Page 1
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-
Page 2
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).
Page 3
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
Page 4
4
-2
-1.5 -1
-0.500.5
11.5
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
QD QD
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↓.
Page 5
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
Page 6
6
-1.4 -0.7
00.7
(ω - εv) / Uc
0.0
0.3
0.5
0.7
-1.4 -0.7
00.7
(ω + εv) / Uc
100
10000
α(ω)
-1
0
1
23
45
(ω - ω0) / TK
AbsorptionEmission Emission
Uexc / Uc =
εc / Uc = -0.5
Uv / Uc = 1.0
Γ / Uc = 0.15
FIG. 5: Absorption and emission spectra for different values of Uexc (ǫc/Uc = −0.5,Uv/Uc = 1.0 and Γ/Uc = 0.15). An increase
in Uexc results in an increase in the slope of the divergence at the threshold-energy ω0 and in a monotonic shift of ω0 of the
absorption and emission spectra. The right panel shows the divergence of the emission spectrum at the threshold energy ω0
normalized to TK [TK/Uc = 0.020, extracted from Eq. (9)].
fact, we find that the spectra diverge at the threshold en-
ergy ω0, the energy below which no photon is absorbed or
emitted, respectively, in close analogy to the well-known
X-ray edge absorption problem. Secondly, the threshold
energy ω0shows a marked, monotonic shift as a function
of the exciton binding energy Uexc.
A.Exponent of the power-law divergence
Let us first study the divergence of the spectral peak at
threshold. For any Uexc?= 0, we find a power-law diver-
gence for both the absorption and the emission spectra19,
for energies ω near the threshold energy ω0:
α(ω) ∼
?
1
ω − ω0
?β
, ω → ω0.(18)
Examples of this behavior are shown in Fig. 6, where
the absorption spectrum is plotted for several different
values of Uexc on a double logarithmic plot leading to
nice straight lines for energies (ω − ω0) < TK, i. e.
in the regime where Kondo correlations can build up.
The slope of such a line yields the exponent β.
markably, we find that the exponent so determined de-
pends only on the change ∆n in the local occupation,
∆n ≡ ±(?nc?f− ?nc?i) (’+’ for absorption, ’−’ for emis-
sion) , to be called ’screening charge’, and obeys the fol-
lowing relation20(who’s origin will be discussed below):
Re-
0.0010.01 0.1
1
10
(ω-ω0) / TK
1000
10000
αG(ω)
0.0
0.2
0.4
0.6
0.8
1.0
Uexc/Uc=
εc =-0.5 Uc
Uv = 1.0 Uc
Γ = 0.3 Uc
FIG. 6: Asymptotic behavior (ω → ω0) of the shifted absorp-
tion spectra normalized to TK [TK/Uc = 0.10, extracted from
Eq. (9)]. For energies (ω − ω0) < TK, in the regime where
Kondo correlation build up, we find the power-law behavior
as predicted in Eq. (18). The exponent β increases as Uexc is
increased. The lower bound of αG(ω) is set by the number
of NRG iterations (here: (ω − ω0) ≈ 10−3TK). The asymp-
totic behavior of the emission spectra (not shown here) looks
identical to that of the absorption spectra.
β = ∆n −(∆n)2
2
.(19)
Since for ω → ω0the relevant transitions in the case of
absorption and emission are |G? → |g? and |g? → |G?, re-
Page 7
7
0 0.2 0.40.60.8
1
∆n
0
0.1
0.2
0.3
0.4
0.5
β
y = x - x2/2
εc =-0.5 Uc
εc =-0.3 Uc
εc =-0.1 Uc
εc = 0.1 Uc
εc = 0.3 Uc
εc = 0.5 Uc
Uv = 1.0 Uc
Γ = 0.3 Uc
FIG. 7:
tracted from the NRG results for different values of ǫc (sym-
bols) coincides very well with the formula for β given by Eq.
(19) (solid line), indicating that β is fully determined by ∆n.
Here ∆n has been varied between 0 and ∼ 0.8 by varying Uexc
between 0Uc and 1Uc in steps of 0.1Uc.
The exponent β of the power-law divergence ex-
spectively, ∆n is the same for both types of transitions,
implying the same exponent β for both absorption and
emission for a given choice of parameters. In particular,
for the absorption spectra whose asymptotic behavior is
shown in Fig. 6, we have (for ω → ω0) ?nc?i = ?nc?G
and ?nc?f= ?nc?gwhere ?nc?gand ?nc?Gdenote the av-
erage occupation of the states |g? and |G?, respectively,
see Eq. (14). At Uexc= 0, we have ∆n = 0, since there
|G?c+WL = |g?c+WL, see Section III. As Uexc increase,
?nc?gand thus ∆n also increases, since the Coulomb at-
traction between the hole and the electrons in the c-level
pulls down the c-level to an effective value ˜ ǫc= ǫc−Uexc
[note that |g? is an eigenstate of Hvσ, whereas |G? is an
eigenstate of Hv↑↓and thus independent of Uexc].
We have extracted the exponent β for several different
values of ǫc. For each value of ǫc we have varied ∆n
between 0 and ∼ 0.8 by varying Uexcbetween 0 and Uc.
The results are shown in Fig. 7. We find a very good
agreement between the results extracted from the NRG
and the universal behavior predicted in Eq. (19): all data
points nicely collapse onto the curve predicted by Eq.
(19).
The numerical results presented in Fig. 5 should thus
be interpreted in the following way: for Uexc= 0 we have
∆n = 0 and thus β = 0, which gives a finite height of the
absorption and the emission spectrum at the threshold
[in fact the height is 2π times the height of the corre-
sponding LDOS, see Eq. (17)]. As soon as we choose
values of Uexc > 0, we find β > 0, leading to an infi-
nite height of the absorption and the emission spectral
peaks. Of course, the infinite peak height is not resolved
by our numerical data, for which α(ω − ω0) is always fi-
nite. However, with increasing Uexcthe exponent β also
increases, resulting in a steeper slope of the peak at the
threshold, which leads to a higher peak in the numerical
results.
An explanation for the universal behavior given by Eq.
(19) can be given by studying the analogy between the
physics presented in this paper and the well-known X-
ray edge absorption problem. A result analogous to Eq.
(18) was found by Schotte and Schotte21, where the ab-
sorption spectrum was studied for the X-ray edge prob-
lem. [In Ref.21all results are presented for the absorp-
tion spectrum. However, by rewriting Eq. (7) of Ref.21
for emission, their results can be applied to the emission
spectrum as well. Keeping that in mind, we will focus
only on the absorption spectrum in the following, but
the argumentation can easily be applied to the emission
spectrum as well.] In Ref.21, Fermi-liquid arguments re-
lating phase shifts and local screening charges are used
to derive an expression for the exponent β, namely
β = 1 −
?
σ
N2
σ,(20)
where Nσis the ’effective’ number of spin-σ electrons [not
necessarily an integer] which flow away from the local
level in the absorption process. Eq. (20) is known as
“Hopfield’s Rule of Thumb”22. We can use this result
to analyze our absorption spectra, too, since the system
behaves like a Fermi liquid for T = 0. Thus arguments
based on the relation between phase shifts and screening
charges do apply. In experiments, we expect to find the
behavior (20) for A|γ|2,T ≪ ω − ω0≪ TK, where A|γ|2
is the optical line width.
To see that Eqs. (19) and (20) are equivalent, we will
now analyze the absorption process |G? → |g? [relevant
absorption process at threshold] and count the charges
Nσ. It is helpful to consider an example for the process
|G? → |g?, shown in Fig. 8, where the initial state |G? is
the strongly correlated Kondo ground state with a singly
occupied c-level. The state |G? is a coherent superpo-
sition of states with different occupation of the c-level,
where the contribution of the state with empty c-level is
small but finite [depicted in Fig. 8(b)]. If the operator
c†
σvσ [the part of Hpert corresponding to absorption] is
applied to |G?, this contribution results in a state with
one hole, a singly occupied c-level and one extra spin-σ
electron in the conduction band, illustrated in blue in Fig.
8(a). The latter subsequently flows away from the QD,
making a contribution of +1 to Nσ. This contribution
to the final state c†
σvσ|G? also is a part of the state |g?,
which likewise has contributions from states with empty,
singly and doubly occupied c-level [Fig. 8(c)]. The weight
of the contribution with singly occupied c-level to |g? de-
pends on Uexc: the Coulomb attraction of the hole in
|g? pulls down the c-level to the effective value ˜ ǫcresult-
ing in an increase of the average occupation ?nc?gof the
c-level by ∆n compared to ?nc?G, see above. As Uexcis
increased, the charge ∆n [which screens the Coulomb po-
tential of the hole] increases and thus the relative weight
of the contribution to |g? with doubly occupied c-level
Page 8
8
FIG. 8: (a) Illustration of an example for the absorption process |G? → |g?, the relevant absorption process for energies ω
close to the threshold ω0 for the case ǫc = −Uc/2, for which the average occupations of the v- and c-levels are ?nv?G = 2 and
?nc?G = 1. Figs. (b) and (c) schematically depict the initial and final states |G? and |g?, respectively. Both |G? and |g? are
coherent superpositions of very many different components, whose c-level can be either empty, singly or doubly occupied; for
|G?, the v-level is doubly occupied, and the components with empty and doubly occupied c-level have a very small but finite
weight. For |g?, the v-level is singly occupied, and since the electron-hole attraction lowers the energy of each c-level electron by
Uexc, the weight of the components with singly or doubly occupied c-level depends on Uexc. In the absorbtion process depicted
in (a) and described by the matrix element ?g|c†
the c-level (middle panel). This leads to a transition from a |G?-component with empty c-level and an extra electron in the
conduction band (which subsequently flows away from the QD, making a contribution of +1 to Nσ), to a state with one v-hole
and a singly occupied c-level, which in turn is a component of |g? [see (c)]. The Coulomb attraction in the state |g? between
the v-hole and the c-level electrons pulls down the c-level from ǫc to an effective value ˜ ǫc = ǫc− Uexc, resulting in an increase
of the average c-level occupation ?nc?g by ∆n compared to ?nc?G. The screening charge ∆n = ?nc?g−?nc?G flows towards the
QD, thus making a contribution ∆n/2 to both Nσ and N¯ σ (with ¯ σ = {↓,↑} for σ = {↑,↓}).
σvσ|G?, a photon causes the promotion of a spin-σ electron from the v-level to
increases, too, whereas the relative weight of the state
with singly occupied c-level decreases.
charge ∆n flows towards the QD, making equal contri-
butions −∆n/2 to both Nσ and N¯ σ. (with ¯ σ = {↓,↑}
for σ = {↑,↓}).[Another possibility for a transition
form |G? to |g? [not depicted in Fig. 8(a)] starts from
a component of |G? with a singly occupied c-level and
ends up in a contribution of |g? with doubly occupied
c-level. One obtains the same results for Nσ and N¯ σ if
one argues that one unit of charge with spin σ has to
leave the doubly occupied c-level and the charge ∆n has
to flow into the c-level to reach the average occupation
?nc?g= ?nc?G+ ∆n.] Collecting all contributions to Nσ
and N¯ σ, we find Nσ = 1 − ∆n/2 and N¯ σ = −∆n/2,
which, when inserted into (20), yields Eq. (19).
The screening
A similar argument has been used in18,23, where the
local spectral function of the Anderson was studied.
B. Behavior of the threshold energy ω0
Let us now consider the second effect observed in Fig.
5, the monotonic shift of the threshold energy ω0. The
threshold energy for both absorption and emission is
given by ω0 = Eg− EG, where H0|G? = EG|G? and
H0|g? = Eg|g?, as explained in Section III. The shift in
ω0 can be understood by considering a mean-field esti-
mate of the relevant energies EGand Eg:
EG ≃ 2ǫv+ ǫc?nc?G+ Uc
?1
2nc
2nc
?2
G,
Eg ≃ ǫv+ ǫc?nc?g+ Uc
?1
?2
g− Uexc?nc?g. (21)
Here a correlation energy of the order of TKhas been ne-
glected. The average occupations ?nc?Gand ?nc?gcan be
calculated by NRG. Eq. (21) allows for a rough estimate
of the threshold energy ω0:
ω0 = Eg− EG
Page 9
9
0 0.5
1 1.5
Uexc / Uc
-1.5
-1
-0.5
0
(ω0 - εv) / Uc
0
0.5
1
1.5
2
〈nc〉g
εc = 0.357 Uc
Uv = 0.714 Uc
Γ = 0.057 Uc
NRG
Mean-field
III III
FIG. 9: Behavior of the threshold energy as a function of
Uexc. Upper panel: average occupation ?nc?g of the c-level
for the state |g?, see Eq. (14). Three distinct regimes can be
identified: empty orbital (I), LMR (II) and full orbital (III)
regime, where the c-level is empty, singly or doubly occupied,
respectively. Since the Coulomb attraction between a hole in
the v-level and the electrons in the c-level ’pulls down’ the
c-level [˜ ǫc = ǫc − Uexc], an increase in Uexc bears the same
effect for ?nc?g as a decrease in ǫc. Lower panel: threshold
energy ω0 versus Uexc extracted from the NRG results (solid)
and obtained from the mean-field estimate (circles), Eq. (22),
where ?nc?g from the upper panel has been used.
≃ −ǫv+ ǫc(?nc?g− ?nc?G)
+1
4Uc
??nc?2
g− ?nc?2
G
?− Uexc?nc?g.(22)
The results for ω0shown in Fig. 9 reveal a good agree-
ment between the threshold energy extracted from the
absorption and emission spectra calculated with NRG
(solid line) and the estimation given by Eq. (22). In the
latter approach ?nc?Gand ?nc?gwere determined via the
NRG, see top panel of Fig. 9 [note that ?nc?Gdoes not
depend on Uexcand that ?nc?G= ?nc?gfor Uexc=0]. We
find a linear behavior of ω0as a function of Uexcfor those
values of Uexcwhere ?nc?gstays approximately constant.
For this purpose three regions of constant occupation can
be identified, region I (?nc?g∼ 0), II (?nc?g∼ 1) and III
(?nc?g ∼ 2). As expected by considering the last term
in Eq. (22), we observe the slope of ω0(Uexc) to be 0 in
region I, to be −1 in region II and to be −2 in region III,
respectively.
The cross-over regions (dotted lines in Fig. 9), where
?nc?gchanges between 0 and 1 (I → II) or between 1 and
2 (II → III), on the other hand, show non-trivial behavior
as a function of ω0. In these regions the terms in Eq. (9)
compete with each other, which explains the non-linear
behavior (since in these regions ?nc?gitself is a function
of Uexc, too).
V.CONCLUSIONS
Motivated by experimental studies of excitons in QDs
coupled to a wetting layer1,2, the aim of this paper was to
calculate the absorption and emission spectra of a QD in
the strongly correlated Kondo ground-state. We studied
an extended Anderson model, including a local valence
band level and a local conduction band level which is
coupled to a Fermi-sea (2DEG), see Section II. For the
academic limiting case of a vanishing exciton binding en-
ergy, Uexc= 0, we could relate the absorption and emis-
sion spectrum to the well known local density of states of
the local conduction band, see Section III. Starting from
this limiting case, we used the NRG to study the spectra
for arbitrary values of Uexc. Our main results are summa-
rized in Fig. 5 which shows two rather dramatic features:
Firstly, an increase in the slope of the divergence of the
absorption and emission spectrum as Uexc is increased.
In fact, the spectra show a power-law divergence at the
threshold energy. Remarkably, the exponent of the diver-
gence depends only on ∆n, the difference in occupation
of the local conduction band level between the inital and
final states for transitions at the threshold. We showed
that the universal behavior of the exponent can be ex-
plained by considering the X-ray edge problem, which
stands in close analogy to the physics presented in this
paper. Secondly, increasing Uexcproduces a marked shift
of the threshold energy, which can be understood rather
simply on a mean field level.
In the present paper we considered, for simplicity,
a model which contains only a single local conduc-
tion band level. However, in the present generation of
experiments1,2, the wetting layer forms a 2DEG only for
values of the gate voltage Vgfor which several local con-
duction band levels are occupied (not only one, as as-
sumed in the present paper). Nevertheless, we expect24,
that future generations of samples could be produced for
which the assumptions of our model, namely one c-level
with presence of a 2DEG, are fulfilled. Of course it would
be very interesting to generalize our considerations to
more general models, including several local conduction
band levels.
Acknowledgments
We wish to thank R. Bulla, T. Costi, A. Govorov, A.
H¨ ogele, K. Karrai, M. Kroner, A. Rosch P. Schmitteck-
ert and S. Seidel for helpfull discussions. This work was
supported by the DFG under the SFB 631 and under the
CFN, ’Spintronics’ RT Network of the EC RTN2-2001-
00440. L.B. acknowledges support by Hungarian Grants
No. OTKA D048665 and T048782.
Page 10
10
1R. J. Warburton, C. Sch¨ aflein, D. Haft, F. Bickel, A. Lorke,
K. Karrai, J. M. Garcia, W. Schoenfeld, and P. M. Petroff,
Nature 405, 926 (2000).
2K. Karrai, R. J. Warburton, C. Schulhauser, A. H¨ ogele,
B. Urbaszek, E. J. McGhee, A. O. Govorov, J. M. Garcia,
B. D. Geradot, and P. M. Petroff, Nature 427, 135 (2004).
3D.Goldhaber-Gordon,H.
D. Abusch-Magder, U. Meirav, and M. A. Kastner, Na-
ture 391, 156 (1998).
4S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen-
hoven, Sience 281, 540 (1998).
5T. V. Shahbazyan, I. E. Perakis, and M. E. Raikh, Phys.
Rev. Lett. 84, 5896 (2000).
6K. Kikoin and Y. Avishai, Phys. Rev. B 62, 4647 (2000).
7Y. Chen and J. Kroha, Phys. Rev. B 46, 1332 (1992).
8A. O. Govorov, K. Karrai, and R. J. Warburton, Phys.
Rev. B. 67, 241307 (2003).
9P. W. Anderson, Phys. Rev. 124, 41 (1961).
10K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
11P. Nozieres and C. T. de Dominicis, Phys. Rev. 178, 1097
(1969).
12R. J. Warburton, B. T. Miller, C. S. D¨ urr, C. B¨ odefeld,
K. Karrai, J. P. Kotthaus, G. Medeiros-Ribeiro, P. M.
Petroff, and S. Huant, Phys. Rev. B 58, 16221 (1998).
13A. M. Tsvelick and P. B. Wiegmann, Adv. Phys. 32, 453
(1983).
14T. A. Costi, A. C. Hewson, and V. Zlatic, J. Phys. Con-
dens. Matter 6, 2519 (1994).
15D. C. Langreth, Phys. Rev. 150, 516 (1966).
16For our numerical calculations, we use |ǫc|,Uc ≪ D, but
this is not an essential assumption to realize the Kondo
Shtrikman,D. Mahalu,
effect, which can also occur for Uc ≫ D, as long as ?ˆ nc? ≃
1.
17To obtain a continuous spectrum αi(ω), the δ-functions in
Eq. (11), associated with the sum over discrete final states,
have to be properly ’broadened’, see Refs.14,25.
18T. A. Costi, P. Schmitteckert, J. Kroha, and P. W¨ olfle,
Phys. Rev. Lett. 73, 1275 (1994).
19One might expect that the absorption and emission spec-
trum are related by the particle-hole symmetry. However,
the Hamitonian H0, Eq. (2), has no particle-hole symme-
try: The operators J+≡ c†
?J+,J−?
is proportional to the charge operator and thus commutes
with the Hamiltonian. Particle-hole symmetry exists if J+
commutes with the Hamiltonian. In our case, however,
?H0,J+?
Hamiltonian can only be particle-hole symmetric for one
particular choice of nvσ, i. e. different blocks of H0, see Eq.
(12) cannot be particle-hole symmetric at the same time.
20The fact that the exponent depends only on the local
charge of the QD is an artefact of the particle-hole sym-
metry in the WL. In general a sum over the charges in the
WL will also contribute to the exponent.
21K. D. Schotte and U. Schotte, Phys. Rev. 185, 509 (1969).
22J. J. Hopfield, Comments Solid State Phys. 2, 40 (1969).
23T. A. Costi, J. Kroha, and P. W¨ olfle, Phys. Rev. B 53,
1850 (1996).
24K. Karrai, private communications.
25R. Bulla, T. A. Costi, and D. Vollhardt, Phys. Rev. B 64,
045103 (2001).
↑c†
↓, J−= (J+)†and Jz =
form a SU(2) algebra, where Jz = c†
↑c↑+ c†
↓c↓
= (−2ǫc−Uc+2Uexc?
σ(1−nvσ))c†
↑c†
↓. Thus the
Download full-text