.0 5.85
0.8
(x 10 a.u.)
5.70 5.65 5.60 5.55
0
2
4
6
8
10
Position of Impurity ( Rx a.u.)
b
-5
6 5 (x 10 a.u.)
5.75
-4
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
5.80
Half-Life Period (x 10 a.u.)
(x 10-7 a.u.)
Half-Life Period
0.7
-5
0.8 0.4
0.9
4 3 2 1 0
1.2
-0.8
-0.4
0.0
0.4
0.8
R
Amplitude ( V0 a.u.) Fig. 5. A plot of half-life (in a.u.) versus V R0 when V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states.
8 7 6 5 0
2
4
6
8
10
Position of Impurity ( Rx a.u.) Fig. 6. (a) A plot of half-life versus Rx with V I0 ¼ 0:004 a.u., x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me at (SC) V R0 ¼ 0:8 a.u. for the ground, at (ST) V R0 ¼ 0:8 a.u. for the first excited and (SD) V R0 ¼ 0:8 a.u. for the eighth excited states. (b) A plot of half-life versus Rx with V I0 ¼ 0:004 a.u., x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me at V R0 ¼ 0:8 a.u. for the ground state.
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
ground state increases with increase in the real part of the impurity potential. The repulsive scatterer forces the carrier electron density towards the centre of the confinement potential and the state progressively becomes similar to the unperturbed ground state. The attractive scatterer, on the other hand progressively draws the carrier electron to itself as jV R0 j increases causing a drop in the life-time. ‘s versus xc ’ plot for the first excited state shows a reverse trend. Half-life of the ‘eighth excited’ state appears to increase with V R0 , irrespective of its sign. It reaches a minimum value for an electron dopant strength of V R0 ¼ 0:21 a.u. The behaviour can be rationalized from the density plots of the concerned states of the unperturbed dot. The position of the impurity centre also has an impact on the life-span of the quasi-stable dot states. In the presence of a hole doping impurity half-life of the ground state increases as impurity centre moves away from the origin along x-axis (Fig. 6a (SC)). In the first and eighth excited states, however, it decreases to a steady value beyond Rx ¼ 7 a.u. Fig. 6a (ST and SD). The further the impurity centre from the origin, the lesser is its interaction with the carrier electron, and lesser is the metastability of the impurity perturbed dot states. As the electron doped impurity centre moves further and further away from the origin, it stabilizes the ground state as expected (Fig. 6b). The life-time changes therefore directly mirror fine changes in the wave functions of the carrier electron in the presence of metastable impurity centres of either an attractive or a repulsive type. It would be nice to have experimental data for further advancement in the modelling. 4. Conclusion The presence of a complex impurity potential makes all the states of a 2D quantum dot metastable with half-lives that depend on the nature (attractive or repulsive) and location of the centre of
221
the impurity and the relative magnitudes of the real and the imaginary parts of the potential. The strength of the transverse magnetic field significantly modulates the lifetimes or decay rates which essentially reflect the changes in the electron density distribution of the unperturbed dot states, which mix with the metastable impurity states. Acknowledgements Thanks are due to the CSIR, Government of India, New Delhi for the award of Senior Research Fellowships to RKH and MG.
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L. Jacak, P. Hewrylak, A. Wöjos, Quantum Dots, Springer, 1998. L. Bányai, S.W. Koch, Semiconductor Quantum Dots, World Scientific, 1993. L.L. Chang, L. Esaki, R. Tsu, Appl. Phys. Lett. 24 (1974) 593. R. Dingle, W. Wiegmann, C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. M.A. Herman, Postepy Fizyki 34 (1983) 431. P.M. Petroff, A.C. Gossard, C.H. Henry, Appl. Phys. Lett. 41 (1982) 635. M.A. Reed, R.T. Bate, K. Bradshaw, W.M. Duncan, W.M. Frensley, J.W. Lee, H.D. Smith, J. Vac. Sci. Technol. B 4 (1986) 358. V. Fock, Z. Phys. 47 (1928) 446. C.G. Darwin, Proc. Camb. Philos. Soc. 27 (1930) 86. M. Born, Z. Phys. 40 (1926) 165. M. Born, V. Fock, Z. Phys. 51 (1950) 435. R.T. Skodje, F. Borondo, W.P. Reinherdt, J. Chem. Phys. 82 (1985) 4611. T. Kato, J. Phys. Soc. Jpn 5 (1950) 435. F. Rellich, Mitt. Math. Ann. 113 (1936) 600. H. Gingold, P.K. Hshich, SIAM J. Math. Anal. 18 (1987) 1287. D. Kohen, D.J. Tannor, J. Chem. Phys. 98 (1993) 3168. S. Adhikari, P. Dutta, S.P. Bhattacharyya, Chem. Phys. 206 (1996) 315. H. Gingold, Int. J. Quant. Chem. Z.-Phys. 40 (1991) 3. C. Jaffe, J. Chem. Phys. 86 (1987) 4499; C. Jaffe, J. Chem. Phys. 85 (1986) 2885. C.S. Lent, P.D. Tougaw, Proc. IEEE 85 (1997) 547. V. Fock, Z. Phys. 47 (1928) 446. C. Darwin, Proc. Camb. Philos. Soc. 27 (1930) 86.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Metastable impurity perturbed states of 2D single carrier quantum dots Ram Kuntal Hazra a, Manas Ghosh b, S.P. Bhattacharyya a,* a b
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India Department of Chemistry, Physical Chemistry Section, Visva-Bharati, Santiniketan 731 235, India
a r t i c l e
i n f o
Article history: Received 5 November 2008 In final form 1 December 2008 Available online 7 December 2008
a b s t r a c t We investigate the characteristics of a few metastable states in 2D quantum dots in the presence of complex impurity potentials. The interplay of electrical confinement potential, transverse magnetic field strength, position of the impurity centre, real and imaginary amplitudes of the complex dopant potential and their role in stabilizing or destabilizing the metastable states are analyzed. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Low dimensional semiconductors are objects of great scientific curiosity and practical utility [1,2]. Practical realization of quasitwo dimensional quantum wires and quasi-zero dimensional quantum dot atoms (superatoms) are real break-throughs in electronics and optics [3–7]. In a model 2D quantum dot external electric field (harmonic) may be used to confine carrier electron(s). Shape of such a quantum dot can be controlled by the nature and the geometry of electrostatic gates. Occurrence of impurity is natural in the fabrication of nanoscale objects, but such impurities (dopants) can play profound role in determining the properties of the objects of interest. For example, impurity plays an important role in transistor technology [1]. Researchers are naturally expected to focus their attention on the controllability of the impurity induced effects. For single carrier 2D dots with harmonic confinement, the Schrödinger equation is exactly solvable [8,9]. One can therefore find the impurity perturbed states by diagonalizing the hamiltonian including impurity interactions, in the basis of eigenstates of the unperturbed hamiltonian. Alternatively, one may make use of the idea of quantum adiabatic switching [10– 20] and solve the appropriate evolution equations for the impurity modulated states, starting from the corresponding unperturbed eigenstate of the impurity free dot hamiltonian. A switching function is then used to interpolate continuously between the initial and the final hamiltonians. For adiabaticity of the transition, the switching function must fulfill Kato conditions [13,14]. In reality, it is a challenging task to construct such a function that is computationally efficient without sacrificing adiabaticity of the evolution. The impurity can be a hole or an electron localized in a trap. The impurity may have stable (bound) states or metastable states (autoionizing states, for example). If the impurity supports only metastable states, they could interact with bound dot states making all the states of the composite (dot + impurity) system metasta-
ble. One way to model the effects of metastable impurity states could be to introduce a phenomenological complex impurity-dot interaction potential. Alternatively, we could use a real interaction potential and use the complex coordinate rotation method to generate a non-hermitian hamiltonian for describing the metastable states of the composite system. In the present work, we have investigated the signatures of a complex dopant interaction with the carrier electron and the role of electrical confinement, transverse magnetic field, the location of the impurity centre and the strength of complex dopant potential on the life times of different states of the metastable impurity perturbed dot. To the best of knowledge, no such studies have been so far reported in literature. 2. Model The stationary states of a carrier electron with effective mass m in 2D parabolic dots confined by potential v ðx; yÞ ¼ 12 m x20 ðx2 þ y2 Þ in the presence of transverse magnetic field B (¼ r A) of symmetric gauge f¼ 12 ðB rÞ ¼ 12 ðBy; Bx; 0Þg, correspond to the eigenstates of the hamiltonian [21,22] given by the equation
Hðx; y; x0 ; xc Þ ¼
1 e 2 1 p þ A0 þ m x20 x2 þ y2 ; 2m 0 c 2
ð1Þ
where xc ¼ ðmeB cÞ. The energy levels are obtained by solving timeindependent Schrödinger equation
Hðx; y; x0 ; xc ÞWn ðx; yÞ ¼ En Wn ðx; yÞ
ð2Þ
and elaboration of the hamiltonian leads to the equation
Hðx; y; xc ; x0 ÞUnm ðx; yÞ i o o xc y x Unm ðx; yÞ ¼ Hðx; x0 ; xc Þ þ Hðy; x0 ; xc Þ þ h 2 ox oy ð3Þ
* Corresponding author. Fax: +91 33 2473 2805. E-mail address: [email protected] (S.P. Bhattacharyya). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.12.005
) Hðx; y; xc ; x0 ÞUnm ðx; yÞ ¼ Enm Unm ðx; yÞ;
ð4Þ
217
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
where 2
2
h o 1 þ m X2 x2 ox2 2 2m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Hðx; x0 ; xc Þ ¼
xc 2 and X ¼ fx20 þ 14 ðm Þ g. The system can be viewed as a composite of two harmonic oscillators interacting with each other by the coub int ¼ i hxc ðy oxo x oyo Þ. At very low magnetic field it pling potential V 2 reduces to two separable harmonic oscillators Hðx; x0 ; 0Þ and Hðy; x0 ; 0Þ, respectively oscillating along x- and y-axes having eigenfunctions
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 1 X Un ðx; x0 ; xc ! 0Þ ¼ expðz2 =2ÞHn ðzÞ 2n n! p
ð5Þ
and
z¼
We may expand the wave function w in a Fourier integral over real energy eigenfunctions of H0 , with their time-dependence 0 determined by the exponentials eiE t=h and try to find out the order of magnitude of the energy interval over which the amplitudes of the real energy functions are non-vanishing. Now, the Fourier expansion of the wave function with complex energy Ec is, R
I
eiEc t=h ¼ eCk t=2h eiEk t=h ¼
Xx
ð6Þ
For a non-zero magnetic field strength, an exact solution of the wave equation for doped parabolic 2D superatom is not possible. We have therefore adopted a variational approach for approximately solving the problem by taking,
Wðx; yÞ ¼
X
C n;m /n ðX; xÞ/m ðX; yÞ ¼
ð11Þ
where the amplitude aðEÞ is given by
aðEÞ ¼
Z
1 2ph
1
R
eCk t=2h eiðEk EÞ dt ¼ I
0
X
ck jki
jaðEÞj2 ¼ 4p
2
1 ERk
1.0
where eigenstate jki corresponds to product basis elements f/n ðX; xÞ/m ðX; yÞg. Interaction among these two oscillators can be easily evaluated exactly with the help of standard recurrence relation
0.8
hn; mjV int ðx; y; xc Þjn0 ; m0 i ! rffiffiffiffiffiffi " rffiffiffiffiffiffiffiffiffiffiffiffiffiffi hxc m0 þ 1 m0 dm;m0 þ1 þ ¼i dm;m0 1 2 2 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi n0 n0 þ 1 dn;n0 þ1 dn;n0 1 2 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi n0 þ 1 n0 dn;n0 þ1 þ dn;n0 1 2 2 !# rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi 0 0 m m þ1 dm;m0 þ1 dm;m0 1 2 2
0.4
h
2p
CIk 2
1 i : þ i ERk E
ð12Þ
2
þ
CI2 k
:
ð13Þ
4
þ V p ðx; yÞ 2
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2
1.0
ð8Þ
Norm
0.8
Dynamic Non-dynamic
0.6 0.4 0.2
ð9Þ
CIk
0.0 1.0 Dynamic Non-dynamic
0.8 0.6 0.4 0.2
2
and the corresponding energy eigenfunctions are decaying wave functions. The probability amplitude is then proportional to I R eCk t=2h eiCk t=h . The second factor accounts for the exponential falloff of the amplitude, the first factor being the dynamical phase factor of the amplitude of the initial state which is now damped. The mean decay time (Dt k ) is given by I
0.6
0.0
where V p ðx; yÞ ¼ V 0 ecððxax0 Þ þðyay0 Þ Þ or V 0 ecð ðxax0 Þ þðyay0 Þ Þ with c > 0, V 0 > 0 (electron doping), V 0 < 0 (hole doping) and ðax0 ; ay0 Þ is the centre of impurity. The matrix elements of V p ðx; yÞ may be computed numerically. Let us assume that the amplitude of the impurity potential is I complex V 0 ¼ ðV R0 iV 0 Þ. In that case, H is no longer hermitian. The eigenvalues of H are now generally complex
Ec ¼ Ek ¼ ERk i
Dynamic Non-dynamic
0.2
1 Hp ðx0 ; xc Þ ¼ Hðx; x0 ; xc Þ þ Hðy; x0 ; xc Þ þ V int ðx; y; xc Þ m
1
E
k
In the presence of impurity perturbations, the hamiltonian becomes
0
aðEÞeiEt=h dE;
1
ð7Þ
n;m
Z
þ1
The square of the amplitude, i.e., the probability is
pffiffiffiffi
Dt k ¼
Z
eCk t=h dt ¼
h
CIk
:
ð10Þ
0.0 0
1
2
3
4
5
Time x 10 -6 (a.u.) Fig. 1. A plot of norm versus time of (solid circle (SC)) the ground state with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me , (solid triangle (ST)) the ground state with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me and (solid diamond (SD)) the first excited state with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me .
218
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
Table 1 Half-lives of different states (V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u., m ¼ 1:5me ) in impurity doped dots. Quantum states
j0i j1i j8i
s1=2 for hole doping (V R0 ¼ 0:8 a.u.)
s1=2 for electron doping (V R0 ¼ 0:8 a.u.)
Dynamic (a.u.)
Nondynamic (a.u.)
Dynamic (a.u.)
Nondynamic (a.u.)
5:616 104 3:464 107 6:931 106
5:62 104 3:45 107 6:92 106
5:691 105 7:719 106 9:418 106
5:70 105 7:72 106 9:42 106
Clearly,
1 jaðEÞj2 2 CI as ERk E ! k : 2
jaðEÞj2 !
The energy width is therefore CIk and its reciprocal is linked to the mean decay time. From definition, it is evident that the dimension of the amplitude aðEÞ is (energy)1. Therefore, if we integrate the square of the amplitude over energy we obtain a quantity whose dimension is also (energy)1. But it is also the total area of jaðEÞj2 versus E curve in the interval in which amplitudes of the expansion are non-zero. If we denote this interval as DE, we obtain, by definition
Z þ1 1 ¼ jaðEÞj2 dE DE 1 Z þ1 1 1 dE ) ¼ 2 2 CI2 DE 4p 1 R Ek E þ 4k
ð14Þ ð15Þ
1 1 ¼ DE 2pCIk
Comparing Eqs. (10) and (16), we have,
DEDt ¼ 2ph
ð17Þ
The relation Eq. (17) should be taken to imply the energy of a state existing for a limited time ‘Dt’ is determinate to within the accuracy h=DtÞ. We have therefore two options at our of the order ð2p disposal: (a) we diagonalize the impurity perturbed complex non-hermitian hamiltonian (H) in the basis of eigenstates of H0 , find the imaginary parts of the eigenvalues and from that, the level width (C), the inverse (s) of which gives the decay rate (K C ). We may call it the static way of computing lifetime or decay rate; (b) we solve the time-dependent Schrödinger equation for H numerically in the basis of eigenstates of H0 , by an accurate Adams–Moulton Predictor-corrector algorithm or 6th order Runge–Kutta–Fehlberg method with small enough time steps and numerically compute the rate at which jwj2 decay, yielding directly the decay rate ðdtd jwj2 ¼ K C Þ. We may term it the dynamic way of computing decay rate or the lifetime. The questions that we wish to probe are: (i) if the two ways of computing decay rate (lifetime) produce same results, (ii) if the strength of the transverse magnetic field affects the computed decay rates or lifetimes, and if so, in what ways. (iii) if the location, the type and strength of the impurity potential has significant effects.
ð16Þ Probability Density X 10 10
1.6
c
1.2
(x 10-8 a.u.)
8
0.8 0.4
6
0.0 0 2 4 6 8 10
1.2 0.9 0.6 0.3
4 2
-100 -50
0
8
50 100
0 50 100 X
-100 -50
0 Y
0 45 90 X
-45
0 Y
45
-90
0 -60 -30 Y
30
b
10 8 6 4 2 0 2 4 6 8 10
10 (x 10-7 a.u.)
Half-Life Period
4
2.5 2.0 1.5 1.0 0.5
6 4
-90-45
2 12
6.0
90
-4
(x 10 a.u.)
5.6
10
5.2
a
4.8
0 2 4 6 8 10
8
5 4 3 2 1
6 4
-60 0
2
4
6
8
10
ωc (x 10 a.u.) 4
Fig. 2. A plot of half-life (in a.u.) versus xc with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), x0 ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states.
-30
0 30 60 X
60
Fig. 3. Probability density map of (a) the ground, (b) the first excited and (c) the eighth excited states with xc ¼ 0:000001 a.u., x0 ¼ 0:0005 a.u., m ¼ 1:5me , V R0 ¼ 0:8 a.u., V I0 ¼ 0:0 a.u and impurity centre R = (3, 0).
219
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
3. Results and discussion We have investigated the time-profiles of the norms for the quasi-stable states in the presence of both electron and hole doping impurities in 2D single quantum dots (Fig. 1). The impurity is located at the point (3, 0). Fig. 1 (SC and ST) shows how the norm decays in the metastable ground states of both the hole and electron impurity perturbed quantum dots (V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., xc ¼ 0:0005 a.u., x0 ¼ 0:0005 a.u. and m ¼ 1:5me Þ and Fig. 1 (SD) clearly highlights the decay phenomenon of the first excited state in the case of a electron impurity perturbed dot. The half-lives estimated from the exponential decay of the norms turn out to be 5:616 104 a.u., 3:464 107 a.u. and 6:93 106 a.u. of time for the ground, first, and eighth excited states, respectively. On the other hand, the imaginary part of the corresponding complex eigenvalue (obtained by diagonalizing H in 36-dimensional basis) yields the half-lives of 5:62 104 a.u., 3:47 107 a.u. and 6:94 106 a.u. of time. The agreement between the two sets is quite close. Table 1 reports the dynamically and non-dynamically computed half-lives (s1=2 ) of several states of the same system. The agreement between the two sets of values is satisfactory. Since half-lives of the first and the eighth excited states in hole-doped single dots are rather high, the time scale of final decay becomes very large and we refrain from presenting the corresponding figures. Fig. 2 displays how the computed half-lives of hole-doped states vary with the cyclotron frequency or the strength of the transverse magnetic field. For the ground state, half-life is high when xc is small and it registers a moderate decrease as xc increases for the hole-doped impurity with the amplitudes of the real and the imag-
inary parts of the impurity potential V R0 ¼ 0:8 a.u. and V I0 ¼ 0:004 a.u. (Fig. 2 (SC)). The half-life of the first excited state mildly oscillates with increase in xc (Fig. 2 (ST)). The half-life of the eighth excited state decreases to a rather low value at xc ¼ 4:0 1044 a.u., then rises abruptly around xc ¼ ½6:0 104 7:0 10 a.u. (Fig. 2 (SD)) and saturates thereafter. The unperturbed dot states have infinite life-times either at low or at high cyclotron frequencies. At low xc , the ground state is described by a rather diffuse wave function which overlaps and interacts only weakly with the metastable hole impurity states. The weak mixing causes significant decrease in the life-time relative to the impurity free ground state. At higher xc , the ground state wave function is more compact around the centre of the potential and interacts more strongly with the metastable impurity states. The density plots of the relevant states of the impurity free dot are shown in Fig. 3a–c at low xc (=0.000001 a.u.). The impurity perturbed states are generally seen to have much lower life-times than the corresponding free states. The density plots (Fig. 3a–c) show that the impurity free excited states have considerable accumulation of electron density in the neighborhood of the impurity centre (3, 0). The mixing with the metastable impurity states is thus more pronounced in the excited states causing decrease in the life-times. The variation of s1=2 with xc reflects the changes in the electron density distribution in the free dot as xc varies. The behaviour of the ‘half-life period versus xc ’ plot of the ground state in an electron doped system with the amplitude of V R0 ¼ 0:8 a.u. and V I0 ¼ 0:004 a.u. (Fig. 4a (SC)) is quite different from what was observed in the previous case (Fig. 2 (SC)). Thus for a 2D single quantum dot with complex impurity potential, hole doping progressively destabilizes the ground state as xc increases although the extent of destabilization is small. The repulsive elec-
1.6
-7
(x 10 a.u.)
(x 10-8 a.u.)
3 1.2 0.8
1
0.4
0 0
2
4
6
8
10
6
0.6
Half-Life Period
-9
(x 10 a.u.)
0.8
0.4 0.2 0.0 1.0
4
2
0 1.2
0.8 (x 10-6 a.u.)
-7
(x 10 a.u.)
Half-Life Period
1.0
(x 10-7 a.u.)
0.0 1.2
2
0.6 0.4 0.2 0.0
0.9 0.6 0.3 0.0
0
2
4
6
ωc(x 10 a.u.) 4
8
10
-0.8
-0.4
0.0
0.4
0.8
R
Amplitude ( V0 a.u. )
Fig. 4. (a) A plot of half-life (in a.u.) versus xc with V R0 ¼ 0:8 a.u., V I0 ¼ 0:004 a.u., impurity centre R = (3,0), x0 ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states. (b) A plot of half-life (in a.u.) versus V R0 with V I0 ¼ 0:008jV R0 j a.u., impurity centre R = (3, 0), x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states.
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
tron doping potential scatters the carrier electron, reducing its lifetime which remains virtually constant over a range of xc . For much larger xc , the carrier electron remains strongly localized around the centre of the confinement potential, reducing the scattering induced effect and the life-time increases. For the first excited state, the half-lives computed by the diagonalization method nicely agree with the respective dynamically computed values. The excited state life-times are generally two orders of magnitudes lower than the life-times of the metastable ground state. The density plot for the first excited state of the impurity free dot shows (Fig. 3b) build up of carrier electron density near the impurity centre which is strongly scattered by the repulsive impurity potential leading to a general decrease in life-time. When xc increases substantially the carrier electron density is drawn closer to the centre of the potential, reducing the effects of scattering from the repulsive metastable impurity potentials. These values decrease initially as xc increases and finally saturate (Fig. 4a (ST)). For the eighth excited state, the plot of ‘half-life versus xc ’ reveals oscillations which get damped reducing s1=2 to near zero value (Fig. 4a (SD)) as xc increases. This feature can be similarly explained from the nature of variations in the density distribution in the unperturbed eighth excited state of the dot as xc changes. The details will be presented elsewhere. When the magnitude of the real part of the (repulsive) electron impurity potential is gradually enhanced, with a proportional increase the imaginary part, the computed half-life of the ground state passes through a clear minimum at V R0 ¼ 0:45 a.u.; but for the hole-doped system it diminishes to an almost vanishing value making the ground state appear very much like a transient state (Fig. 4b (SC)). The minimum appears presumably due to a delicate balance between the effects of repulsive scatterer at (3, 0) (tending
to push the carrier electron towards the centre of the confining potential) and the confinement potential which also tends to draw the carrier electron to itself. There is a clear discontinuity in the computed half-lives at V R0 . Since in this case V R0 ¼ 0 implies V I0 ¼ 0, the life-time of any state for V R0 ¼ 0 is infinite explaining the discontinuity in the ‘s1=2 versus V R0 ’ plots. The computed halflife (s1=2 ) remains almost constant for the first excited state when V R0 < 0. For V R > 0, s1=2 decreases with increasing amplitude of the real part of the perturbation potential (Fig. 4b (ST)) and there is a discontinuity at V R ¼ 0. In the case of the eighth excited state (chosen randomly) the pattern of the ‘s1=2 versus V R0 ’ plot is similar to what was found for the ground state and s1=2 reaches a minimum at V R0 = 0.42 a.u. (Fig. 4b (SD)). One can explain the observed features using arguments already provided for the ground state. If the imaginary part of the impurity potential is held fixed at a small value, V I0 ¼ 0:004 a.u., s1=2 -V R0 profiles of the quasi-stable states in a 2D single dot do not reveal any discontinuous jump near V R0 ¼ 0 (Fig. 5) as expected since V I0 –0 even when V R0 ¼ 0, so that the life-time remains finite even at V R0 ¼ 0. The life-time of the
a
6 5
(x 10-7 a.u.)
220
4 3 2 1 0
1.4
-7
(x 10 a.u.)
1.3 1.2 1.1 1.0
(x 10-8 a.u.)
Half-Life Period
1.6
0.0 5.85
0.8
(x 10 a.u.)
5.70 5.65 5.60 5.55
0
2
4
6
8
10
Position of Impurity ( Rx a.u.)
b
-5
6 5 (x 10 a.u.)
5.75
-4
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
5.80
Half-Life Period (x 10 a.u.)
(x 10-7 a.u.)
Half-Life Period
0.7
-5
0.8 0.4
0.9
4 3 2 1 0
1.2
-0.8
-0.4
0.0
0.4
0.8
R
Amplitude ( V0 a.u.) Fig. 5. A plot of half-life (in a.u.) versus V R0 when V I0 ¼ 0:004 a.u., impurity centre R = (3, 0), x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me of (SC) the ground, (ST) the first excited and (SD) the eighth excited states.
8 7 6 5 0
2
4
6
8
10
Position of Impurity ( Rx a.u.) Fig. 6. (a) A plot of half-life versus Rx with V I0 ¼ 0:004 a.u., x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me at (SC) V R0 ¼ 0:8 a.u. for the ground, at (ST) V R0 ¼ 0:8 a.u. for the first excited and (SD) V R0 ¼ 0:8 a.u. for the eighth excited states. (b) A plot of half-life versus Rx with V I0 ¼ 0:004 a.u., x0 ¼ 0:0005 a.u., xc ¼ 0:0005 a.u. and m ¼ 1:5me at V R0 ¼ 0:8 a.u. for the ground state.
R.K. Hazra et al. / Chemical Physics Letters 468 (2009) 216–221
ground state increases with increase in the real part of the impurity potential. The repulsive scatterer forces the carrier electron density towards the centre of the confinement potential and the state progressively becomes similar to the unperturbed ground state. The attractive scatterer, on the other hand progressively draws the carrier electron to itself as jV R0 j increases causing a drop in the life-time. ‘s versus xc ’ plot for the first excited state shows a reverse trend. Half-life of the ‘eighth excited’ state appears to increase with V R0 , irrespective of its sign. It reaches a minimum value for an electron dopant strength of V R0 ¼ 0:21 a.u. The behaviour can be rationalized from the density plots of the concerned states of the unperturbed dot. The position of the impurity centre also has an impact on the life-span of the quasi-stable dot states. In the presence of a hole doping impurity half-life of the ground state increases as impurity centre moves away from the origin along x-axis (Fig. 6a (SC)). In the first and eighth excited states, however, it decreases to a steady value beyond Rx ¼ 7 a.u. Fig. 6a (ST and SD). The further the impurity centre from the origin, the lesser is its interaction with the carrier electron, and lesser is the metastability of the impurity perturbed dot states. As the electron doped impurity centre moves further and further away from the origin, it stabilizes the ground state as expected (Fig. 6b). The life-time changes therefore directly mirror fine changes in the wave functions of the carrier electron in the presence of metastable impurity centres of either an attractive or a repulsive type. It would be nice to have experimental data for further advancement in the modelling. 4. Conclusion The presence of a complex impurity potential makes all the states of a 2D quantum dot metastable with half-lives that depend on the nature (attractive or repulsive) and location of the centre of
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the impurity and the relative magnitudes of the real and the imaginary parts of the potential. The strength of the transverse magnetic field significantly modulates the lifetimes or decay rates which essentially reflect the changes in the electron density distribution of the unperturbed dot states, which mix with the metastable impurity states. Acknowledgements Thanks are due to the CSIR, Government of India, New Delhi for the award of Senior Research Fellowships to RKH and MG.
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