Absorption and Emission in quantum dots: Fermi surface effects of Anderson excitons

Abstract

Recent experiments measuring the emission of exciton recombination in a self-organized single quantum dot (QD) have revealed that novel effects occur when the wetting layer surrounding the QD becomes filled with electrons, because the resulting Fermi sea can hybridize with the local electron levels on the dot. Motivated by these experiments, we study an extended Anderson model, which describes a local conduction band level coupled to a Fermi sea, but also includes a local valence band level. We are interested, in particular, on how many-body correlations resulting from the presence of the Fermi sea affect the absorption and emission spectra. Using Wilson's numerical renormalization group method, we calculate the zero-temperature absorption (emission) spectrum of a QD which starts from (ends up in) a strongly correlated Kondo ground state. We predict two features: Firstly, we find that the spectrum shows a power law divergence close to the threshold, with an exponent that can be understood by analogy to the well-known X-ray edge absorption problem. Secondly, the threshold energy $\omega_0$ - below which no photon is absorbed (above which no photon is emitted) - shows a marked, monotonic shift as a function of the exciton binding energy $U_{\rm exc}$

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arXiv:cond-mat/0502329v1 [cond-mat.str-el] 14 Feb 2005
Absorption and Emission in quantum dots: Fermi surface effects of Anderson excitons
R. W. Helmes,1M. Sindel,1L. Borda,1,2and J. von Delft1
1Physics Department, Arnold Sommerfeld Center for Theoretical Physics,
and Center for NanoScience, Ludwig-Maximilians-Universit¨ at M¨ unchen, 80333 M¨ unchen, Germany
2Research Group “Theory of Condensed Matter” of the Hungarian Academy of
Sciences and Theoretical Physics Department, TU Budapest, H-1521, Hungary
(Dated: February 12, 2005)
Recent experiments measuring the emission of exciton recombination in a self-organized single
quantum dot (QD) have revealed that novel effects occur when the wetting layer surrounding the
QD becomes filled with electrons, because the resulting Fermi sea can hybridize with the local
electron levels on the dot. Motivated by these experiments, we study an extended Anderson model,
which describes a local conduction band level coupled to a Fermi sea, but also includes a local
valence band level. We are interested, in particular, on how many-body correlations resulting from
the presence of the Fermi sea affect the absorption and emission spectra. Using Wilson’s numerical
renormalization group method, we calculate the zero-temperature absorption (emission) spectrum
of a QD which starts from (ends up in) a strongly correlated Kondo ground state. We predict two
features: Firstly, we find that the spectrum shows a power law divergence close to the threshold,
with an exponent that can be understood by analogy to the well-known X-ray edge absorption
problem. Secondly, the threshold energy ω0 - below which no photon is absorbed (above which no
photon is emitted) - shows a marked, monotonic shift as a function of the exciton binding energy
Uexc.
PACS numbers: 73.21.La, 78.55.Cr, 78.67.Hc
I. INTRODUCTION
Recent optical experiments1,2using self-assembled
InAs quantum dots (QDs), embedded in GaAs, showed
that it is feasible to measure the absorption and emis-
sion spectrum of a single QD. In absorption spectrum
measurements photons are absorbed inside the QD by
electron-hole pair (exciton) excitation. In emission spec-
trum measurements, on the other hand, an exciton
created by laser excitation recombines inside the QD,
whereby a photon is emitted which is measured.
Due to spatial confinement, the QD possesses a charg-
ing energy and a discrete energy level structure, which
can be rigidly shifted with respect to the Fermi energy
EF by varying an external gate voltage Vg. Therefore
Vg allows for an experimental control of the number of
electrons in the QD, which in turn determines the en-
ergy of the absorbed and emitted photons. Indeed, the
optical data reveal a distinct Vg-dependence and justify
the assumption of a discrete energy level structure of the
QD1.
In the experimental set-up, depicted in Fig. 1, the InAs
QDs are surrounded by an InAs mono-layer, called ’wet-
ting layer’ (WL), like islands in an ocean. Above a certain
value of Vg, the conduction band of delocalized states of
this WL begins to be filled, forming a two-dimensional
Fermi sea of delocalized electrons, i. e. a two-dimensional
electron gas (2DEG). The 2DEG hybridizes with local-
ized states of the QD, leading to anomalous emission
spectra which could not be explained by only considering
the discrete level structure of the QD1.
Motivated by these experiments, we investigate here
the optical properties of a QD coupled to a Fermi sea, at
temperatures sufficiently small that Kondo correlations
can occur (T=0). The Kondo effect in a QD has already
been detected in transport experiments3,4, where it leads
to an enhanced linear conductance. So far the Kondo
effect in QDs has been studied almost exclusively in rela-
tion to transport properties. The experiments of Refs.1,2
open the exciting possibility to study the Kondo effect in
optical experiments.
In optics, the effect of Kondo correlations on QDs has
to the best of our knowledge been discussed theoretically
only with respect to non-linear and shake-up processes in
a QD5,6. The influence of disorder in heavy-fermion sys-
tems on the x-ray-photoemission has been studied e. g.
in Ref.7In this paper we investigate the absorption and
emission spectra of a QD. We are especially interested
in optical transitions (examples are shown in Fig. 4 be-
low) for which the QD starts in or ends up in a strongly
correlated Kondo ground state, and will investigate how
the Kondo correlations affect the observed line shapes.
In Ref.8the emission spectrum in the Kondo regime has
already been studied, however with methods which only
produce qualitative results.
The paper is organized as follows: In Section II, we ex-
tend the standard Anderson model9by including a local
valence band level (henceforth called v-level) containing
the holes. In contrast to Refs.1,2we consider only one
local conduction band level (henceforth called c-level),
to simplify the calculations. In Section III, we explain
how Wilson’s numerical renormalization group (NRG)
method10can be adapted to calculate the absorption and
emission spectrum of the QD. In Section IV, we present
the results of our calculations and predict two rather dra-
matic new features. Firstly, the absorption and emission
spectra show a tremendous increase in peak height as the
exciton binding energy Uexcis increased. In fact, the ab-

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2
FIG. 1: Right hand side: experimental setup used in Refs.1,2
(Picture: courtesy of the group of K. Karrai.) [bottom to top:
GaAs substrate (2000 nm), highly doped GaAs back contact
(20 nm, zero-point of x-axis), GaAs tunnel barrier (20 nm),
InAs mono-layer, forming the wetting layer, together with the
QDs, GaAs layer (30 nm), AlAs/GaAs tunnel barrier (∼ 100
nm), GaAs (4nm), NiCr top gate]. The gate voltage Vg, ap-
plied between the back contact and the top gate, drives no
current through the system, since the contacts are separated
by two tunnel barriers. Left hand side: position-dependence
(in x-direction) and energy-dependence of the lower conduc-
tion band edge of the layered structure for two different gate
voltages Va
rial result in jumps at the corresponding interfaces.
the InAs band gap is smaller than that of GaAs, there is a
dip in the band gap at the GaAs-InAs interface, resulting in
QDs with localized conduction and valence band states. The
number of localized electrons trapped in the QD can be con-
trolled by Vg, which shifts the energy levels with respect to
the Fermi energy EF (set by the back contact). Inset: holes
can be trapped as well due to the bump of the upper band
edge of the valence band at the position of the QDs.
radiation by laser light excites electron-hole pairs (excitons)
inside the GaAs layer, which migrate and become trapped in
the InAs QDs. Finally they recombine by emitting photons,
whose emission spectrum is detected.
g and Vb
g; the different band gaps of each mate-
Since
Ir-
sorption spectrum shows a power law divergence at the
threshold energy ω0, in close analogy to the well-known
X-ray edge absorption problem11. Exploiting analogies
to the latter, we propose and numerically verify an ana-
lytical expression for the exponent that governs this di-
vergence, in terms of the absorption (emission)-induced
change in the average occupation of the c-level. Secondly,
the threshold energy below which no photon is absorbed
or above which no photon is emitted, respectively, say ω0,
shows a marked, monotonic shift as a function of Uexc; we
give a qualitative explanation of this behaviour by con-
sidering the interplay of various relevant energy scales.
Conclusions are given in Section V.
????????????
2DEG
????????????
?????? ??????
????????????
????????????
εv
Uexc
εc
V
Uc
γ
EF
Quantum Dot
FIG. 2:
level and one v-level, with energies ǫc and ǫv, respectively.
The Coulomb repulsion of two electrons in the c-level has the
strength Uc. The coupling between the c-level and the 2DEG
is parametrized by the tunnelling matrix element V . Crucial
for the model is the Coulomb attraction between holes in the
v-level and electrons in the c-level, which has a strength Uexc.
The excitation of electrons from the v-level to the c-level (by
photon absorption) and the relaxation of electrons from the
c-level to the v-level (by photon emission) is considered as a
perturbation of strength γ.
Model of a semiconductor QD, consisting of one c-
II.MODEL
The experimental setup used in Refs.1,2, which inspired
our analysis, is depicted in Fig. 1 (see Fig. caption for de-
tails). To model this system, we consider an Anderson-
like model9for a QD, with localized conduction and va-
lence band levels, coupled to a band of delocalized con-
duction electrons stemming from the WL. Our model is
similar in spirit, if not in detail, to that proposed in
Refs.8,12. It consists of six terms, illustrated in Fig. 2:
H = H0+ Hpert, (1)
where
H0= Hc+ Hv+ HUexc+ HWL+ Hc−WL. (2)
We consider one c-level with energy ǫc and one v-level
with energy ǫv, originating from the conduction or va-
lence band of the InAs QD, respectively. Note that ǫv
is smaller than ǫc by the order of the band gap; since
this difference is at least two orders of magnitude larger
than all other relevant energy scales, its precise value is
not important, except for setting the overall scale for the
threshold for absorption or emission processes.
Since one c-level is sufficient to produce the effects of
present interest, we will, in contrast to the experimen-
tal situation realized in Refs.1,2, disregard further local
levels to simplify the calculations (the experimental sit-
uation realized in Refs.1,2will be considered in a future
publication).

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3
The c-level and the v-level are described by Hc and
Hv, respectively,
Hc =
?
σ
?
σ
ǫcˆ ncσ+ Ucˆ nc↑ˆ nc↓,
Hv =
ǫvˆ nvσ+ Uv(1 − ˆ nv↑)(1 − ˆ nv↓), (3)
where ˆ ncσ ≡ c†
operators c†
level or in the v-level, respectively. The parameters Uc
and Uvare Coulomb repulsion energies which have to be
paid if the c-level is occupied by two electrons or if the
v-level is empty, respectively. Since states with two holes
are very highly excited states independent of the value of
Uv(due to the band gap), the actual value of Uvhas no
influence on the results. The term
σcσ and ˆ nvσ ≡ v†
σand v†
σvσ. Here the Fermi
σcreate a spin-σ electron in the c-
HUexc= −
?
σ,σ′
Uexcˆ ncσ(1 − ˆ nvσ′) (4)
accounts for the exciton binding energy: the Coulomb
attraction between each electron in the c-level and each
hole in the v-level lowers the energy of the system by
Uexc.
The 2DEG formed in the WL is described by
HWL=
?
k,σ
ǫkl†
kσlkσ, (5)
where the Fermi operator l†
electron with wave vector k. The hybridization between
the c-level and the 2DEG is described by
kσcreates a delocalized spin-σ
Hc−WL=
?
k,σ
V
?
l†
kσcσ+ c†
σlkσ
?
, (6)
where the tunneling matrix element V is assumed to be
real and energy-independent. The hybridization between
the c-level and the 2DEG is henceforth parametrized by
Γ ≡ πρFV2, where ρF is the density of states (DOS)
of the 2DEG at EF; we assume a flat and normal-
ized DOS with bandwidth D. Since in the considered
experiments1,2, the mass of the (heavy) holes is signif-
icantly larger than the mass of the electrons, we neglect
the hybridization between the v-level and the 2DEG.
The last part of the Hamiltonian,
˜ Hpert=
?
k,σ
?
γkˆ ake−iωktc†
σvσ+ γ∗
kˆ a†
keiωktv†
σcσ
?
, (7)
describing the excitation (first term) and the annihila-
tion (second term) of excitons in the QD by photon ab-
sorption or photon emission, respectively, is considered
as a perturbation of the system. Here ˆ ak (ˆ a†
(creates) a photon of the laser field with wave vector k,
where the laser has the frequency ωk = c|k|. The cou-
pling is given by γk = e(¯ hωk/2ǫ0V )
k) destroys
1
2? ǫk·?D, with the
elementary charge e, the dielectric constant ǫ0, the quan-
tization volume V , the orientation of the laser field ? ǫk,
and the dipole moment of the QD transition?D. Here we
assume γ to be independent of k, since we consider the
laser to be approximately monochromatic. Note that for
simplicity, we neglect in the present study terms of the
form?
transitions between the 2DEG and the v-level. Such tran-
sitions will lead to Fano-type effects, which we choose not
to consider here, but will be the subject of future work.
Treating˜ Hpertperturbatively is valid as long as the op-
tical line width |γ|2A, where A is the density of states of
the photon field, is small compared to the Kondo tem-
perature TK(defined below), the smallest energy scale in
our studies: |γ|2A ≪ TK.
In the following considerations, the quantized nature
of the photon field will not play any role in our consider-
ations; to calculate emission and absorption line shapes,
all that we shall be concerned with are the matrix el-
ements of the operators c†
notation, we shall therefore henceforth write the pertur-
bation term simply as
kσγk
?
l†
kσvσ+ v†
σlkσ
?
, describing photon-induced
σvσ+ v†
σcσ. For simplicity of
Hpert= γ
?
σ
?c†
σvσ+ v†
σcσ
?. (8)
For the scenario of a local spinfull level coupled to a
Fermi sea, the Kondo effect occurs if the temperature
T < TKand the average occupancy of the local level is
roughly one, i. e. in our case ?ˆ nc? =?
as the ’local moment regime’ (LMR). Here TK is given
by
σ?ˆ ncσ? ≃ 1, known
TK≡ (UcΓ/2)1/2eπǫc(ǫc+Uc)/2ΓUc, (9)
see Ref.13. If T < TK, TKis the only relevant energy scale
in the problem. The Kondo effect introduces a quasi-
particle peak, the Kondo resonance, at the Fermi energy
EFin the local density of states (LDOS) Ac(ω),
Ac(ω) =
?
˜f,σ
?????˜f|c†
????˜f|cσ|˜G?
σ|˜G?
???
δ?ω + (E˜f− E˜G)??
2
δ?ω − (E˜f− E˜G)?
+
???
2
,(10)
see Fig. 3. Here |˜G? and |˜f? are eigenstates of H0 with
energy E˜Gand E˜f, respectively, where |˜G? is the ground
state. The LDOS Ac(ω) of the c-level is a well-known
function, which was calculated with the NRG, e. g., by
Costi et al.14, and has been studied frequently since.
In transport experiments at T < TK, the Kondo effect
causes the ’zero bias anomaly’, an enhanced conductance
due to the quasi particle peak at EF. Here we will inves-
tigate how the Kondo effect affects the absorption and
emission spectrum16.
Fig. 4(a) and Fig. 4(b) show examples of absorption
and emission processes to be studied in this paper. For

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4
-2
-1.5-1
-0.50 0.5
1 1.5
2
ω / Uc
0
0.2
0.4
0.6
0.8
1
Ac(ω) / (2/πΓ)
εc / Uc =-0.5
Γ / Uc = 0.06
FIG. 3: The normalized15local density of states Ac(ω) of the
c-level in the Kondo regime, with ǫc = −Uc/2. The Kondo
effect results in a resonance at the Fermi energy EF. There
are side peaks of the singly (doubly) occupied local level at
ω = ∓Uc/2 of a level width 2Γ.
both examples the QD is tuned such that the c-level is
initially singly occupied, ?ˆ nc? = 1, i. e. in the LMR and
therefore gives rise to a strongly correlated Kondo state
for T<
∼TK.
In the absorption process, Fig. 4(a), a photon excites
an electron from the v-level into the c-level. Due to the
exciton binding energy, the c-level is ’pulled down’ by
the value of Uexc. Thus the occupation of the c-level in
the final state can have any value between one and two,
depending on the value of Uexc relative to the charging
energy Ucof the c-level. If the final occupation is not in
the LMR, the Kondo-state is lost.
In the emission process, Fig. 4(b), an electron from
the c-level recombines with a hole in the v-level, thereby
emitting a photon. In contrast to the absorption process,
here the occupation of the c-level decreases since the ex-
citon binding energy is lost in the final state. Again the
Kondo state is lost if the final occupation is not in the
LMR.
III. METHOD
The absorption and emission spectra can be calculated
using Fermi’s Golden Rule for the transition rate out of
an initial state |i?, which is proportional to
αi(ω) =
2π
|γ|2
?
f
|?f|Hpert|i?|2δ (ω − (Ef− Ei)),(11)
where |i? and the possible final states |f? are eigenstates
of H0, cf. Eq. (1), with energy Eiand Ef, respectively.
No analytical method is known to calculate both the
eigenenergies of H0 and all matrix elements ?i|Hpert|f?
exactly. Here we calculate them with Wilson’s NRG
method10, a numerically essentially exact method17.
????????????
Kondo state
????????????
2DEG
????????????
?????? ??????
????? ?????
????? ?????
2DEG
????? ?????
??????????
????? ?????
????????????
Kondo state
????????????
2DEG
????????????
????????????
??????????
??????????
2DEG
????? ?????
??????????
εc
εv
εc
εv
εv
exc
U
εv
εc
Ucεc
QD
Uc
Uc
QDQD
QD
photon
EF
EF
Uexc
photon
a)
b)
FIG. 4:
tially (left hand side) singly occupied. (a) Photon absorption
process, inducing a transition from a state with no hole in the
v-level and a singly occupied c-level (Kondo state) to a state
with a v-level hole and a doubly occupied c-level (non-Kondo
state). As indicated, the occupation of the c-level in the final
state is determined by the value of ǫc− Uexc+ Uc relative to
EF. (b) Photon emission process, inducing a transition be-
tween a state with a v-level hole and a singly occupied c-level
(Kondo state) to a state without v-level hole and an empty
c-level.
QD (cf. Fig. 2) tuned such that the c-level is ini-
A. Block structure of Hamiltonian
Since H0commutes with ˆ nvσ, the number of holes in
the v-level is conserved. Thus it is convenient to write
the unperturbed Hamiltonian H0in the basis |Ψ?c+WL⊗
|Ψ?v, where |Ψ?c+WL denotes a product state of the c-
level and the 2DEG in the WL, and |Ψ?vdenotes a state
of the v-level. In this particular basis the unperturbed
Hamiltonian H0reads
H0=



|0?v
Hv0
0
0
0
| ↑?v
0
Hv↑
0
0
| ↓?v
0
0
Hv↓
0
| ↑↓?v
0
0
0
Hv↑↓


, (12)
where the Hamiltonians
Hv0 = Hc−WL+ HWL+ Hc−
?
σ
?
σ
2Uexcˆ ncσ+ Uv,
Hv↑ = Hc−WL+ HWL+ Hc−Uexcˆ ncσ+ ǫv,
Hv↑↓ = Hc−WL+ HWL+ Hc+ 2ǫv
(13)
act only on states |Ψ?c+WL. Since we have not included
a magnetic field in our model, Hv↑= Hv↓.

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5
Absorption (A) and emission (E) processes (see Fig. 4)
involve transitions between different blocks of Eq. (12):
A :
E :
|G? = |G?c+WL⊗ |↑↓?v→ |f? = |f?c+WL⊗ |σ?v
|g? = |g?c+WL⊗ |σ?v→ |f? = |f?c+WL⊗ |↑↓?v,
(14)
where |G? is the ground state of H0, |G?c+WLthe corre-
sponding ground state of Hv↑↓and |g?c+WLis the ground
state of Hvσ, with σ =↑,↓. For absorption, which is gov-
erned by c†
σvσ, |f? is a state of the block Hvσ. For emis-
sion, which is governed by v†
σcσ, |f? is a state of the block
Hv↑↓.
To calculate the absorption spectrum, cf. Fig. 4(a),
we insert |G? for |i? in Eq. (11). Then αG(ω) gives the
probability per unit time for the transition from |G? to
any final state |f? of Hvσ [containing one hole], equiva-
lent to the probability per unit time that a photon with
frequency ω is absorbed, which is the desired absorption
spectrum αG(ω), divided by |γ|2. The actual value of
γ is not important, since it does not affect the shape of
the absorption function, but only its height. The same
argument applies to the emission spectrum, cf. Fig. 4(b).
Here, one needs to insert |g? for |i? in Eq. (11).
To employ the NRG to calculate αi(ω) via Eq. (11),
one has to overcome a technical problem. The NRG is
a numerical iterative procedure, where the energy spec-
trum is truncated in each iteration [besides the first few
iterations]. In standard NRG implementations, transi-
tions from or to highly excited states can only be calcu-
lated qualitatively rather than quantitatively. In our case
we need to compute transitions to or from states of the
blocks Hvσ, see Eq. (14), which are highly excited since
they are separated by the order of the band gap from
states of Hv↑↓, see Section II. We solve this problem by
keeping the same number of states for the blocks Hvσ
and Hv↑↓ in each NRG iteration, which is in principle
the same as running two NRG iterations for both blocks
at the same time. This approach is similar to the one
used by Costi et al.18, who studied a problem analogous
to ours.
B. Limiting case of vanishing exction binding
energy (Uexc = 0)
To check the accuracy of the modified NRG method,
we begin by considering the limiting case of vanishing
exciton binding energy, Uexc= 0. We will show that for
this particular case the absorption and emission spectra
are related to the local spectral function.
For Uexc= 0 the v-level is decoupled from the c-level
and the 2DEG, see Eq. (4). When decomposing the states
in the same way as above, |Ψ? = |Ψ?c+WL⊗|Ψ?v, the total
energy can be written as a sum, E = Ec+WL+Ev. Thus,
using Eqs. (11) and (14), the absorption and emission
spectrum can be written as
αG(ω) = 2π
?
f,σ
?
?
f,σ
?
??c+WL?f|c†
ω −?Ef,c+WL− EG,c+WL
|c+WL?f|cσ|g?c+WL|2·
σ|G?c+WL
??2·
× δ
?−∆ω
?
,
αg(ω) = 2π
× δω −?Ef,c+WL− EG,c+WL
?+∆ω
?
.
(15)
Here ∆ω ≡ Ef,v−EG,v= −ǫvrepresents a constant shift.
To compare the LDOS with the absorption and emis-
sion spectrum, we divide it as Ac(ω) = A+
with
c(ω) + A−
c(ω),
A+
c(ω) =
?
f,σ
??c+WL?f|c†
× δ
?
?
f,σ
?
σ|G?c+WL
??2
ω −?Ef,c+WL− EG,c+WL
|c+WL?f|cσ|G?c+WL|2
??
for ω > 0,
A−
c(ω) =
× δω +?Ef,c+WL− EG,c+WL
??
for ω < 0.
(16)
Since the operator c†
the sum in Eq. (16) runs only over states |f? of Hv↑↓.
To compare Eqs. (15) with Eqs. (16), note that for
Uexc= 0 the blocks of the Hamiltonian (12) are degener-
ate (aside from a constant shift), Hvσ= Hv↑↓, and thus
|G?c+WL= |g?c+WL. Therefore
σdoes not change the state of the VB,
A :
E :
αG(ω) = 2πA+
αg(ω) = 2πA−
c(ω − ∆ω),
c(−ω − ∆ω).(17)
Thus, for Uexc= 0, we can calculate the absorption and
emission spectra in two different ways: firstly, with the
modified NRG procedure and secondly, via Eq. (17) with
Ac(ω) obtained from the NRG as well. We find an ex-
cellent agreement between both approaches, which serves
as a consistency check that the modified NRG works as
intended.
IV. RESULTS
In Section IIIB we showed that for Uexc= 0 the ab-
sorption or emission spectra are related to the LDOS.
Starting from this well-understood limiting case, let us
now study how the absorption and emission spectra be-
have upon increasing Uexc. We use the modified NRG
procedure, described in Section III, to calculate the ab-
sorption and emission spectra αi(ω) from Eq. (11). The
results are shown in Fig. 5. We see two striking be-
haviors: Firstly, there is a tremendous increase in peak
height for both the absorption and emission spectra. In