LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES dimensional... · LOW DIMENSIONAL SYSTEMS AND...

2. Electronic states and quantum confined systems

Nerea Zabala

Fall 2007

1

LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES

• From last lecture...

“Top-down” approach------update of solid state physics

• Not bad for many metals and doped semiconductors

• Shows qualitative features that hold true in detailed treatments.

• Successive approximations:

- “Free particles” –no external potential - Independent electron approximation - Assumes many-particle system can be modeled by starting from single-particle case. Beyond these approximations...Density Functional Theory (DFT), Quantum Montecarlo.....

First, find allowed single-particle states and energies....

2

•Contents:

• Electrons in solids: approaches

• Independent electrons

• Electrons in a 1d box: confinement

• 3D electrons gas. Filling states. The density of states

• 2D electron gas

• Electrons in 1D

•Quantum dot

• DOS in 3, 2,1D

• Crystal structure and effective mass approximation. Semiconductors

•Quantum size effects

• Some useful confining potentials

• Summary

LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES

2.Electronic states and quantum confined systems

3

• Electrons in solids: approaches

4

Metal and conduction electrons

Pseudopotentials,Jellium models

Ions smeared out into a positive

background

Pseudopotentials, jellium models

free atoms a solid

valence

electrons

nuclei

core

electrons

• Independent electrons

Time-independent SchrÖdinger equation:

Solve, consistent with boundary conditions.

, traveling plane waves with wave vector

Energies : No restriction for allowed values of k or E, continuous.

Free ! V= 0

Solutions, electron wave functions:

5

! !2

2m"2! + V (!r)! = "!

! = Aei!k·!r + Be!i!k·!r

!k

k = 2!/" and " is the wave length

!(k) =!2k2

2m

!(k)

k


1d particle in a box potential:

2

!!! EVm

=+"# )(2

22

r

Time-independent Schroedinger equation (TISE):

Must solve, consistent with boundary conditions. Free

particle means V = 0. Solutions are of the form

rkrk $#$+

iiBeAe~!

These are traveling waves with wavevector k.

|k| = 2%/&, where & is wavelength.

Energies are then

m

kE

2)(

22

=k

Dispersion relation

No restrictions here on allowed values of k or E.

E

k

Now try a 1d particle in a box potential:

'()

<<=

Lxx

LxV

;0,

0,0

0 L

V

Inside eht box, V= 0, so solution must look like superposition of plane waves, but " must vanish at walls.

3

Interior of box, V = 0, so solution must look

like superposition of plane waves, but ! must

vanish at walls. Answer:

nxLL

xn

"# sin

2)( =

3,2,1=n

So, allowed k values are

3,2,1, == nL

nk

"

meaning allowed energy values are

2

22222

22)(

mL

n

m

kE

"==k 0 L

V

0

20

40

60

80

100

0 2 4 6 8 10

k [" /L ]

0

20

40

60

80

100

0 2 4 6 8 10

0

20

40

60

80

100

Finite sample size drastically alters allowed energy levels!

0

20

40

60

80

100

E [h

2/2

mL

2]

Dispersion

relation plot

(E vs k)

Energy level diagram

(allowed E values)

0 2 4 6 8 10 0 2 4 6 8 10

k-space plot

(allowed k values)

E [h

2/2

mL

2]

E [h

2/2

mL

2]

E [h

2/2

mL

2]

k [" /L ]

k [" /L ] k [" /L ]

Allowed k values:

3




nxLL

xn

"# sin

2)( =

3,2,1=n


3,2,1, == nL

nk

"


2

22222

22)(

mL

n

m

kE

"==k 0 L

V

0

20

40

60

80

100

0 2 4 6 8 10

k [" /L ]

0

20

40

60

80

100

0 2 4 6 8 10

0

20

40

60

80

100


0

20

40

60

80

100

E [h

2/2

mL

2]

Dispersion

relation plot

(E vs k)


(allowed E values)

0 2 4 6 8 10 0 2 4 6 8 10

k-space plot

(allowed k values)

E [h

2/2

mL

2]

E [h

2/2

mL

2]

E [h

2/2

mL

2]

k [" /L ]

k [" /L ] k [" /L ]

Allowed energy values

3




nxLL

xn

"# sin

2)( =

3,2,1=n


3,2,1, == nL

nk

"


2

22222

22)(

mL

n

m

kE

"==k 0 L

V

0

20

40

60

80

100

0 2 4 6 8 10

k [" /L ]

0

20

40

60

80

100

0 2 4 6 8 10

0

20

40

60

80

100


0

20

40

60

80

100

E [h

2/2

mL

2]

Dispersion

relation plot

(E vs k)


(allowed E values)

0 2 4 6 8 10 0 2 4 6 8 10

k-space plot

(allowed k values)

E [h

2/2

mL

2]

E [h

2/2

mL

2]

E [h

2/2

mL

2]

k [" /L ]

k [" /L ] k [" /L ]

Infinite wall potential

6

V

0 L

V

0 L


Dispersion relation plot

Energy level diagram (allowed energy values)

k-space plot (allowed k values)

! Energy spectrum is now discrete rather than continuous.

! Allowed wavevectors are uniformly spaced in k-space with a separation of #/L.

! Sample size L determines spacing of allowed wavevectors and single-particle energies, with a smaller box giving larger spacings.


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!(k)

k

!(k)

Ene

rgy

Ene

rgy

k[!/L]

!(k)

k

k[!/L]


Non-interacting many-electron systems

• Interested in ground state of many-electron system.

• No interactions $many-body eigenstates should be linear combinations of products of single-particle eigenstates.

• They obey Pauli principle $correct total wavefunction should be antisymmetric under exchange of any two particles. Many-body eigenstates should be linear combinations of Slater determinants built out of single- particle eigenstates. Approximation (or shorthand):

start filling each single-particle state from the lowest energy, each with one spin-up and one spin-down electron.

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! !2

2m

!!2

!x2+

!2

!y2+

!2

!z2

"!!k("r) = #!k("r)!!k("r)

Confined in a cube of size L ! !!k = 0 at the boundaries

traveling waves and energies

•Boundary conditions:

!!k(!r) =1V

ei!k·!r

! allowed momentum values and standing waves

kx = ±2!nx

Lx, ky = ±2!ny

Ly, kz = ±2!nz

Lz

nx, ny, nz = 0, 1, 2, 3...

!!k =!2k2

2m

9

Fermi energy Fermi momentum/velocity


•Filling states:

Fermi sphere•Counting states:

!F =!2k2

F

2m

kF =1!!

2m!F

1 state! (2!)3

Vvolume in "k space

!

k

! V

(2!)3

"d"k

N = 2V

(2!)3

! kF

04!k2dk =

V

3!2k3

F

•Electron density n =N

V=

k3F

3!2

For spin

10

•Fermi energy and momentum increase with density of electrons!

•All single-particle states with below the Fermi energy are occupied at T=0. These states are called the Fermi Sea. The set of points in k-space that divides empty and full states is the Fermi surface.

•Exactly how the Fermi energy depends on density depends drastically on dimensionality.

!F =!2

2m(3"2)2/3n2/3 pF = !kF = !(3!2n)1/3

•Usually dimensionless parameter , radius of sphere containing one electron:

4!

3(rsa0)3 =

1n


11Bohr


•Some values (Kittel)

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•Some numbers for Cu


•Hall measurements in macroscopic samples yield the electron density of Cu (it can be also estimated from from interatomic distances) :

n = 8.47! 1028m!3

Calculate kF ,!F , "F and vF

Note that the Fermi velocity is less than a percent of the velocity of light, so no relativistic treatment is needed.

13

The density of states (DOS), The number of allowed single-particle states with energies

between E and E+dE,in an element of length/area/volume.

! From our expressions for n(E),nd is the spatial electron density in d dimensions

! From this definition, we can find the spacing of single particle levels in a piece of material!

! Higher DOS means levels are more closely-spaced.

!F =!2

2m(3"2)2/3n2/3 n(!) =

13"2

!2m

!2

"3/2

!3/2

!(") =dn

d"!(") =

12#2

!2m

!2

"3/2

"1/2


14

!(")

!


Some numbers...

15

! Estimate the energy level spacing in 1 cm3 of Na

Consider one valence electron per Na atom.We use the density and fermi energy from the table

n ! 2.65" 1022cm!3

!F = 3.2eV = 5.2! 10!19J

!("F ) ! 8" 1046J!1m!3

Energy level spacing: !! = 1/"(!F )V ! 10!41J The particle energy levels are continuous


Some numbers...

16

!Now suppose we have 1nm3 of Na, instead

A similar calculation yields the energy level spacing:

This is actually measurable!

!! ! 3 meV

At low temperatures, the individual electronic levels in a piece of metal can dominate many properties, something that doesn´t happen at macroscopic sizes

•3D electrons gas. Filling states. The density of states

The density of states of the free electron gas at finite temperature

T = 0

T != 0

! Fermi-Dirac distribution function (fermions)

T ! 0" f(!)! "(!F # !)

n =!

n(!)f(!, T )d!

17

f(!) =1

e(!!µ)/kT + 1

, step function

• 2D electron gas

D ! !F

4

2. Quantum Well States (QWS) and Quantum Size

Effects

Qualitative explanation…

yikxik

nyx eezzyx )(),,( !="

2

kk

2),,(

2

y

2

x

2

22 ++=

D

nkknE yx

!

zk

xk

yk

Electronic structure

in parabolic subbands

D

•Confinement in z direction, free in x and y

!n,!k!(!r) = "n(z)ei!k!·!r!

!k! = (kx, ky) Paraboloidal subbands

!F

!1

!2

!3

•Strictly 2D if !1 < !F < !2

•In infinite well confining potential (1D):

!n =!2

2m

!n"

D

"2

•Discrete continuous

18

!n("k) = !n +(!k!)2

2m

• 2D electron gas

•Filling states in 2D, occupation of subbands

•Periodic boundary conditions!allowed kx = ±2!nx

Lx, ky = ±2!ny

Ly

•Consider T=0 1 state! (2!)2

L2surface

•As a function of energy:

•Density (per surface)n2D =

!

nfilled

k2F

2!

19

N2D = 2!

nfilled

L2

(2!)2

" kF

02!kdk =

!

nfilled

12!

L2k2F

n2D =!

nfilled

(!F ! !n)m

"!2

•Density of states in 2D:

• 2D electron gas

!(") =dn

d" !2D(") =m

#!2

!

n

$("! "n)

step or Heaviside function

Infinitely deep square well

(GaAs, D=10 nm)

, energy levels

Subbands,

transverse kinetic energy

Steplike DOS of a quasi 2D system

Parabolid density of states for unconfined

3D electrons

20

n2D(!) =!

nfilled

(!n ! !)m

"!2

m

!!2

•Electrons in 1D

•Further confinement (in 2D), x and y : quantum wire

or electron wave guide

•Parabolic subbands

•Density (per unit length)

•As a function of energy:

(Use the dispersion relation for each subband)

!m,n,kz (!r) = "m,n(x, y)eikzz

!m,n(kz) = !m,n +(!kz)2

2m

N1D = 2!

m,nfilled

2L

2!

" kF

0dk = 2

!

m,nfilled

L

!kF

n1D = 2!

m,nfilled

kF

!

n1D = 2!

m,nfilled

"2m(!F ! !m,n)

!"

z

Lx, Ly ! !F

21

•Electrons in 1D

•Density of states in 1D:

n1D = 2!

m,nfilled

"2m(!! !m,n)

!"

DOS of a quasi 1D system,

GaAs, 9!11 nm infinitle deep well

Parabolid density of states for unconfined

3D electrons

22

!1D(") =!

m,n

(2m)1/2("! "m,n)!1/2

!#$("! "m,n)

•Electrons in 0D•Further confinement (in 3D): quantum dot or

artificial atom

•Discrete energy levels, as in atoms

•DOS is just asum of delta functions

discrete eigenenergies of the system

Lx, Ly, Lz ! !F

23

!0D(") = 2!

j

#("! "j)

!(")

!

• DOS in 3, 2,1D

Quantum Confinement and Dimensionality

24

•Crystal structure and effective mass approximation. Semiconductors

•Electrons in periodic potential (Nearly free electron model)

Electrons in a periodic potential

un!k(!r + !R) = un!k(!r),

Bloch's theorem:

!n!k("r) = exp(i"k · "r)un!k("r)n: band index

Lattice vector

Standing waves

25

!!k(!r + !R) = ei!k·!R!!k(!r)

ion

core

R

periodic

potential

Probability

density for standing

waves produced

V (!r) = V (!r + !R)

Bloch wave functions

Two wave vectors and the solutions of Schrödinger equation are related to each other. This leads to equal eigenvalues

and equal wave functions

!n("k) = !n("k + "K)

!n!k("r) = !n!k+ !K("r).

ei !K·!R = 1

Each energy branch has the same period as the reciprocal lattice. As the functions are periodic, they have maxima and minima which determine the width of the bands.

The wave vector k can always be chosen in a way to belong to the first Brillouin zone because !k! = !K + !k

Reciprocal lattice vector


26

•Exercise: Solve 1D periodic square-well potential model: Kronig-Penney

Kittel


27

Nearly quadratic

•The effective mass approximation: m*


28

First Brillouin zone

Gap opening

(valid for low electron momenta)

•Occupation of energy bands: type of materials

InsulatorMetal or semimetal

(if band overlap small)

Metal

Sketch


29

Effective Mass

from band dispersion

m!e =

h̄2

!2"/!k2


30

Band structure of some semiconductors

Ge Si GaAs


31

•Positive and negative effective mass:

Negative m*, holes


32

•In summary: we can consider also semiconductors including m* in the Schrodinger

equation.

Then, for the density of states in 0,1,2,3D one can extrapolate the results

!0D = 2!

j

"(#! #j)

!2D(") =m!

#!2

!

n

$("! "n)

!1D =!

m,n

(2m!)1/2("! "m,n)"1/2

!#$("! "m,n)

•Signature of dimensionality


33

!(") =1

2#2

!2m!

!2

"3/2

"1/2


•Confinement !Discrete Quantum Well States!Oscillations in the physical properties (as energy,

Fermi level....) as a function of size

34

Condition of QWS existence

,...3,2,1 2 === nnDL !

,...3,2,1 2

2

22

== nD

nE

"

Energy of system

(per electron)

#F


35

•An example: magic heights of Pb islands on Cu(111) studied with STS

Covered area

Courtesy: Rodolfo Miranda

Number of islandsBuilding island heights

histograms

R. Otero, A. L. Vázquez de Parga and R. Miranda PRB 66, 115401 (2002)


36

The model seems to give reasonable results

(fortunately)

1D potential model (self-consistent calculation)

Island stability (II)

• Very good agreement with the

experiments.

• Shell and supershell structure like

for nanowires and clusters.

• It can now be observed indirectly

in the experiments.

?

Ogando, Zabala, Chulkov, Puska, Phys. Rev.B 69, 153410 (2004)


37

•Find other examples of quantum size effects in the literature


38

•Some useful confining potentials

•Square well of finite depth -quantum wells, thin slabs-

•Parabolic well -quantum dots-

•Triangular well -heterojunctions-

•Cylindrical well -quantum corral, metallic nanowire-

•Spherical well -clusters, quantum dots-

See for example J.H. Davies

Also in next lectures

39


•Triangular well -heterojunctions-

•Confining potential

(z perpendicular to 2DEG)

Introduce dimensionless variable

equation:

solutions with boundary conditions (finite at infinity an zero at z=0)

40


solutions

with

Subbands

m, effective mass41

Airy functions

•Summary

• In another course: phonons, plasmons, excitations in low dimensions

• Electron interactions in low D have not been considered but they may very important in many problems, for example to explain superconductivity, magnetism, quantum Hall effect... In low D screening, response etc... is different

• The effects of confinement have been studied qualitatively starting from non-interacting electrons confined in potential wells.

•Confinement produces discreteness of allowed electron energies giving rise to quantum size effects.

•The characteristic pattern of the density of states in one, two and three dimensions has been obtained.

•The conclusions are valid for metals and semiconductors, when the effective mass approximation is considered.

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