Author's Accepted Manuscript
Stark effect of optical properties of excitons in a quantum nanorod with parabolic confinement S.K. Lyo
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S0022-2313(13)00553-X http://dx.doi.org/10.1016/j.jlumin.2013.08.064 LUMIN12143
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Journal of Luminescence
Received date: 14 June 2013 Revised date: 21 August 2013 Accepted date: 27 August 2013 Cite this article as: S.K. Lyo, Stark effect of optical properties of excitons in a quantum nanorod with parabolic confinement, Journal of Luminescence, http: //dx.doi.org/10.1016/j.jlumin.2013.08.064 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Stark Effect of Optical Properties of Excitons in a Quantum Nanorod with Parabolic Confinement S. K. Lyo Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA (Dated: August 30, 2013)
Abstract We study the quantum Stark effect of optical properties of a quasi-one-dimensional quantum rod with parabolic confinement. Interplays between the competing/cooperative forces from confinement, electron-hole (e-h) attraction, and an external field are examined by studying the binding energy, the oscillator strength, and the root-mean -square (RMS) average of the e-h separation in a nonlinear electric field. In a long rod with weak confinement, the e-h interaction dominates over the confinement effect, yielding an abrupt drop of the exciton binding energy, oscillator strength, and a sudden increase of the RMS average e-h separation as the excitons are dissociated at the threshold field as the field increases. The exciton-dissociation transition is gradual in a short rod, where the confinement force dominates over the e-h attraction. We show that a DC field can induce an optically active excited exciton state in a narrow field range, causing a sharp peak in the oscillator strength and a dip in the RMS average of the e-h separation as the field increases. The Stark effects are also investigated as a function of the linear confinement length (i.e., rod length) at fixed fields. Phone number: 949-800-7101 Fax: 949-824-1159 email:
[email protected]
1
I.
INTRODUCTION
Quantum nanostructures have received increasing attention recently due to their many useful optical and electronic properties that can be controlled for device applications as well as for their novel physical properties. It is now possible to fabricate a nanorod of a desired length along the axial (i.e. x) direction with a narrow confinement radius in perpendicular direction.1–4 Quantum nanorods are expected to play an important role for modern tunable solid-state-lighting devices, where the wave length and the oscillator strength are determined by their size, intrinsic electron-hole (e-h) interaction, and an applied DC field. In quantum wells (QWs), the confinement potential along the x direction can be made parabolic by using a graded composition of a large bandgap material in the medium of a smaller gap QW material.5,6 In this paper, we study the Stark effect of optical properties of quantum rods with parabolic confinement by examining the competing/cooperative role of the well confinement, e-h attraction, and the external field in determining the wave length and the oscillator strength of nanorods. Electronic, excitonic, and optical properties of QWs and quantum dots with parabolic confinement have been studied in the past7–16 owing to the fact that some many-body effects can be solved exactly in this model. Some interesting exact results were obtained for magneto-optical7 and electronic properties11 in two-dimensional QWs, magneto-optical properties of two-dimensional quantum dots8 for the electrons in the conduction band, and for excitons in two dimensional quantum dots.10 The effect of an electric field on excitonic optical properties on three-dimensional quantum dots,13,14,16 two-dimensional quantum disks,15 and one-dimensional quantum rod4 has been investigated in the past by employing approximate methods such as a variational theory,13,15,16 a configurational interaction approach,14 as well as a self-consistent Schr¨odinger-Poisson numerical simulation.4 An exact solution for the exciton wave functions in a confined space is difficult to obtain, because this is basically a three-body problem, where the electron and hole interact with their confinement potentials while interacting each other through Coulomb attraction. We examine the interplay between the confinement effect, e-h attraction, and an external field in a quasi-one-dimensional quantum rod with parabolic confinements in the conduction and valence bands by studying the binding energy, oscillator strength, and root-mean-square (RMS) average of the e-h separation as a function of the rod length and the field strength at
2
zero temperature. An exact solution is possible for the special case of parabolic confinement potentials where the interaction-free level separations in the harmonic energy ladders of the conduction (¯hωe ) and valence (¯hωh ) bands are equal (i.e, ωe = ωh ) in the absence of the DC field.10 We present a solution for the energies and the wave functions in a parabolic quantum rod for the case ωe = ωh as well as for a general more realistic case where the energy-level separation is not equal (i.e., ωe 6= ωh ). We find that the physical properties for the general case is qualitatively the same as those of the special case.
II.
BASIC FORMALISM A.
Exciton wave functions and energies
Consider an electron and a hole in a quasi-one-dimensional (1D) parabolic potential in an external field Fex along the rod (i.e., in the x direction). The Hamiltonian is given by
where
˜ =H ˜e + H ˜ h + V˜eh , H
(1)
¯ 2 d2 1 ˜σ = − h + V˜σ , V˜σ = mσ ωσ2 x˜2σ + esσ F x˜σ , H 2 2mσ d˜ xσ 2
(2)
e2 V˜eh = − √ 2 , x˜ = x˜e − x˜h . κ x˜ + ρ˜2
(3)
and
Here, mσ is the average effective mass of the electron (σ = e) and the hole (σ = h), se = 1, sh = −1, κ is the bulk dielectric constant, F = Fex /κ is the internal field, x˜σ is the coordinate along the rod, and ρ˜ = [(˜ ye − y˜h )2 + (˜ ze − z˜h )2 ]1/2 , where y˜σ , z˜σ are perpendicular coordinates. Hereafter, we express the energy in units of the three-dimensional (3D) Bohr energy εB = h ¯ 2 /(2µa2B ) and the length in units of the Bohr radius aB = κ¯h2 /µe2 , where µ is the reduced ˜ B and x = x˜/aB , xσ = x˜σ /aB mass. The dimensionless quantities are defined as H = H/ε etc. by removing the tilde symbols. The Hamiltonian is then rewritten as H = He + Hh + Heh , He = −αh
νe2 2 d2 + x + Fxe , dx2e αh `4 e 3
(4) (5)
Hh = −αe
νh2 2 d2 + x − Fxh , dx2h αe `4 h
(6)
2 , x2 + ρ 2
(7)
Veh = − √
where ασ = mσ /M, M = me + mh , the reduced field F is defined as F= and
s
˜ B , `˜ = ` = `/a
F εB , , F∗ = ∗ F eaB
(8)
√ ωσ h ¯ , ω = ωe ωh , νσ = . µω ω
(9)
It is convenient to rewrite the Hamiltonian in terms of the relative coordinate x and the center-of-mass (CM) coordinate X = αe xe + αh xh : H = Hx + HX + HxX ,
(10)
where d2 1 + 4 (αh νe2 + αe νh2 )x2 + Fx + Veh , 2 dx ` µ d2 M HX = − + 4 (αe νe2 + αh νh2 )X 2 , 2 M dX µ` 2 HxX = γxX, γ = 4 (νe2 − νh2 ). `
Hx = −
(11) (12) (13)
The eigenfunctions φm (X) and eigenvalues εcm m of the CM Hamiltonian is given by HX φm (X) = εcm m φm (X), where
√ φm (X) = q
λ
√ exp(− 2m m! π
(λX)2 )Hm (λX) 2
(14)
(15)
and εcm m =
(αe νe2 + αh νh2 )1/2 (2m + 1), m = 0, 1, · · · , `2
(16)
and λ=
(αe νe2 + αh νh2 )1/4 . √ αe αh `
(17)
For a thin quantum rod, only the ground state is relevant in the perpendicular direction at low temperatures. The level separations in the lateral direction is very large for a narrow quantum rod. We set the origin of the energy at the ground sublevel for convenience, 4
defining ”energy” to be related only to the degree of freedom along the rod, unless otherwise mentioned. We assume the ground state in the lateral direction is given by ψ⊥ = C exp(−
2 2 r⊥e + r⊥h ), 2`2⊥
relevant to a parabolic confinement, where r⊥e = (ye , ze ), r⊥h = (yh , zh ), and C, `⊥ are constants. Defining ρ = r⊥e − r⊥h , R⊥ = r⊥e + r⊥h , we find that the probability distribution for ρ is given by P(ρ) =
ρ2 1 exp(− ). π`2⊥ `2⊥
(18)
We replace the quantity Veh in Eq. (7) with its average over this distribution, obtaining √ 2 π Veh = − exp[(x/`⊥ )2 ]efrc(x/`⊥ ), (19) `⊥ where `⊥ is the effective dimensionless radius of the rod (assumed to be much smaller than `) and efrc(x) is the complementary error function. The quantity Veh reduces to Veh = −2/|x| √ for x `⊥ as expected and to Veh = −2 π/`⊥ in the opposite limit. The eigenfunctions and eigenvalues of Hx are given by Hx ψk (x) = εk ψk (x)
(20)
and can be calculated to a desired accuracy. For the special case ωe = ωh , the coupling term in Eq. (13) vanishes (i.e., HxX = 0) and the total wave function is simply given by Ψmk (x, X) = φm (X)ψk (x)
(21)
with the energy given by E = εk + εcm m . In this case, the field-dependent properties of the wave functions (e.g., oscillator strengths) and the energies are contained in ψk (x) and εk determined from Eq. (20). In general, we expect ωe 6= ωh . In this case, the total wave function is expanded as HΨ(x, X) = EΨ(x, X), Ψ(x, X) =
X
Amk Ψmk (x, X),
(22)
k,m
yielding X
Ak0 ,m0 [δk0 ,k δm0 ,m (εk + εcm m − E) + γ < ψk |x|ψk0 >< φm |X|φm0 >] = 0.
k0 ,m0
5
(23)
The wave function Ψ is given by the coefficients Amk , which can be represented by a column vector of dimension Nψ Nφ : A = col(A0 , A1 , · · · , ANφ −1 ),
(24)
where Am is a subcolumn vector of dimension Nψ : Am = col(A1,m , A2,m , · · · , ANψ ,m ).
(25)
The wave equation in Eq. (23) is then rewritten as HA = EA.
(26)
The Hamiltonian H is a tri-diagonal array of Nψ × Nψ symmetric submatrices Hm,m and 0
Hm±1,m , while O is a null matrix otherwise. Here, the submatrix Hm,m consists of the elements: m,m cm Hk,k 0 = δk,k 0 εk + εm ,
(27)
and m> ,m< Hk,k 0
=
m< ,m> Hk,k 0
γ = λ
m> δm> ,m< +1 < ψk |x|ψk0 >, 2
r
(28)
where m> (m< ) is the greater (lesser) of the two indeces (i.e., m> > m< ). Other elements 0
m ,m Hk,k which do not satisfy m0 = m ± 1 or m = m0 vanish. In the limit of γ = 0, Eq. (26) 0
yields Eq. (21). Namely, Amk = 1 for only one set of m = mp , k = kp for an exciton state p and Amk = 0 for all other m and k.
B.
Oscillator strengths and second moments
Application of a strong longitudinal DC field induces dissociation of an exciton and yields a striking effect on the oscillator strength and the second moment of the e-h separation. The oscillator strength is given, except for a proportionality constant, by the quantity10,17 f≡
Z ∞
Ψ(x =
−∞
2 0, X)dX
=
X Z ∞ A ψ (0) k,m k
−∞
k,m
2 φm (X)dX ,
(29)
yielding, in view of Eq. (17), √
f = QLf0 , Q =
6
παe αh , (αe νe2 + αh νh2 )1/4
(30)
where L = 2` arises from the CM motion and f0 ≡
X ∞ X ψ (0) k k
q
Ak,2m
m=0
(2m)! 2
2m m!
(31)
represents the contribution from the e-h relative motion only. The above mentioned proportionality constant for the oscillator strength depends on the material parameters such as the interband dipole moment and optical energy gap but is independent of the field and the exciton wave function.10,17 Therefore, we will simply define f as the oscillator strength for convenience in this paper. For the special case ωe = ωh (i.e., γ = 0), we find Q =
√
παe αh , while the relative and
CM degrees of freedom are decoupled, yielding f=
√
παe αh Lf0 : f0 = |ψk (0)|2
(2m)! . (2m m!)2
(32)
Here, k = 1, m = 0 for the ground state. The ratio of f and f0 depends on the mass of me and mh . The RMS average of the e-h separation xeh is given in terms of the second moment < x2 > by
√ xeh =
< x2 > : < x2 >=
X
Ak,m Ak0 ,m < ψk |x2 |ψk0 > .
(33)
k,k0 ,m
In the limit of γ = 0 (i.e., ωe = ωh ), the second moment becomes independent of the CM function and depends only on the given eigenstate (e.g., ψk ) of the relative coordinate and the above equation reduces to < x2 >γ=0 =< ψk |x2 |ψk > .
C.
(34)
Analytic result for noninteracting e-h pair
In a strong field, the electron and the hole are separated far apart with negligible Coulomb interaction. In the absence of e-h interaction, the oscillator strength, the energy, and the second moment can be calculated analytically, providing a useful check for the accuracy of the numerical results for the energy, oscillator strength and the second moment calculated from Eqs. (23), (30) and (33) in the asymptotic limit of a strong field. In this case, the electron and hole wave functions are calculated from Eq. (2): F ψσ,n (xσ ) = ψσ,nσ (xσ + sσ xF σ ), n = 0, 1, 2, . . . σ
7
(35)
where
2
2
e−(xσ /2`σ ) ψσ,nσ (xσ ) = q √ Hnσ (xσ /`σ ) 2nσ nσ !`σ π
(36)
is the harmonic wave function, xF σ is the field-induced displacement of the centroid of the electron (σ = e) and the hole (σ = h) wave function xF e =
Fαh `4 Fαe `4 , , x = F h 2νe2 2νh2
(37)
and s
s
1 q αh αe `σ = , `h = ` . h ¯ /mσ ωσ : `e = ` aB νe νh
(38)
The energy is given by E=
1 F`2 2 ) [αe νe2 + αh νh2 ]. [ν (2n + 1) + ν (2n + 1)] − ( e e h h `2 2
(39)
The exciton wave function is the product of the electron (ψene ) and the hole (ψhnh ) wave functions, yielding for the oscillator strength in Eq. (29) f∞ = |Seh |2 : Seh =
Z ∞ −∞
ψene (x)ψhnh (x)dx2 ,
(40)
where the subscript ∞ signifies a noninteracting e-h pair at a high-field limit. For the transition between the ground states, Seh equals Seh
Z ∞ 1 1 (x + xF e )2 (x − xF h )2 ≡√ + exp[− ( )]dx, 2 `2e `2h `e `h π −∞
(41)
√ 2 αe αh x2F eh = exp[− ], αh νh + αe νe [αh νh + αe νe ]`2k
(42)
yielding f∞ where
xF eh = xF e + xF h =
F`4 (αe νe2 + αh νh2 ). 2
(43)
. The second moment of the e-h separation gives a useful indication of e-h binding. It is defined by < x2 >∞ =
Z
2 2 ψe,n (xe + xF e )x2 ψh,n (xh − xF h )dxe dxh , e h
(44)
where x = xe − xh , yielding for ne = nh = 0 1 < x2 >∞ = (αh νh + αe νe )`2 + x2F eh . 2 8
(45)
III.
NUMERICAL STUDIES AND DISCUSSIONS
In this section, we examine the field dependence of the various quantities such as the energy level, the oscillator strength, and the second moment studied in the previous section. Since universal dimensionless units are employed for these quantities, viz. the bulk exciton binding energy for the energy, the Bohr radius for the length, and F ∗ for the field, we show the estimates of these quantities in Table I along with other parameters for GaAs and CdSe structures. We will study the simple special exact case ωe = ωh first and then investigate general cases ωe 6= ωh . The latter more general situation affects only the field-dependent properties quantitavely, while the basic physical properties of the problem is essentially contained in the special case. In the first step of the numerical evaluation, the eigenfunctions ψk (x) and the eigenvalues εk of Hx in Eq. (20) are calculated accurately by choosing a domain of x extending sufficiently beyond the classical turning points for the energies of interest, subdividing the domain into sufficiently small sections, and adopting a standard five-point difference equation approach.18 The center of the chosen x-domain was shifted by ∆x = −F`4 /[2(αh νe2 + αe νh2 )] according to Eq. (11) for convenience as the field is increased.
A.
Special case ωe = ωh :
We study this case in detail first, because the result to be discussed here is universal in the sense that it is independent of specific material parameters. In this case, the exciton wave function is given by Eqs. (15), (20), (21) and the energy by cm Ek,m = εk + εcm m , εm =
1 (2m + 1), m = 0, 1, · · · . `2
(46)
Since the CM contribution to the energy εcm m is independent of the field, we study only the field dependence of εk . The ground-state energy εk=1 (black solid curves) and the lowest excited level εk=2 (blue dashed-dotted curves) are displayed in Fig. 1(a) as a function of the reduced field F for ` = 1 and the wire radius `⊥ = 0.1. The ground-state energy εk=1 is also displayed for a longer quantum rod ` = 2 for the same radius `⊥ = 0.1 in red dotted curves. The thick bold curves are obtained with e-h interaction turned on, while the thin curves are with the interaction turned off. The energy differences between the thick curves and the concomitant thin curves are the binding energies. The ground-state exciton binding 9
is clearly seen from the nearly flat thick black solid and red dotted curves for εk=1 below the critical fields F = 6.2 and F = 1.5, respectively. Beyond these fields, they join the thin interaction-free curves, signaling that the exciton becomes dissociated. For the red thick dotted curve for a longer rod with ` = 2, the exciton dissociation is relatively sharp as seen from the rapid rise of the curve near F = 1.5 to join the thin interaction-free dotted curve. On the other hand, the black thick solid curve for shorter ` = 1 joins the thin interaction-free solid curve very gradually. Here, we note that the scale of the field is much smaller for the longer rod, which has larger potential-energy drop over the length of the rod. We also found that exciton binding becomes stronger for a narrower rod radius `⊥ as expected. Note that the e-h pair is not bound for k = 2 in most of the field range shown in Fig.1 except in the range 6.4 < F < 6.8 in the sense discussed for k = 1 in view of the absence of a flat plateau for the dashed-dotted curves outside this region: Here, the role of e-h attraction is simply to lower the energy. However, the k = 2 exciton state is bound in the above cited narrow range 6.4 < F < 6.8 where the energy curve becomes flat as a function of the field, while the oscillator strength peaks sharply. The physical origin of this interesting field-induced exciton binding will be discussed later in Fig. 2. The field dependence of the oscillator strength f = QLf0 for the state k is given by Eq. (32). While the quantity Lf0 is universal, being independent of the parameters of the √ material such as effectives masses, the quantity Q = παe αh depends on the electron and hole effective masses. Table I yields QCdSe ' 0.709 for the numerical purpose and QGaAs ' 0.6. Figure 1(b) displays the quantity f = QCdSe Lf0 = QCdSe L|ψk (0)|2 for the ground CM level m = 0 as a function of the reduced field F for the ground level k = 1 (black solid curves) and the lowest excited level k = 2 (blue dashed-dotted curves) for ` = 1 and `⊥ = 0.1. The quantity f is also displayed for the ground state k = 1 for a longer quantum rod ` = 2 for `⊥ = 0.1 in red dotted curves. The thick bold curves are obtained with e-h interaction turned on, while the thin curves are with the interaction turned off. It is clearly seen again that the thick red dotted curve for a longer rod with ` = 2 drops more abruptly beyond the critical field of exciton dissociation than the thick black solid curve before joining the thin interaction-free curves as discussed for Fig. 1(a). It is also seen that the enhancement of the oscillator strength due to exciton binding is much larger for the longer rod with ` = 2 than that of the shorter rod with ` = 1. Again, the scale of the field is much smaller for the longer rod with ` = 2. 10
The oscillator strength for the p-like excited level k = 2 in Fig. 1 (b) vanishes at zero field due to the odd parity. The DC field breaks the inversion symmetry and increases the oscillator strength gradually for k = 2. However, the main role of the field is not merely to break the inversion symmetry but to induce a bound state as will be discussed in the following. The rate of the increase of the oscillator strength is slowed considerably by e-h attraction as seen in Fig. 1(b) upon comparing the thin and the thick dashed-dotted curves. In the presence of e-h attraction, the oscillator strength grows to a sharp peak at around F = 6.6 where k = 2 forms an s-like bound state as shown by the thick blue dashed-dotted curve. This kind of novel field-induced otherwise forbidden optical transition was apparently observed previously from a single InAs/InP quantum dot under a lateral electric field.14 In order to explain the physical origin of the interesting field-induced bound state, we plot in Fig. 2 the wave functions for k = 1 (black solid curve), k = 2 (blue dashed curve) on the left axis and the total potential energy profile in a red dotted curve on the right axis as a function of the coordinate x for three reduced fields: (a) F = 0, (b) F = 6, and (c) F = 6.6. It is seen there that the center of the background parabolic potential is shifted, with increasing field, gradually to the left of the center of the Coulomb potential at x = 0, dragging the main part of the k = 2 wave function with it and transferring the amplitude from the right to the left lobe of the p-function, while the s-like k = 1 ground state function is pinned (i.e., bound) up to F = 6 at x = 0. Beyond F = 6, the ground-state wave function slides rapidly to the center of the parabolic potential before the field reaches F = 6.6, where the right lobe of the p-wave function is shifted to the sharp potential-energy minimum at x = 0 and a bound state is formed. The formation of this bound state arises by back-transferring the amplitude from the left to the right lobe of the severely distorted p-like k = 2 wave function until it deforms to an s-like wave function pinned at x = 0 through e-h attraction. Beyond F = 6.6, the centroid of the k = 2 state slides rapidly to the center (i.e., energy minimum) of the background parabolic potential just like in (a) except that the sharp minimum of the Coulomb potential is shifted to the far right away from the center of the parabolic potential, explaining the abrupt drop of the k = 2 oscillator strength above F = 7 in Fig. 1(b). We showed in Fig. 1 that a confined exciton is dissociated by a strong DC field. However, the confinement force tries to keep the electron and the hole together, preventing e-h separation and exciton dissociation. The opposing competition between the DC field and the confinement force is clearly demonstrated in Fig. 3, where the ground level energy εk=1 11
(black curves, left axis) and the oscillator strength f (red curves for QCdSe =0.709, right axis) are shown for `⊥ = 0.1 as a function of the reduced field F for several values of ` = 3 (dashed curves), ` = 2.3 (solid curves) and ` = 2 (dotted curves). For each set of the curves, the thick (thin) curves are with (without) e-h Coulomb interaction. The thick curves, for the energy and the oscillator strength, clearly show that a stronger confinement (i.e., shorter `) requires a stronger DC field for the exciton dissociation, which occurs, from the left to the right, near F = 0.43 (dashed curves), F = 0.96 (solid curves) and F = 1.43 (dotted curves) in the order of decreasing `. It is seen that the exciton dissociates at these fields, resulting in an abrupt drop of the oscillator strength and a sudden rise of the level energy to that of a noninteracting e-h pair. Note that the thick as well as thin energy curves for three different values of ` converge together nearly to one point at zero field because the ground-level energy reaches a steady plateau near large ` > 2 as shown in Fig. 4 (a): In this weak-confinement region, an exciton is formed, free from the confining walls and a further increase of ` has no effect on the ground-level energy. For a much stronger confinement with smaller ` = 0.5, however, the energy levels are much higher as shown by a solid (open) triangle for interacting (noninteracting) e-h pair on the left vertical axis at zero field: The energy level for the interacting case is nearly flat in this case as a function of the reduced field in the range shown. An electron and a hole is forced into proximity by both 1) binding via Coulomb attraction and 2) confinements inside a well. We expect that the confinement effect will be dominant in a narrow QW, while Coulomb interaction will prevail in a wide well. In order to demonstrate the boundary between the dominance of the confinement effect and that of e-h interaction, we study, in Fig. 4, the dependence of (a) the energy in the direction of the rod and (b) the oscillator strength on the rod length (defined as L ≡ 2`) for F = 0. In (a), the ground-level energy εk=1 decreases gradually to zero as a function of the rod length in the absence of e-h interaction (thin dashed-dotted curve) as expected. In the presence of e-h interaction, however, a confinement-free exciton is formed roughly beyond the rod length L > 2. As a result, the energy saturates to a plateau corresponding to the the negative exciton-binding energies εk=1 = −9.9 for `⊥ = 0.1 (thick solid curve) and εk=1 = −14.8 for `⊥ = 0.05 (thick dashed curve), indicating that the binding energy for the narrower rod with `⊥ = 0.05 is larger than that of the wider rod with `⊥ = 0.1 due to a closer e-h proximity in the perpendicular direction. It is to be remarked here that, for `⊥ = 0.05 in (a), a large constant 12
lateral energy shift ε⊥ = 600 is to be added to the dashed as well as to the dashed-dotted curve for the total energy. The energy shift ε⊥ is relative to the ground level of the rod with `⊥ = 0.1 in the lateral direction. The dominance of the confinement effect in a short rod is clearly seen in (b), where the oscillator strength f is displayed for a CdSe rod without e-h interaction (thin dashed-dotted curve at the bottom) and with e-h interaction for `⊥ = 0.1 (thick solid curve) and for `⊥ = 0.05 (thick dashed curve). For a GaAs rod, f should simply be multiplied by a factor QGaAs /QCdSe ' 0.85. Note that the oscillator strengths for the three curves converge to one point near small `, because e-h interaction plays a subsidiary role in this strong-confinement regime. Here, the oscillator strength without e-h interaction is independent of the rod length, while it increases with the length when e-h interaction is present. Again, the oscillator strength is larger for the narrower rod (thick dashed curve) than the wider rod (thick solid curve) due to a tighter binding for the former. We mention here that the contribution f0 from the e-h relative motion in Eq. (32) behaves as f0 ∝ 1/L in the absence of e-h interaction, yielding the dashed-dotted curve for f in (b), while it saturates to a constant value for L 1 in the presence of e-h interaction like the energy in (a), meaning that the solid and dashed curves in (b) grow linearly with L for large L. Figure 5 displays (a) the ground-level energy (black curves, left axis) and (b) the oscillator strength (red curves, right axis) of an e-h pair in a CdSe rod for `⊥ = 0.1 as a function of the rod length L for several reduced fields F = 0.43 (dashed curves), F = 1 (solid curves), and F = 1.43 (dotted curves). Again, f should be multiplied by a factor QGaAs /QCdSe ' 0.85 for a GaAs rod. The thick (thin) curves are with e-h interaction turned on (off). The abrupt rise of the energy and the concomitant sudden drop of f from the thick curves to the thin curves indicate field-induced exciton dissociation. It is seen there that, at a low field F = 0.43, exciton dissociation does not occur until the rod length reaches L = 6.04 (i.e., ` = 3.02) corresponding to a weak confinement. For an intermediate field F = 1, however, the exciton dissociates at L = 4.54 (i.e., ` = 2.27) corresponding to an intermediate confinement, while, for a strong field F = 1.43, the exciton dissociates at a shorter rod length L = 4 (i.e., ` = 2) in the stronger confinement regime. It is also seen in Fig. 5 that the three thin interaction-free black curves for the energy merge together in a short-length (strongconfinement) regime below L ' 1.2 as the rod length decreases, indicating that the effect of the DC field is negligible in the strong confinement regime. A similar behavior is also noted for the three thin interaction-free red curves for f . In contrast, the field-induced dispersion 13
of the thick curves for the energy and f occurs at much longer rod lengths in the presence of e-h binding, showing that the effect of the field is much smaller in the bound state. The black solid (open) circle indicates the energy level with (without) e-h interaction for a very strong field F = 7 at a rod length L = 2. The oscillator strengths are negligibly small in this case with or without e-h interaction. Another measure of exciton binding is given by the RMS average of the e-h separation. This quantity is displayed in Fig. 6 for the ground level for ` = 1 (solid curves), ` = 1.5 (dashed curves), ` = 2 (dashed-double-dotted curves) and for the k = 2 level for ` = 1 (dashed-dotted curves) as a function of the reduced field in the presence (thick curves) and absence (thin curves) of e-h interaction. The relationships between ` and F at the exciton dissociation points in this figure are consistent with those discussed earlier. The dip in the thick dashed-dotted curve around F = 6.6 represents the field-induced s-like k = 2 e-h bound state discussed earlier for Figs. 1 and 2.
B.
General case ωe 6= ωh :
For a numerical study of a more general case ωe 6= ωh , we consider an interesting case, where the harmonic ground levels of the conduction and valence bands are fitted to those of an infinitely deep rectangular QW of width W : h ¯ ωe /2 = h ¯ 2 π 2 /(2me W 2 ) and h ¯ ωh /2 = h ¯ 2 π 2 /(2mh W 2 ). This assumption yields ωe me = ωh mh , `e = `h = W/π and s
νe =
1 αh = , W = π(αe αh )1/4 `. αe νh
(47)
In this case, Eq. (30) can simply be written as f = (παe αh )1/2 Lf0
(48)
as in Eq. (32), while f0 here defined in Eq. (31) is more general than that in Eq. (32). As will be shown below, the case ωe 6= ωh under consideration offers qualitatively similar results in dimensionless units to the universal results presented in the previous subsection. Eqations (22) and (26) are diagonalized by choosing sufficiently large number of states Nψ = 50 and Nφ = 151, where the numerical results converge. GaAs and CdSe rods are studied with the parameters given in Table I, which, according to Eq. (48), yields WGaAs = 1.82`, WCdSe = 1.99`. 14
The well width here is roughly W ∼ L = 2` defined previously. Both W and L are defined as the distance between the two classical turning points for the ground state. Figure 7 displays the energy level (black curves, left axis) and oscillator strength f (red curves, right axis) of the ground state for GaAs (dashed curves) and CdSe (solid curves) quantum rod for `⊥ = 0.1 as a function of the reduced field for (a) ` = 1 and a longer rod (b) ` = 2. The thick (thin) curves are with e-h interaction turned on (off). Exciton dissociation signaled by the abrupt drop of f occurs at a much smaller field for the longer rod in (b). The behavior in this figure is similar to that shown in Fig. 1. The thick dashed-dotted solid curves in Fig. 7 (a) show the energy (black curve, left axis) and the oscillator strength (red curve, right axis) of the lowest excited level for ` = 1 for CdSe. Here, the flatness of the energy curve and vanishing oscillator strength below F = 5.5 indicate that this level is a p-like e-h bound state with a nod at xe = xh . As the field increases, the energy curve drops until it becomes flat again and the oscillator strength attains a sharp peak in the narrow range of the field 6.4 < F < 6.7, indicating that the p-like state is deformed into an s-like state. The exciton is dissociated above this field range. Figure 8 shows the energy level (black solid curves, left axis) and oscillator strength f (red dashed curves, right axis) of the ground state for a CdSe quantum rod for `⊥ = 0.1 as a function of the rod length L for F = 1. The thick (thin) curves are with e-h interaction turned on (off). The behavior in this figure is similar to that shown in Fig. 5. Finally, Fig. 9 shows the RMS average of the e-h separation for a CdSe rod or `⊥ = 0.1 for ` = 1 (solid curves), ` = 1.3 (dashed-dotted curves), ` = 1.5 (dotted curves), and ` = 2 (dashed-double-dotted curves) as a function of the reduced field in the presence (thick curves) and absence (thin curves) of e-h interaction. The behavior in this figure is similar to that in Fig. 6 except for one difference to be discussed below. The thick dashed curve near the bottom of Fig. 9 stands for the first excited level for ` = 1. The field dependence of this curve is consistent with that in Fig. 7 (a), showing a tightly bound p-like state below F = 5.5 and a field-induced s-like bound state in the range 6.4 < F < 6.7. In Fig. 6, the p-like excited state is unbound at low fields unlike in Fig. 7 (a) and therefore has a large RMS average of the e-h separation.
15
IV.
CONCLUSIONS
We calculated the exciton binding energy, the oscillator strength, and the RMS average of the e-h separation in a strong longitudinal electric field in a quasi-one-dimensional nanorod with parabolic confinement in the conduction and valence band. Interplay between the three competing forces from barrier confinement, e-h attraction and the field-induced e-h separation for exciton binding was examined. We found that, for a long nanorod with weak confinement, the exciton binding energy as well as the oscillator strength drops abruptly as a function of the field near the exciton dissociation point, while the RMS average of the e-h separation rises rapidly. For shorter rods, the transition is more gradual due to strong e-h Coulomb attraction caused by the proximity of the electron and the hole. We also showed that a DC field can induce an s-like exciton bound state out of the p-like wave function by shifting the centroid of the p-function and massively transferring the amplitude from one lobe to the other in a narrow field range, causing a sharp peak in the oscillator strength and a dip in the RMS average of the e-h separation. These interesting field-induced optical properties are not limited to parabolic confinements: Recently, we obtained very similar properties for quantum rods confined by high rectangular barriers.19 The dependence of the optical properties on the rod length was also investigated for varying fields.
Acknowledgments
The author acknowledges helpful discussions with Dr. J. A. Tsao at Sandia National Labs and Prof. D. F. Kelley at UC, Merced. This work was supported by U.S. DOE, Office of Science, Office of Basic Energy Sciences through the Energy Frontier Research Center (EFRC) for Solid-State Lighting Science at Sandia National Laboratories under contract DE-AC04-94AL85000. Computational resources were provided by the DOE NERSC facility.
16
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17
TABLE I: Effective masses me , mh in units of the free-electron mass m0 , dielectric constants κ, bulk exciton Bohr radius aB , binding energy εB , the internal field unit F ∗ = εB /(eaB ), and κF ∗ . Materials me /m0 mh /m0
κ
aB (˚ A) εB (meV) F ∗ (KV/cm) κF ∗ (KV/cm)
GaAs
0.067
0.45
12.9
117
4.77
4.08
52.6
CdSe
0.11
0.44
8
48.1
18.7
38.9
311
FIG. 1: (a) Energy εk and (b) the oscillator strength f are displayed for the ground state k = 1 (black solid curves) and the first excited level k = 2 (blue dashed-dotted curves) as a function of the reduced field F for ` = 1, ` = 2, and `⊥ = 0.1. The red dotted curves show (a) the energy εk=1 and (b) the oscillator strength f for the ground level for a longer quantum rod with ` = 2 and `⊥ = 0.1. The thick bold (thin) curves are with e-h interaction turned on (off). A prefactor QCdSe = 0.709 is used for the oscillator strength.
FIG. 2: The wave functions (left axis) of the two lowest levels k = 1 (black solid curves) and k = 2 (blue dashed curves) and the total potential energy (red dotted curves, right axis) for three different reduced fields (a) F = 0, (b) F = 6, and (c) F = 6.6.
FIG. 3: The energy (black curves, left axis) and the oscillator strength f (red curves, right axis) of the ground level k = 1 for `⊥ = 0.1 as a function of the reduced field for three different lengths (a) ` = 2 (dotted curves), (b) ` = 2.3 (solid curves), and (c) ` = 3 (dashed curves). The thick (thin) curves are with e-h interaction turned on (off). A prefactor QCdSe = 0.709 is used for the oscillator strength. The solid (open) triangle on the left axis indicates the energy for ` = 0.5 with (without) e-h interaction at F = 0.
FIG. 4: (a) The energy along the rod and (b) the oscillator strength f of the ground level k = 1 at zero field as a function of the rod length L ≡ 2` for `⊥ = 0.1 (thick solid curves) and `⊥ = 0.05 (thick dashed curves) in the presence of e-h interaction. The thin dashed-dotted curves show the result without e-h interaction. A prefactor QCdSe = 0.709 is used for the oscillator strength. For the case of `⊥ = 0.05 in (a), a large constant lateral energy shift ε⊥ = 600 is to be added to the dashed as well as to the dashed-dotted curve for the total energy. The energy shift ε⊥ is relative to the ground level of the rod with `⊥ = 0.1 in the lateral direction.
18
FIG. 5: The ground level energy εk=1 (black curves, left axis) and the oscillator strength f (red curves, right axis) as a function of the rod length L ≡ 2` for three reduced fields (a) F = 0.43 (dashed curves), (b) F = 1 (solid curves), and (c) F = 1.43 (dotted curves). A prefactor QCdSe = 0.709 is used for the oscillator strength. The thick (thin) curves are with e-h interaction turned on (off). The black solid (open) circle indicates the energy level with (without) e-h interaction for a very strong field F = 7 at a rod length L = 2.
FIG. 6: The root-mean-square average of the e-h separation of the ground state as a function of the reduced field for ` = 1 (solid curve), ` = 1.5 (dashed curve), and ` = 2 (dashed-double-dotted curve). The dashed-dotted curves near the bottom represent ` = 1, k = 2. The thick (thin) curves are with e-h interaction turned on (off).
FIG. 7: The ground level energy εk=1 (black curves, left axis) and the oscillator strength f (red curves, right axis) for a GaAs rod (dashed curves) and a CdSe rod (solid curves) for `⊥ = 0.1 as a function of the reduced field for (a) ` = 1 and (b) ` = 2. Other parameters are given in the text. The thick (thin) curves are with e-h interaction turned on (off). The thick dashed-dotted curves in (a) show the energy (blue curve, left axis) and the oscillator strength (green curve, right axis) for the first excited level for CdSe: Here, the energy curve is nearly flat and the oscillator srength has a sharp peak in the narrow range 6.4 < F < 6.7, indicating a field-induced bound state.
FIG. 8: The ground level energy εk=1 (black solid curves, left axis) and the oscillator strength f (red dashed curves, right axis) for a CdSe rod for `⊥ = 0.1 as a function of the rod length L ≡ 2` for F = 1. The thick (thin) curves are with e-h interaction turned on (off).
FIG. 9: The root-mean-square average of the e-h separation as a function of the reduced field for a CdSe rod for `⊥ = 0.1 for ` = 1 (solid curve), ` = 1.3 (dashed-dotted curve), ` = 1.5 (dotted curve), and ` = 2 (dashed-double-dotted curve). The thick (thin) curves are with e-h interaction turned on (off). The thick dashed curve near the bottom stands for the first excited level for ` = 1.
19
field-‐induced bound state
field-‐induced bound state
Fig. 1!
Fig. 2!
Fig. 3!
Fig. 4!
Fig. 5!
Fig. 6!
Fig. 7!
Fig. 8!
Fig. 9!