Główna Physical Review B Defect-assisted relaxation in quantum dots at low temperature

Defect-assisted relaxation in quantum dots at low temperature

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Physical Review B
July, 1996
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15 JULY 1996-I

Defect-assisted relaxation in quantum dots at low temperature
Darrell F. Schroeter* and David J. Griffiths
Department of Physics, Reed College, Portland, Oregon 97202

Peter C. Sercel
Department of Physics and Materials Science Institute, University of Oregon, Eugene, Oregon 97403
~Received 13 October 1995; revised manuscript received 14 March 1996!
A model for electron relaxation in a quantum dot, including a nonradiative pathway through a point defect,
is presented, using time-dependent perturbation theory. The results obtained here extend previous work @Phys.
Rev. B 51, 14 532 ~1995!# to the experimentally relevant low-temperature regime. It is found that relaxation
through defects may circumvent the phonon bottleneck predicted for ideal nanometer-scale quantum dot
structures even at low temperatures. @S0163-1829~96!06828-2#

For nanometer scale quantum dots, the separation between adjacent electron energy levels may exceed the LO
phonon energy. In this regime, single phonon emission cannot account for electron relaxation between adjacent energy
levels. The alternatives, multiple phonon and radiative processes, occur much more slowly, resulting in prolonged lifetimes for electrons in excited states of the quantum dot.1,2
This effect is known as the ‘‘phonon bottleneck.’’ While this
effect appears to stand on firm theoretical ground, it has not
been observed in recent experiments on high-quality quantum dots.3
Several models have been proposed to solve the ‘‘phonon
bottleneck’’ problem. Inoshita and Sakaki4 have shown that
processes involving one LO phonon and one LA phonon can
provide a pathway for rapid relaxation between electronic
states separated by \ v LO 6 \ v LA , however, this process is
inoperative for quantum dots where the intraband spacing is
substantially greater than the LO phonon energy. Other investigators have examined the role of Auger-like processes
in overcoming the phonon bottleneck.5,6 This type of mechanism could allow for rel; axation on picosecond time scales.
An alternate model wherein electrons thermalize by coupling to nearby traps or interfacial defects has been proposed
by Sercel.7 In this model, relaxation occurs by the following
sequential process: an electron makes a transition from the
quantum dot to the defect, the defect relaxes by multiphonon
emission, and the electron makes a second transition to a
lower-lying energy level of the quantum dot. The defect
functions as a sort of ‘‘elevator’’ carrying the electron from
the upper level in the quantum dot to the lower level, providing a channel for nonradiative intraband carrier relaxation. Sercel’s analysis is based upon a semiclassical approximation for the lattice, strictly valid only when the
thermal energy in the system is greater than the height of the
activation barrier for a dot-defect transition. The conclusion
of Sercel’s study was that defect-assisted relaxation could
play a role in breaking the phonon bottleneck effect in
nanometer-scale quantum dots. However, the majority of experiments performed on quantum dots are carried out at
cryogenic temperatures where the semiclassical approximation fails.


In the present paper the low-temperature rates are calculated fully quantum mechanically, and it is found that the
defect-assisted relaxation rates may remain large even at low
temperatures. The analysis shows that electron relaxation via
the defect-coupling channel circumvents the phononbottleneck effect in qualitative agreement with experiments
reported in Ref. 3.
To conform with the analysis of Sercel,7 we consider a
model spherical In 0.5Ga 0.5As/GaAs quantum dot with a radius of 5 nm, and a point defect located within the GaAs
matrix. At this radius, the dot has only two bound conduction
states, the ground state C0, and first excited state C1. The
valence-band states are assumed to be thermalized owing to
the smallness of the level spacings relative to the conduction
band. We begin by examining the transition rates between
the trap state T and the two quantum dot states C0 and
For either of the quantum dot states (i5C0, C1) the total
energy is a sum of electronic and vibrational ~thermal! energy:
V i 5E i 1 12 m v 2 Q 2 ,


where it is assumed that the configuration of the interaction
mode may be specified by a single coordinate Q. The total
wave function is a product of an electronic and a vibrational
wave function: c i (r,Q)5 w i (r) x i (Q). The electronic wave
function for these conduction states is calculated using a
single-band model in the effective-mass approximation.8 The
potential energy V representing the band offset between the
conduction bands of GaAs and In 0.5Ga 0.5As, has been calculated as 360 meV,9 for conditions of uniaxial strain. Solving
for the energy of these states using material parameters given
by Shur10 shows that E C0 5170 meV and E C1 5332 meV, as
measured from the bottom of the In 0.5Ga 0.5As conduction
For simplicity, we assume that the defect has only a single
bound state. The Dirac d function potential satisfies this criterion and we follow Sercel7 in using this potential to model
the defect. The wave function is again a product of electronic
and vibrational wave functions, with the electronic portion
given by

© 1996 The American Physical Society



w T5





a e 2 a u r 2b u
2 p u rW 2bW u


where a [ A2m*E 0 /\ 2 is the wave number and bW is the
distance from the center of the quantum dot to the defect.
The electron-phonon coupling of the defect state is taken to
be linear, so that the total energy of an electron in the trap
takes the form of a displaced parabola
V T 5E T 1 21 m v 2 ~ Q2Q 0 ! 2 ,


where E T [E 0 2m v 2 Q 20 /2. It is helpful to introduce the
Huang-Rhys factor S,
S\ v 5 21 m v 2 Q 20 ,


and the ‘‘number of phonons’’ p l defined as the difference
between the minima of the two energy curves corresponding
to quantum dot state Cl and the defect state:
p l \ v 5E Cl 2E T .


For comparison with the semiclassical rates derived
previously,7 it is necessary to obtain an explicit formula for
the coordinate of the crossing point as well as the height of
the barrier between the two wells. These can be obtained
from Eqs. ~1! and ~3!:
Q Cl 5


~ S2p l ! 2 \
2m v S

EA Cl 5

~ S2p l ! 2 \ v




x i ~ Q ! * x j ~ Q ! dQ 5

W~ E !5


The energy EA Cl is measured from the bottom of the parabola which the electron occupies in its initial state. Figure
1 shows a configuration coordinate diagram depicting totalenergy curves for both of the quantum dot states C0 and
C1, the trap state T, as well as the quantities given in Eqs.
Transitions between the dot and the defect fall into two
different categories, depending on temperature. In the first
case the thermal energy exceeds EA Cl , the crossing point
energy. This is the case that was treated by Sercel,7 and for
which the semiclassical approach is valid. The second case is
when the thermal energy in the system is less than EA Cl . In
that case the electron must tunnel through the barrier in configuration space in order to make a transition between the dot
and the defect. In both of these cases ELECTRONIC tunneling
occurs, since the dot and the defect are separated spatially. It
is a second type of tunneling, through configuration space
~nuclear tunneling!, that is at issue in the low-temperature
case. In this section we treat this second kind of tunneling
using Fermi’s golden rule, which gives the transition rate at
a given energy E in terms of a matrix element M , and the
density of final states N f :


FIG. 1. Total-energy curves for the quantum dot states C0,
C1, and the trap state T as a function of the configuration coordinate Q.

u M u2N f .


The density of final states takes the form 1/\ v , 11 so that the
problem reduces to a calculation of the matrix element M .
The matrix element is the integral over the initial state,
the Hamiltonian that causes the transition ~the perturbation!,
and the final state:





x i ~ Q ! * w i ~ r ! * H int~ r,Q ! x j ~ Q ! w j ~ r ! drdQ.

Performing the integral over r gives the electronic matrix
element V(Q). Since the overlap of the vibrational components is sharply peaked at the crossing point Q c , 11 the electronic matrix element can be pulled outside of the integral:
M 5V ~ Q c !




x i ~ Q ! * x j ~ Q ! dQ.


The integral in Eq. ~9! is the overlap of two harmonic oscillator wave functions of arbitrary quantum numbers i and j,
separated by a distance Q 0 . This integral can be calculated
exactly, the result is

i! m 2 v 2 Q 20
2\ 2



e 2m

2 v 2 Q 2 /2\ 2

L ij2i

m 2 v 2 Q 20
2\ 2





This gives us the rate for transitions out of a single state with energy E, W(E); to obtain the more pertinent quantity, the
total transition rate at a given temperature, we must perform a summation over all states, with each weighted by the appropriate




Boltzmann factor. This sum can be calculated exactly, and gives an equation for the transition rate at low temperatures with
no approximations other than the ones latent in Fermi’s golden rule. The result is
W~ T !5



In this equation, p\ v denotes the ground-state energy difference between the defect and the relevant quantum dot
state, defined in ~5!, while the Huang-Rhys parameter S,
which characterizes the defect is defined in Eq. ~4!. As it
stands this equation is extremely general. For example, the
same result turns up in the study of electron transfer in biological systems.12 It remains only to calculate the electronic
matrix element V(Q c ).
The electronic matrix element is calculated in the tightbinding approximation.8 This assumption is valid if the separation between the dot and defect is large enough. The total
Hamiltonian may be written as


2 p u V~ Q c !u 2
exp 2Scoth
I Scsch
\ 2v
2kT p

p̂ 2
2m *


where the first two terms together make up Ĥ TRAP , the
Hamiltonian whose eigenstate is the defect wave function.
The electronic matrix element can be written as



state (C1-T), and between the quantum dot ground state and
the defect state (C0-T). These values, which are plotted as a
function of separation between the quantum dot and the defect, were calculated numerically using the band-structure
parameters chosen above.
Using the results of Fig. 2, we are able to calculate the
total transition rate as a function of temperature. This is done
for a separation of 10 nm in Fig. 3~a! and for a separation of
20 nm in Fig. 3~b!. In order to perform this calculation a
number of parameters must be specified. The Huang-Rhys
factor S is chosen such that S\ v 5100 meV when the frequency v is taken to be that of the TA phonon (\ v 510
meV!. The energy of the unoccupied trap is taken to be 275
meV. Therefore, for transitions from C1 to T the number of
phonons p516, and for transitions from T to C0 it is 1. In
order to obtain the transition rates for the opposite direction,
the sign of p is reversed. These are values typical of the
electron trap M 1 common to GaAs grown by molecularbeam epitaxy.13,14

V ~ Q ! 5 ^ w DOTu H TRAPu w TRAP& 1 ^ w DOTu V DOTu w TRAP&
5E ^ w DOTu w TRAP& 1 ^ w DOTu V DOTu w TRAP& .


Since we are interested in the value of this matrix element at
the crossing point V(Q c ) where the energies of the two states
are equal, the energy E in Eq. ~13! can be taken to be the
electronic energy of the quantum dot state involved in the
transistion. Figure 2 shows the electronic matrix element between the first excited state of the quantum dot and the defect

FIG. 2. Electronic matrix element V(Q) for transitions between
the defect and the ground quantum dot state (C0-T), and for transitions between the defect and the first excited quantum dot state
(C1-T). Separation measures the distance between the defect and
the center of the quantum dot. Parameters used are an effective
electron mass m * 50.041m e and a band offset of V5360 meV,
giving rise to the bound-state energies E C0 5170 meV and
E C1 5332 meV.

FIG. 3. Total transition rates between the quantum dot and the
defect at ~a! a dot-defect separation of 10 nm and ~b! a dot-defect
separation of 20 nm. Parameters used are S510, \ v 510 meV,
p 0 51, and p 1 516.



Figure 3~a! shows that for a dot-defect separation of 10
nm the rate C1→T is on the order of 1013 s 21 , and the rate
T→C0 is on the order of 1010 s 21 , at 0 K. The reverse rates
are in fact ‘‘frozen out’’ at low temperatures, since such
transitions would require an influx of energy into the system.
If the distance between the dot and the defect is increased to
20 nm, a substantial reduction is seen in the transition rates
@Fig. 3~b!#. The fact that the T to C0 rate drops to approximately 105 s 21 means that the defect-assisted tunneling
mechanism would not serve to overcome the ‘‘phonon
bottleneck’’ at such a large separation.
In this work we have modeled the effect of coupling to a
deep-level trap on electron relaxation in quantum dots in the
low-temperature limit. The main conclusion of the present
analysis is that the presence of point defects may serve to
enhance the luminescence efficiency of quantum dot material, even at low temperature, a regime in which defectrelated processes might have been expected to be frozen out.
The persistence of the defect relaxation process at low tem-

*Present address: Department of Physics, 117 Clark Hall, Cornell
University, Ithaca, NY 14853-2501.
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perature is due to tunneling through the defect activation
barrier for capture and emission. As discussed in Ref. 7, the
physical situation described in this paper could only arise if
the spatial distribution of defects is strongly correlated with
that of the quantum dot structures, e.g., through formation of
interface states or point defects as a consequence of the
growth process. That this situation commonly exists in
MBE-grown material is demonstrated, for example, by the
observation of resonant enhancement of nonradiative processes involving point growth defects in MBE-grown
AlxGa12xAs/GaAs quantum wells.15 With this caveat, the
proposed mechanism may thus explain the failure to observe
a significant phonon bottleneck effect in recent work on
InxGa12xAs quantum dot structures.
This material is based upon work supported by the National Science Foundation under Grant No. DMR 9304537.
Support by the Oregon Joint Centers for Graduate Schools in
Engineering is gratefully acknowledged.


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