low dimensional systems and nanostructures dimensional... · low dimensional systems and...
TRANSCRIPT
2. Electronic states and quantum confined systems
Nerea Zabala
Fall 2007
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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:
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! !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
• Electrons in a 1d box: confinement
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
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 energy values
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 ]
Infinite wall potential
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V
0 L
V
0 L
Finite sample size drastically alters allowed energy levels!
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.
• Electrons in a 1d box: confinement
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!(k)
k
!(k)
Ene
rgy
Ene
rgy
k[!/L]
!(k)
k
k[!/L]
• 3D electrons gas. Filling states. The density of states
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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• 3D electrons gas. Filling states. The density of states
! !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
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Fermi energy Fermi momentum/velocity
• 3D electrons gas. Filling states. The density of states
•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
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•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
• 3D electrons gas. Filling states. The density of states
11Bohr
• 3D electrons gas. Filling states. The density of states
•Some values (Kittel)
12
•Some numbers for Cu
• 3D electrons gas. Filling states. The density of states
•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.
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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
• 3D electrons gas. Filling states. The density of states
14
!(")
!
• 3D electrons gas. Filling states. The density of states
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
• 3D electrons gas. Filling states. The density of states
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!
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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
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!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!
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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
$("! "n)
step or Heaviside function
Infinitely deep square well
(GaAs, D=10 nm)
, energy levels
Subbands,
transverse kinetic energy
Steplike DOS of a quasi 2D system
Parabolid density of states for unconfined
3D electrons
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n2D(!) =!
nfilled
(!n ! !)m
"!2
m
!!2
•Electrons in 1D
•Further confinement (in 2D), x and y : quantum wire
or electron wave guide
•Parabolic subbands
•Density (per unit length)
•As a function of energy:
(Use the dispersion relation for each subband)
!m,n,kz (!r) = "m,n(x, y)eikzz
!m,n(kz) = !m,n +(!kz)2
2m
N1D = 2!
m,nfilled
2L
2!
" kF
0dk = 2
!
m,nfilled
L
!kF
n1D = 2!
m,nfilled
kF
!
n1D = 2!
m,nfilled
"2m(!F ! !m,n)
!"
z
Lx, Ly ! !F
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•Electrons in 1D
•Density of states in 1D:
n1D = 2!
m,nfilled
"2m(!! !m,n)
!"
DOS of a quasi 1D system,
GaAs, 9!11 nm infinitle deep well
Parabolid density of states for unconfined
3D electrons
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!1D(") =!
m,n
(2m)1/2("! "m,n)!1/2
!#$("! "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
•Crystal structure and effective mass approximation. Semiconductors
26
•Exercise: Solve 1D periodic square-well potential model: Kronig-Penney
Kittel
•Crystal structure and effective mass approximation. Semiconductors
27
Nearly quadratic
•The effective mass approximation: m*
•Crystal structure and effective mass approximation. Semiconductors
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
•Crystal structure and effective mass approximation. Semiconductors
29
Effective Mass
from band dispersion
m!e =
h̄2
!2"/!k2
•Crystal structure and effective mass approximation. Semiconductors
30
Band structure of some semiconductors
Ge Si GaAs
•Crystal structure and effective mass approximation. Semiconductors
31
•Positive and negative effective mass:
Negative m*, holes
•Crystal structure and effective mass approximation. Semiconductors
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
•Crystal structure and effective mass approximation. Semiconductors
33
!(") =1
2#2
!2m!
!2
"3/2
"1/2
•Quantum size effects
•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
•Quantum size effects
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)
•Quantum size effects
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)
•Quantum size effects
37
•Find other examples of quantum size effects in the literature
•Quantum size effects
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
•Some useful confining potentials
•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
•Some useful confining potentials
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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