HAL Id: hal-00596639

https://hal.archives-ouvertes.fr/hal-00596639

Submitted on 30 May 2011

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-

entic research documents, whether they are pub-

lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diusion de documents

scientiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Breather mode in the many-electron dynamics of

semiconductor quantum wells

F. Haas, Giovanni Manfredi, P.K. Shukla, Paul-Antoine Hervieux

To cite this version:

F. Haas, Giovanni Manfredi, P.K. Shukla, Paul-Antoine Hervieux. Breather mode in the many-

electron dynamics of semiconductor quantum wells. Physical Review B: Condensed Matter and

Materials Physics (1998-2015), American Physical Society, 2009, 80 (7), pp.073301-1-073301-4.

�10.1103/PhysRevB.80.073301�. �hal-00596639�

Breather mode in the many-electron dynamics of semiconductor quantum wells

F. Haas,

1

G. Manfredi,

2

P. K. Shukla,

1

and P.-A. Hervieux

2

1

Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany

2

Institut de Physique et Chimie des Matériaux de Strasbourg, BP 43, F-67034 Strasbourg, France

共Received 22 July 2009; published 14 August 2009

兲

We demonstrate the existence of a breather mode in the self-consistent electron dynamics of a semiconductor

quantum well. A nonperturbative variational method based on quantum hydrodynamics is used to determine the

salient features of the electron breather mode. Numerical simulations of the time-dependent Wigner-Poisson or

Hartree equations are shown to be in excellent agreement with our analytical results. For asymmetric quantum

wells, a signature of the breather mode is observed in the dipole response, which can be detected by standard

optical means.

DOI: 10.1103/PhysRevB.80.073301 PACS number共s兲: 73.63.Hs, 73.43.Lp, 78.67.De

I. INTRODUCTION

The many-electron dynamics in nanoscale semiconductor

devices, such as quantum wells and quantum dots, has re-

cently attracted a great deal of interest, mainly in view of

possible applications to the growing ﬁeld of quantum

computing.

1

Particular attention has been devoted to inter-

subband transitions, which involve excitation frequencies of

the order of the terahertz.

2

On this time scale, various collec-

tive electronic modes can be excited. For instance, the elec-

tric dipole response is dominated by a strong resonance at

the effective plasmon frequency. This resonance 共known as

the Kohn mode兲

3

is characterized by rigid oscillations of the

electron gas, which, for perfectly parabolic conﬁnement, are

decoupled from the internal degrees of freedom.

In this Brief Report, we show the existence of a new

distinct resonance—a monopole or “breather” mode—which

corresponds to coherent oscillations of the size of the elec-

tron gas around a self-consistent equilibrium. Breather

modes have been described in many areas of physics, such as

nuclear matter

4,5

共where they are known as giant monopole

resonances兲 and ultracold atom dynamics.

6,7

In experiments

on metallic nanoparticles, monopole oscillations of the ionic

structure have been observed, which manifest themselves as

slow modulations of the surface plasmon.

8

However, to the

best of our knowledge, previous investigations have not ad-

dressed the features of the breather mode in the self-

consistent dynamics of a conﬁned electron gas. Although

quantum wells constitute a typical instance of such conﬁned

systems, the present approach should equally apply to metal

nanoparticles and carbon-based systems such as fullerenes.

II. MODEL

Because of the translational symmetry in the transverse

plane, the problem reduces to a one-dimensional 共1D兲 one in

the x direction.

9,10

To model the electron dynamics, we use a

self-consistent quantum hydrodynamic model 共QHM兲 that

was originally derived for quantum plasmas

11,12

and metallic

nanostructures.

13

In the QHM, the evolution of the electron

density n共x ,t兲 and mean velocity u共x , t兲 is governed by the

continuity and momentum equations

n

t

+

x

共nu兲 =0, 共1兲

u

t

+ u

u

x

=−

1

m

ⴱ

n

P

x

−

1

m

ⴱ

V

eff

x

+

ប

2

2m

ⴱ

2

x

冉

x

2

冑

n

冑

n

冊

, 共2兲

where m

ⴱ

is the effective electron mass, ប is the reduced

Planck constant, P共x,t兲 is the electron pressure, and V

eff

=V

conf

共x兲+ V

H

共x , t兲 is the effective potential, which is com-

posed of a conﬁning and a Hartree term. The Hartree poten-

tial obeys the Poisson equation, namely V

H

⬙

=−e

2

n/ , where e

is the magnitude of the electron charge and is the effective

dielectric permeability of the material. The term proportional

to ប

2

on the right-hand side of Eq. 共2兲 represents the quantum

force due to the so-called Bohm potential.

14

The above QHM can be derived from the self-consistent

Hartree equations

12

— or equivalently from the phase-space

Wigner-Poisson equations

15

—in the limit of long wave-

lengths compared to the Thomas-Fermi screening length. For

the sake of simplicity, we shall neglect exchange/correlation

corrections and assume Boltzmann statistics, which is a rea-

sonable approximation at moderate electron temperatures

T.

9,16

We also stress that the 1D model relies on the separa-

tion of the transverse and longitudinal directions, which may

be broken by collisional effects. However, such effects

should not be dominant on the fast time scales considered

here.

17

The pressure P共x,t兲 in Eq. 共2兲 must be related to the elec-

tron density n via an equation of state 共EOS兲 in order to close

our system of electron ﬂuid equations. We take a polytropic

relation P = n

¯

k

B

T共n/ n

¯

兲

␥

, where k

B

is the Boltzmann constant,

␥

=3 is the 1D polytropic exponent, and n

¯

is a mean electron

density. For a homogeneous system 共where n

¯

=n

0

兲, this EOS

correctly reproduces the Bohm-Gross dispersion relation

12

in

quantum plasmas. For the inhomogeneous electron gas con-

sidered here, the choice of n

¯

is subtler and will be discussed

later.

We assume a parabolic conﬁnement, with V

conf

=

1

2

0

2

m

ⴱ

x

2

, where the frequency

0

can be related to a ﬁcti-

tious homogeneous positive charge of density n

0

via the re-

lation

0

=共e

2

n

0

/ m

ⴱ

⑀

兲

1/2

. We then normalize time to

0

−1

;

space to L

0

=共k

B

T/ m

ⴱ

兲

1/2

/

0

; velocity to L

0

0

; energy to

PHYSICAL REVIEW B 80, 073301 共2009兲

1098-0121/2009/80共7兲/073301共4兲 ©2009 The American Physical Society073301-1

k

B

T; and the electron number density to n

0

. Quantum effects

are measured by the dimensionless parameter H = ប

0

/ k

B

T.

We shall use typical parameters that are representative of

semiconductor quantum wells:

10

the effective electron mass

and the effective dielectric permeability are, respectively,

m

ⴱ

=0.067 m

e

and =13

0

, the equilibrium density is n

0

=4.7 ⫻10

22

m

−3

, and the ﬁlling fraction n

¯

/ n

0

=0.5. These

values yield an effective plasmon energy ប

0

=8.62 meV, a

characteristic length L

0

=16.2 nm, a Fermi temperature T

F

=51.8 K, and a typical time scale

0

−1

=76 fs. An electron

temperature T= 200 K then corresponds to a value H=0.5.

III. LAGRANGIAN APPROACH

In order to derive a closed system of differential equations

describing the breather mode, we ﬁst express the quantum

hydrodynamical equations in a Lagrangian formalism. We

stress that this approach is not based on a perturbative ex-

pansion, and thus is not restricted to the linear regime. The

Lagrangian density corresponding to the system of Eqs. 共1兲

and 共2兲 reads as 共normalized units are used from now on兲

L =

1

2

冉

V

H

x

冊

2

− nV

H

− n

t

−

冕

n

W共n

⬘

兲dn

⬘

−

1

2

冉

n

冋

x

册

2

+

H

2

4n

冋

n

x

册

2

冊

− nV

conf

, 共3兲

where the independent ﬁelds are taken to be n,

, and V

H

.

The velocity ﬁeld follows from the auxiliary function

共x , t兲

through u =

/

x. The quantity W共n兲 in Eq. 共3兲 originates

from the pressure, W ⬅兰

n

dP

dn

⬘

dn

⬘

n

⬘

=共3/ 2兲共n/ n

¯

兲

2

. Taking the

variational derivatives of the action S =兰Ldxdt with respect

to n,

, and V

H

, we obtain the Eqs. 共1兲 and 共2兲, as well as the

Poisson equation for V

H

.

The existence of a pertinent variational formalism can be

used to derive approximate solutions via the time-dependent

Rayleigh-Ritz trial-function method.

18

For this purpose, we

assume the electron density to have a

GAUSSIAN proﬁle

n共x,t兲 =

A

exp

冋

−

共x − d兲

2

2

2

册

, 共4兲

where d共t兲 and

共t兲 are time-dependent functions that repre-

sent the center-of-mass 共dipole兲 and the spatial dispersion of

the electron gas, respectively. The constant A =N/

冑

2

,is

related to the total number of electrons in the well, N

=兰ndx. The above Ansatz is a natural one, because for a

negligible Hartree energy, V

eff

reduces to a harmonic oscilla-

tor potential.

The other ﬁelds to be inserted in the action functional are

and V

H

. The natural way to choose them is by requiring

that the continuity and Poisson equations are automatically

satisﬁed. The continuity equation is solved with n given by

Eq. 共4兲 together with u= d

˙

+共

˙

/

兲

, which leads to

=共

˙

/ 2

兲

2

+d

˙

, where

⬅x − d. An irrelevant gauge func-

tion was discarded in the calculation of

. The solution of the

Poisson equation with a

GAUSSIAN electron density is

V

H

=−A

e

−

2

/2

2

− A

冑

2

Erf

冉

冑

2

冊

+ const, 共5兲

where Erf is the error function. The integration constant is

chosen so that V

H

共⫾a兲= 0, with 2a being the total size of the

system, and letting a → ⬁ at the end of the calculation. As the

potential V

H

is not bounded, a divergence appears in the

Lagrangian density when integrated over space. However,

the divergent term does not depend on the dynamical vari-

ables d and

, so that it can be ignored.

Using the above Ansatz, one obtains the Lagrangian

L ⬅

1

冑

2

A

冕

Ldx =

d

˙

2

+

˙

2

2

−

d

2

+

2

2

+

冑

2

2

A

−

冑

3A

2

6n

¯

2

2

−

H

2

8

2

, 共6兲

which only depends on two degrees of freedom, namely the

dipole d and the variance

. The Euler-Lagrange equations

corresponding to the Lagrangian L read as

d

¨

+ d =0 共7兲

¨

+

=

冑

2A

2

+

冑

3A

2

3n

¯

2

3

+

H

2

4

3

. 共8兲

The quantum-well potential V

conf

manifests itself in the

harmonic forces on the left-hand side of both Eqs. 共7兲 and

共8兲. As expected, the equations for d and

decouple for

purely harmonic conﬁnement. Equation 共7兲 describes rigid

oscillations of the electron gas at the effective plasmonic

frequency, i.e., the Kohn mode.

3,19

Equation 共8兲 describes the

dynamics of the breather mode, which features coherent os-

cillations of the width of the electron density. The three

terms in the right-hand side of Eq. 共8兲 correspond to the

Coulomb repulsion 共Hartree term兲, the electron pressure, and

the quantum Bohm potential, respectively. The breather Eq.

共8兲 can be written as

¨

=−dU / d

, where U共

兲 is a pseudo-

potential deﬁned as U=

2

/ 2−

冑

2A

/ 2+

冑

3A

2

/ 共6n

¯

2

2

兲

+H

2

/ 共8

2

兲. From the shape of the pseudopotential 共Fig. 1兲,it

FIG. 1. Proﬁles of the pseudopotential U共

兲, for H = 0.5, A =0

共solid line兲 and H = 0.5, A =1 共dashed line兲. The ﬁxed points are

0

=1.03 共A=0兲 and

0

=1.43 共A=1兲.

BRIEF REPORTS PHYSICAL REVIEW B 80, 073301 共2009兲

073301-2

follows that

will always execute nonlinear oscillations

around a stable ﬁxed point

0

共A , H兲, which is a solution of

the algebraic equation U

⬘

共

0

兲=0.

So far, we have not speciﬁed the value of the average

density n

¯

that appears in the EOS, P = n

3

/ n

¯

2

, written in

normalized units. It is natural to assume that n

¯

takes some

value smaller than the peak density at equilibrium A /

0

. The

correct way to compute this value is to average the square of

the density using n itself, i.e., n

¯

2

⬅具n

2

典= 兰n

3

dx/ 兰ndx

=A

2

/ 共

冑

3

0

2

兲. A useful check can be performed by plugging

this expression into Eq. 共8兲 and neglecting the Hartree poten-

tial, which yields the equilibrium variance

0

=关1+共1

+H

2

兲

1/2

兴

1/2

/

冑

2. This expression displays the correct low- and

high-temperature limits for the quantum harmonic oscillator:

0

→ 1, for H→ 0; and

0

⬃

冑

H/ 2, for H Ⰷ 1.

With the above prescription for n

¯

, the pseudopotential be-

comes U =

2

/ 2−

冑

2A

/ 2+

0

2

/ 共2

2

兲+ H

2

/ 共8

2

兲. The fre-

quency ⍀ = ⍀共A , H兲 of the breather mode, corresponding to

the oscillations of

, can be obtained by linearizing the equa-

tion of motion 关Eq. 共8兲兴 in the vicinity of the stable ﬁxed

point of U共

兲. The dependence of the breather frequency

with A 共i.e., with the electron density兲 is shown in Fig. 2. For

A=0 共i.e., without the Coulomb interaction兲 the exact fre-

quency is ⍀ =2

0

. For ﬁnite A, the breather frequency de-

creases and approaches ⍀ =

0

, for A→ ⬁. The latter limit

can be understood by noting that for large A the electron

density becomes ﬂatter and ﬂatter, due to the strong Cou-

lomb repulsion. Thus, in the limit A → ⬁ we end up with a

uniform electron density exactly neutralized by the ion den-

sity background. For such a homogeneous system, the

Bohm-Gross dispersion relation holds, which for long wave-

lengths yields ⍀=

0

. Indeed, if one computes the average

density using the prescription used for the EOS, one obtains

具n典= 兰n

2

dx/ 兰ndx= A / 共

冑

2

0

兲→ 1, for A → ⬁ 共see the inset of

Fig. 2兲. Thus, as expected, for large Coulomb effects the

average electron density becomes equal to the ion back-

ground density.

IV. SIMULATIONS

In order to check the validity of the above results, we

performed numerical simulations of the Wigner-Poisson

共WP兲 system, which is equivalent to the time-dependent Har-

tree equations.

15

In the normalized variables, the Wigner

pseudoprobability distribution f共x,

v

,t兲 satisﬁes the evolution

equation

f

t

+

v

f

x

− i

冕

dx

⬘

d

v

⬘

2

H

2

␦

V

eff

e

i共

v

−

v

⬘

兲x

⬘

/H

f共x,

v

⬘

,t兲 =0, 共9兲

and is coupled to the Poisson equation. In Eq. 共9兲,

␦

V

eff

⬅V

eff

共x + x

⬘

/ 2,t兲− V

eff

共x − x

⬘

/ 2,t兲. It is important to note that

this is a microscopic quantum mean-ﬁeld model, much more

general than the hydrodynamic model on which our La-

grangian theory was based.

The initial condition used in the simulations is a quantum

canonical distribution for the harmonic oscillator at ﬁnite

temperature,

20

where the spatial width

0

has been adjusted

to the value obtained from the Lagrangian approach to ac-

count for the Coulomb repulsion. This is very close, but not

quite identical, to an exact equilibrium of the WP equations,

so that the width of the electron density starts to oscillate. We

then compute the evolution of the dispersion 具x

2

典

1/2

=共兰fx

2

dxd

v

/ 兰fdxd

v

兲

1/2

and its frequency spectrum, which

generally shows a sharp peak at a dominant frequency.

The results of the WP simulations are plotted in Fig. 2

共squares兲 and agree very well with the theoretical curve

based on the Lagrangian approach. The agreement slightly

deteriorates for larger values of A, because the electron den-

sity deviates from the

GAUSSIAN proﬁle due to strong Cou-

lomb repulsion. This is clearly visible in Fig. 3, where we

represent the evolved density proﬁles for two values of A.

For A= 1, the proﬁle is still approximately

GAUSSIAN,

whereas for A = 3 an intricate internal structure has devel-

oped. Nevertheless, even in this case, the error on the fre-

quency is still just over 3%.

Table I shows that the breather frequency depends weakly

on the parameter H 共and hence on the electron temperature兲.

Our theoretical results are in good agreement with the simu-

lations, except for H= 1. For this case, the frequency spec-

trum is particularly broad, denoting a signiﬁcant fragmenta-

tion of the resonance.

FIG. 2. The breather frequency ⍀ as a function of A, for H

=0.5. Solid line: analytical results from the Lagrangian method.

Squares: the Wigner-Poisson 共WP兲 simulations. The inset shows the

mean electron density 具n典=A /

冑

2

0

as a function of A.

FIG. 3. Solid lines: electron density proﬁles at

0

t= 150 from

the WP simulations for H=0.5 and two values of A. The dashed

lines represent

GAUSSIAN distributions with the same width and the

same area as the numerical curves.

BRIEF REPORTS PHYSICAL REVIEW B 80, 073301 共2009兲

073301-3

V. NONPARABOLIC WELLS

For a parabolic potential well, there is no coupling be-

tween the breather and dipole modes, which may render the

experimental detection of the breather mode by optical

means difﬁcult to realize in practice.

It can be shown that nonparabolic corrections do not in-

troduce any linear coupling if the conﬁning potential is sym-

metric. A more interesting situation arises for asymmetric

wells,

21

which we model by adding a small cubic term to the

conﬁning potential, V

cub

共x兲= 共K/ 3兲x

3

. The equations of mo-

tion are then

d

¨

+ d =−K共

2

+ d

2

兲, 共10兲

¨

+

=

冑

2A

2

+

冑

3A

2

3n

¯

2

3

+

H

2

4

3

−2K

d. 共11兲

Linearizing Eqs. 共10兲 and 共11兲 around the stable ﬁxed point

共d

0

,

0

兲, we indeed ﬁnd a coupling between the breather and

the dipole, with resonant frequencies ⍀

⫾

.

We model the coupling to the laser ﬁeld by instanta-

neously shifting the initial dipole of a small quantity, i.e.,

d共0兲= d

0

+d

˜

. Figure 4 shows a typical spectrum obtained

from the numerical solution of Eqs. 共10兲 and 共11兲, which is

proportional to the optical absorption spectrum commonly

measured in the experiments. The sharp peaks correspond to

the resonant frequencies ⍀

−

=0.97 and ⍀

+

=1.58, which are

rather close to those obtained for parabolic conﬁnement.

The breather mode can thus be triggered using a purely

dipolar excitation, and a clear signature of the breather fre-

quency can be observed in the optical absorption spectrum.

This opens the way to optically detecting the breather mode

by means of standard pump-probe experiments. Finally, the

methods used here could be readily extended to three-

dimensional nanostructures, and may ﬁnd applications in re-

lated areas such as quantum free-electron lasers.

22

ACKNOWLEDGMENTS

This work was partially supported by the Alexander von

Humboldt Foundation and by the Agence Nationale de la

Recherche.

1

P. Zoller et al., Eur. Phys. J. D 36, 203 共2005兲.

2

J. N. Heyman et al., Appl. Phys. Lett. 72, 644 共1998兲.

3

W. Kohn, Phys. Rev. 123, 1242 共1961兲.

4

M. N. Harakeh, K. van der Borg, T. Ishimatsu, H. P. Morsch, A.

van der Woude, and F. E. Bertrand, Phys. Rev. Lett. 38, 676

共1977兲.

5

D. H. Youngblood, C. M. Rozsa, J. M. Moss, D. R. Brown, and

J. D. Bronson, Phys. Rev. Lett. 39, 1188 共1977兲.

6

K. Góral, M. Brewczyk, and K. Rzazewski, Phys. Rev. A 67,

025601 共2003兲.

7

Y. Zhou and G. Huang, Phys. Rev. A 75, 023611 共2007兲.

8

H. Portales et al., J. Chem. Phys. 115, 3444 共2001兲.

9

M. Santer, B. Mehlig, and M. Moseler, Phys. Rev. Lett. 89,

286801 共2002兲.

10

H. O. Wijewardane and C. A. Ullrich, Appl. Phys. Lett. 84, 3984

共2004兲.

11

G. Manfredi and F. Haas, Phys. Rev. B 64, 075316 共2001兲.

12

G. Manfredi, Fields Inst. Commun. 46, 263 共2005兲.

13

N. Crouseilles, P. A. Hervieux, and G. Manfredi, Phys. Rev. B

78, 155412 共2008兲.

14

C. L. Gardner and C. Ringhofer, Phys. Rev. E 53, 157 共1996兲.

15

G. Manfredi and P.-A. Hervieux, Appl. Phys. Lett. 91, 061108

共2007兲.

16

G. M. Gusev, A. A. Quivy, T. E. Lamas, J. R. Leite, A. K.

Bakarov, A. I. Toropov, O. Estibals, and J. C. Portal, Phys. Rev.

B 65, 205316 共2002兲.

17

J. N. Heyman, K. Unterrainer, K. Craig, B. Galdrikian, M. S.

Sherwin, K. Campman, P. F. Hopkins, and A. C. Gossard, Phys.

Rev. Lett. 74, 2682 共1995兲.

18

M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippen-

bach, Phys. Rev. Lett. 95, 050403 共2005兲.

19

J. F. Dobson, Phys. Rev. Lett. 73, 2244 共1994兲.

20

K. Imre et al., J. Math. Phys. 8, 1097 共1967兲.

21

E. Rosencher and P. Bois, Phys. Rev. B 44, 11315 共1991兲;E.

Dupont, et al., IEEE J. Quantum Electron. 42, 1157 共2006兲.

22

N. Piovella, M. M. Cola, L. Volpe, A. Schiavi, and R. Bonifacio,

Phys. Rev. Lett. 100, 044801 共2008兲.

TABLE I. The breather frequency ⍀ and the equilibrium width

0

for A= 1 and for various values of H.

H

0

⍀ 共theory兲 ⍀ 共sim.兲

0.00 1.41 1.58 1.60

0.50 1.43 1.59 1.60

1.00 1.47 1.60 1.51

1.50 1.52 1.61 1.57

2.00 1.59 1.63 1.63

3.00 1.73 1.66 1.70

FIG. 4. The frequency spectra of the dipole 共solid line兲 and

breather 共dashed兲 modes, for an asymmetric well with A =1, H

=0.5, and K =−0.1.

BRIEF REPORTS PHYSICAL REVIEW B 80, 073301 共2009兲

073301-4