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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