Theory of Polaron Resonance in Quantum Dots and Quantum-Dot
Molecules
K.-M. Hung*
Dept. of Electronics Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan 807, ROC.
Abstract
The theory of exciton coupling to photons and LO phonons in quantum dots (QDs) and quantum-dot molecules (QDMs) is presented. Resonant-round trips of the exciton between the ground (bright) and excited (dark or bright) states mediated by the LO-phonon alter the decay time and yield the Rabi oscillation. The initial distributions of the population in the ground and the excited states dominate the oscillating amplitude and frequency. This property provides a detectable signature to the information stored in a qubit made from QD or QDM for a wide range of temperature T. Our results presented herein provide an explanation to the anomaly on T-dependent decay in self-assembled InGaAs/GaAs QDMs recently reported by experiment.
PACS numbers: 78.67.Hc, 71.35.-y, 71.38.-k
I. Introduction
Charge carriers that move in semiconductor QDs and QDMs provide a larger transition-dipole moment than atomic and molecular systems owing to the interaction with solid-state matter and a spatial variation of the band edge in QDs and QDMs, making applications in quantum information processing [1] and logical operation [2,3] feasible. In such applications, coherent manipulation of the excitonic wavefunction in QDs and QDMs at finite temperature is essential. A longer dephasing time (~1ns in self-assembled InGaAs/GaAs QDs [4-6] and QDMs [6,7]) than the manipulation time (~1ps [8]) is very important, because the coherence of the excitonic transition, or quantum computation, can not be retained in the scope that the dephasing time is comparable to or smaller than the manipulation time.
In QD and QDM systems, the carrier dephasing can be categorized into two parts: (1) the dephasing of spatial wavefunction of exciton and (2) the dephasing of the internal degrees of freedom of exciton, such as degenerate spin states. The former dephasing mainly attributed to, for examples, photon and real phonon scatterings, so called the excitonic decay, because the exciton can not incoherently reside in a spatial-confined state. The second dephasing relaxes the internal degrees of freedom of exciton from one of its degenerate states to the others or changes the phase relation
of the superposition state composed of the degenerate states, while preserves the spatial coherence, i.e., conserves the number of excitons, so called the pure dephasing. In fact, the pure dephasing could also affect the decay time through, for example, the relaxation between spin-dark and spin-bright states according to the selection rule of photon emission [6]. Recently, both experimental and theoretical reports [9,10] have revealed that the virtual-phonon processes for both acoustic and optical phonons result to the second dephasing and exhibit a non-monotonous temperature dependence on rapid initial decoherence (~1ps) [11].
In principle, the charge cancellation for the identical distributions of electron and hole in a strong confining QD reduces the interaction of excitons with LO phonons [12-14], and a large level spacing decreases the strength of exciton scattering from real acoustic phonons. Accordingly, a long decay time is expected. However, the presence of piezoelectric fields [15], QD’s shape and/or size fluctuations [16], the Jahn-Teller effect [17], and charged point defects [18] lead to polarization of the charge distributions, and thus enhance the LO phonon-exciton coupling. As for the electronic polaron [19], the coherent interactions of the electron-hole pair with a polarizable field (a real optical phonon) form excitonic polaron and do not contribute to phase decoherence, because the dressed state is an eigenvector of the interacting exciton-LO-phonon system. It has been shown experimentally that the resonance of excitonic polaron exists as the energy separation between the electronic states differs by one or several LO-phonons [20]. This could occur deficiently in bulk, quantum well, or quantum wire structures according to the phase decoherence through the process of real acoustic phonon, which is strongly suppressed in QDs and QDMs due to a large level spacing.
The decay may result from the photon emission, the coupling of the phonon thermostat that is originated by the anharmonicity of the crystal through the LO-phonon component [19], the thermal emission of carriers out of the dots at high temperature T [6], and the virtual phonon processes [9-11]. With neglecting the second effect, the delta-like density of states of QD prohibited k-space thermalization results in a nearly constant decay time with respect to T during the situations that thermal emission can be ignored and the time scale is far away from that of the pure dephasing, because the photon emission is independent of T. Yet, this feature differs dramatically in self-assembled InGaAs/GaAs QDs [21] and QDMs [6], for which the decay time increases with T. This feature cannot be described as the thermal recycling of the carriers [6,22] because it almost disappears in the InGaAs/GaAs QDs with the same structure as used in the QDMs [6]. Instead, the thermal population of optically inactive states gives a reasonable explanation on the increase of the decay time as increased the lattice temperature.
In this work, a theory of the coupling of excitons to photons and LO phonons associated with the dark or weak-bright excited states is proposed to explain the anomaly. Our theoretical results also reveal the time-dependent Rabi oscillation (RO) for both QD and QDM systems [23]. The oscillating frequency and the oscillating amplitude of the ROs are strongly dependent on the initial distributions of the excitonic population, which depend on the ways of excitation, shown in Fig. 1(a). This property provides a detectable signature to the information stored in a qubit made from QD or QDM with a long lifetime and a wide range of temperature.
In the following discussions, a model and its corresponding Hamiltonian are proposed in Sec. II. Based on the model Hamiltonian, the dynamics of exciton population in QDs/QDMs are derived in Sec. III. The numerical results and discussions for T-dependent decay time of an exciton in QD and QDM systems are presented in Sec. IV. Finally, a conclusion is given in Sec. V.
II. Model and Hamiltonian
II-A. Physical Picture and Model
To describe the basic concept of the proposed theory, consider a three-level system of a QD or QDM embedded in a phonon bath as schematically plotted in Fig. 1(b). The ground (bright) state |g> of exciton is coupled to its vacuum state |vac> via the processes of emission/absorption of a photon, while coupled to the lowest excited state |e> by LO-phonon emission/absorption. The excited state can be a dark (or a weak bright) state in QDMs due to symmetric and anti-symmetric splitting for a symmetric structure [3] (or due to the inter-dot exciton for an asymmetric one) but a bright state in QDs, shown in Fig. 1(c). The same situation has been proposed in the study of exciton-enhanced Raman scattering in bulk semiconductors [24].
In order to explain how the brightness of the excited state associated with the phonon-assisted transition affect the decay time of an exciton in QDs/QDMs, let us imagine that there is an hourglass with a controlled gate at the vent to control the rate of the sand flow (analogous to the optical-transition rate of exciton). There are tow positions for the hourglass one at the upstairs (state |e>) and another at downstairs (state |g>). The gate is assumed to be position dependent, which is open (a bright
exciton) at the downstairs with the flow (optical transition) rate Γg, but is open (bright
exciton) or close (dark or weak-bright exciton) at the upstairs with the rate Γe that
depends on the system one considered. The hourglass could move up and down between the downstairs and upstairs (analogous to the RO of exciton). The dynamics of the hourglass depends on the external force (temperature) that forces the hourglass
(the total coupling strength between exciton and LO-phonon). When the hourglass statically stays in the downstairs (the absence of phonon-assisted transition at low T), the discharged time of the sand (the decay time of the exciton) is solely determined by
the flow rate Γg [25]. When the hourglass moves up and down between the upstairs
and downstairs (the presence of RO), the discharged time will be governed by both the flow rates of the upstairs and downstairs and the dynamical distribution of the hourglass. In the (high T) case that the hourglass has a half time stayed in the downstairs and another half in the upstairs with an equal probability to move up and
down, the averaged flow rate becomes (Γg+Γe)/2, and becomes Γg/2 or double the
discharged (decay) time for Γe=0 or Γe<<Γg (a fully-dark or weak-bright excited state).
Hence, a fully dark-excited state results to the maximum enhancement of the decay time by a factor of two. In moderate temperature, the averaged rate has the form of ~NgNRO(0+)Γg + NeNRO(0+)Γe, where NgNRO (NeNRO) is the initial number of exciton in the state |g> (|e>) with removing RO.
Although this model is simple, it sufficiently describes the mechanism of exciton decay caused by the interaction of exciton and LO-phonon. Spontaneous emission of a LO-phonon by an excited exciton, which is absent in the case that the exciton initially resides in its ground state, makes the T-dependent behavior of exciton very sensitive to its initial population distribution.
II-B. Sudden Approximation
In experiments, there are two excitation approaches, schematically plotted in Fig. 1(a), provided with a low excitation power for preventing charged exciton or biexciton effects often used in photoluminescence (PL) experiments: (i) indirect excitation the excitons are initially created on the substrate or wetting layer and then relaxed into the states of QDs or QDMs. In such an excitation, a nonzero distribution of the exciton population in both the states |g> and |e> is possible. (ii) Direct excitation the distribution of excitonic population is initially determined by the frequency of the excitation light source. The rise time of the number of excitons should depend on the mechanisms of carrier relaxation, optical transition, excitation power, and duration of excitation pulse.
The discussion of the rise time that is, in the most cases, much shorter than the decay time is beyond the scope of this work. To avoid the complicated processes of exciton generation and rapid initial decay we assume that the system could promptly response to an ultra-short pulse excitation, a sudden approximation. In the
approximation, the exciton is abruptly present in the system at the time 0+, before that
exciton-photon are absent [26]. Although the sudden approximation can not occur in a real system, but it is a good approximation to the case that the decay time is much longer than the rise time, such as the exciton decay in InAs/GaAs QDs/QDMs [6], and is useful in the simplification of theoretical derivation.
II-C. Hamiltonian
In bosonized approximation, the interacting Hamiltonian of the exciton coupling to photons and LO-phonons (Fröhlich type) in a three-level system can be expressed as,
∑
∑
∑
+ + + + + = + + + + q e g q q p e p pe p g p pg I a cc a cc d cc H (γ ˆ αˆ . .) (γ ˆ αˆ . .) (γ ˆ αˆ αˆ . .), (1)where aˆ+p (aˆ ), p + q
dˆ (dˆ ), and q + i
αˆ (αˆ ) are the creation (annihilation) operators of i
the photon, the LO phonon, and the exciton, respectively, γq (γpi) is the strength of
coupling of the exciton to the LO phonons (photons). A rotation-less approximation to the photon field is made, because the transition between spin-bright and spin-dark is not a major concern in this work. Notably, the exciton operators utilized in the pairing theory of superconductors [27] satisfy the algebra
) ˆ ˆ 1 ( ] ˆ , ˆ [αi α+j =δi,j −nei −nhi (2)
because of the Pauli principle, where nˆe(h)i is the number operator of electron (hole) in the state i. Although Eq. (2) is not in the form required by Bose-Einstein statistics, the factor nˆei +nˆhi does not affect the physics of the exciton’s dynamics in counting
all orders of resonant LO-phonon (conserved the phonon’s momentum in the exciton loop as shown in Fig. 1(b), detailed in Sec. III-E) and in the second-order
approximation to the photon correction. The definition of the exciton operator αi, a
product of the annihilation operators of the electron (cˆ ) and the hole (i hˆi) in the state i,
gives the useful identities in the derivations of the equations of motion (EOM) of the exciton, . ˆ ] ˆ , ˆ [ ] ˆ , ˆ [ and , ˆ ] ˆ , ˆ [ ] ˆ , ˆ [ , 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ + + + + + + + − = = = = = = = = = = i hi i ei i i hi i ei i i hi i ei hi i ei i i i i i n n n n n n n n α α α α α α α α α α α α α α (3)
In dipole approximation [28], the coupling strength of exciton to photon has the form ) ( ) ( 2 r r r V eE i i i p r i pi ε ω ε φ φ π γ h r r h ⋅ = , (4)
where Ei and φi(r) are the exciton energy of the state i and its associated wavefunction
(in relative coordinate), respectively, εr is the polarization vector of the photon field,
ωp is the photon frequency with the mode p, e is the bare electron charge, and εr is the
relative dielectric constant of dot’s material. The coupling strength Eq. (4) gives the emission rate [28] 2 3 3 2 | | 3 4 2 i i r i i r c e φ φ ε ω r h = Γ , (5)
with ωi = Ei h and the light speed c, Γi is defined in Sec. III-D. In a disc-like
parabolic QD with the identical confinement strengths ωl and ωt to both electron
and hole in the longitudinal and transverse directions, respectively, and assumed that the dot size is much smaller than the Bohr radius of the exciton, i.e. ωl,ωt >>VCoul
with the Coulomb potential VCoul, the exciton’s dipole moment (DM) can be written as
c i i i e r r r el d |≡ | ( ) ( ) |= | r φ rφ , ωµ* t c
l = h is the effective cyclotron radius of exciton,
and µ*
is the exciton’s reduced mass.
To demonstrate the optical properties of the ground and the first-excited states of
a QDM, consider a coupled-QD system with the bare electron (hole) energies Ee1(h1)
and Ee2(h2) to the dots 1 and 2, respectively, and with the tunnel strength te(h) between
the dots. The tunneling t and the Coulomb interaction mix the single-QD states of both electron (|i>e, i denotes the dot 1 or 2) and hole (|i>h). In the regime that the
confinement and the tunnel splitting are predominant over the Coulomb effect, a narrow barrier, the energies and their associated wave-functions of the ground and the excited state for the electron (hole) can be approximated as
] 4 ) [( 2 1 2( ) 2 ) ( 2 ) ( 1 ) ( ), (h g eh eh eh eh e E E E t E = + m ∆ + , (6)
[
( ) ( ) ( )]
2 ) ( ) ( 1 2 1 1 h e h e h e g h e h e g + + = η η , (7) and ] 4 ) [( 2 1 2( ) 2 ) ( 2 ) ( 1 ) ( ), (h ex eh eh eh eh e E E E t E = + ± ∆ + , (8)[
( ) ( ) ( )]
2 ) ( ) ( 1 2 1 1 h e h e h e g h e h e ex + + = χ χ (9) respectively, where Ee(h)1=hωle(h) 2+hωte(h) , ) ( 2 ) ( 2 ) ( ) ( ) ( 2 4 h e h e h e h e h e t t E E ± ∆ + ∆ = η , ) ( 2 ) ( 2 ) ( ) ( ) ( 2 4 h e h e h e h e h e t t E E ∆ + ∆ = mχ , ∆Ee(h) =Ee(h)2−Ee(h)1, and ωle(h) and ωte(h) are the
confinement strengths of electron (hole) in the growth and transverse directions. The
QDM states of the exciton may have the forms of S1 = g e g h, S2 = g e ex h,
h e g ex
S3 = , and S4 = ex e ex h . In double-oscillator approximation and
) ( 2 ) ( 1 ) (h , eh eh e E E
E >>∆ , where ∆Ee(h) is mainly caused by the fluctuation of the dot’s
high, the tunneling strength of the electron (hole) can be estimated by
π α ω α2 ( ) ( ) ) ( ) (h exp( eh ) leh eh e
t = − h with the dimensionless distance αe(h)=a le(h)
between the oscillator centers located at ± , a ( )
* ) ( ) (h ehl leh e m l = h ω , and me*(hl) the
effective mass of electron (hole in growth direction) [28].
The DMs of these states including the effect of indirect exciton can be written as
2 2 2 2 2 2 2 1 1 1 | ] ) ( 2 exp[ ) ( 4 ) 1 ( | | | h e h e h e c c h e S l l a C a l l e d η η η η η η + + + − + + + + = r , (10) 2 2 2 2 2 2 2 2 1 1 | ] ) ( 2 [ exp ) ( 4 ) 1 ( | | | h e h e h e c c h e S l l a C a l l e d χ η χ η χ η + + + − + + + + = r , (11) 2 2 2 2 2 2 2 3 1 1 | ] ) ( 2 [ exp ) ( 4 ) 1 ( | | | h e h e h e c c h e S l l a C a l l e d η χ η χ η χ + + + − + + + + = r , and (12) 2 2 2 2 2 2 2 4 1 1 | ] ) ( 2 [ exp ) ( 4 ) 1 ( | | | h e h e h e c c h e S l l a C a l l e d χ χ χ χ χ χ + + + − + + + + = r , (13)
where the effective confined strength ωt of the exciton can be estimated by
) (
)
( e* te2 *ht th2 e* ht*
t = mω +m ω m +m
mass of hole in the transverse direction. The first term in Eqs. (10)-(13) is resulted from the effect of intra-dot (direct) exciton while the last one from the inter-dot (indirect) exciton. The confinement strengths of electron (hole) in growth and transverse directions can be approximately estimated by ωle(h) = 8Vco(bo) me*(hl)h2
and ωte(h) = 8Vco(bo) me*(ht)R2 , respectively, where Vco(bo) is the conduction (valence)
band-offset, h is dot’s high, and R is dot’s radius.
For me* =0.081m0, mhl* =0.34m0, m*ht =0.153m0, Vco =0.68eV , Vbo =0.1eV ,
eV
Eg(dot) =0.73 , h=3.5nm, R=17nm, and ∆Eh =−∆Ee 5, the calculated results
of the energy levels of exciton and their DMs with respect to ∆ and Ee 2a (the
distance between the dots) are plotted in Figs. 2 and 3, respectively. For a symmetric
structure, ηi=1 and χi=-1, the DMs of the states |S2> and |S3> become zero, a
dark-exciton state, as shown in Fig. 2(a). With increasing its asymmetry, ∆ Ee
increases, the distribution of the ground state of electron and hole tends towards the low energy side, and that of the excited state towards the other side. These inhomogeneous distributions of the electron and hole result to increase the DMs of |S2> and |S3>. For a long inter-dot distance, the coupling between the dots reduces exponentially, Fig. 3(c), and the energy differences between |S1> and |S2> and between |S3> and |S4> are reasonable to reduce to ∆Eh, Fig. 3(b). In this situation, the
exciton states |S1> and |S4> approach the isolated QD’s states and their DMs reduce to the QD’s ones, while the states |S2> and |S3> become a pure indirect exciton and their DMs approach zero, Fig. 3(a), because the inter-dot wavefunction overlap of the electron and hole approaches zero. At short distance, the tunnel splitting that is greater than the energy difference caused by the asymmetry of the dots, Fig. 3(c), dominates the DMs of the states |S2> and |S3>. In this situation, the property of the system tends towards the symmetry one, and the DMs of the states |S2> and |S3> shrink, Fig. 3(a). The increase of the tunneling strengths increases the gaps between these four states, Fig. 3(b). Figure (4) shows the transition rates of these four states with respect to the
inter-dot distance for ∆Ee =15meV and ∆Eeh =3meV. It clearly displays that the
transition rates of both states |S2> and |S3> are one order smaller than the states |S1> and |S4> at whole distance, i.e., these states have the lifetime much longer than the states of |S1> and |S4>. It is worthy to be noted that the transition rate of the state |S1> increases nearly two times that of the single QD (~0.61µeV), which takes the value ~1.16µeV, as deceased the inter-dot distance to ~2.5nm due to the contribution of the indirect exciton. The transition rate of QDM (QD) gives the exciton decay-time
three-level system mediated by one-phonon process gives the maximum enhancement of the decay time by a factor of two. In strong coupling regime, a further consideration of multiple levels with multiple-phonon processes has to be done, since the manifold loops of polaron resonance mediated by multiple phonons could also affect the decay time. For example, there are two types of polaron resonance including two-phonon process as schematically plotted in Fig. 9. In the first one (Fig. 9(a)), the exciton takes more time than three-level system in resonant-round trips between the states |g>, |e1>, and |e2>. At high T and/or strong coupling region, the identical probability (~1/3) of the exciton distribution in these states reduces the decay rate to one-third that of zero-T. While the second (Fig. 9(b)) decreases the decay rate nothing less than the three-level system, because the exciton has the probability of ~1/2 to stay in the state |g> and only the probability of ~1/4 in the states |e1> and |e2>. In conclusion, the increase of the excitonic decay time in QDs/QDMs is resulted from the total effect of multiple weak-bright excited states or from a major state provided with a weak-optical transition and a small detuning energy. Moreover, the population-dependent amplitude and frequency of RO provide a detectable signature to the information stored in QD/QDM systems for a wide range of temperature. This is useful in the application of quantum information processing.
Acknowledgement
This work was supported by the National Science Council of the Republic of China, Taiwan, under Contract No. NSC95-2623-7-151-003-D, and the Ministry of Education of R. O. C.
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* Electronic address: [email protected]
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