Quantum Confinement in Nanostructures

Confined in:

1 Direction: Quantum well (thin film)

Two-dimensional electrons

2 Directions: Quantum wire

One-dimensional electrons

3 Directions: Quantum dot

Zero-dimensional electrons

Each confinement direction converts a continuous k in a discrete quantum number n.

kx

nz

ny

ny

nz

nx

kx

ky

nz

N atomic layers with the spacing a = d/n

N quantized states with kn ≈ n /d ( n = 1,…,N )

Quantization in a Thin Crystal

An energy band with continuous k

is quantized into N discrete points kn

in a thin film with N atomic layers.

n = 2d / n

kn = 2 / n = n

/d

d

E

0 /a/d

EFermi

EVacuum

Photoemission

Inverse Photoemission

Electron Scattering

k= zone

boundary

N atomic layers with spacing a = d/n :

N quantized states with kn ≈ N /d

Quantization in Thin Graphite FilmsE

0 /a/d

EFermi

EVacuum

Photoemission

Lect. 7b, Slide 11

k

1 layer = graphene

2 layers

3 layers

4 layers

layers = graphite

Quantum Well States

in Thin Films

discrete for small N

becoming continuous for N

Paggel et al.Science 283, 1709 (1999)

10

16

16

16

16

16

16

13

14

14

11.5

13

1413

14

h (eV)Ag/Fe(100)

Binding Energy (eV)

012

Photo

emiss

ion In

tensity

(arb.

units)

1

2

3

4

5

6

7

8

9

10

11

13

14

15

12

N

Temperature (K)

100 200 300

Line W

idth (e

V)

0.1

0.2

0.3

(N, n')

(3, 1)(7, 2)(12, 3)(13, 3)

(2, 1)

1

3

2

4

10

16

16

16

16

16

16

13

14

14

11.5

13

1413

14

h (eV)Ag/Fe(100)

Binding Energy (eV)

012

Photo

emiss

ion In

tensity

(arb.

units)

1

2

3

4

5

6

7

8

9

10

11

13

14

15

12

N

Temperature (K)

100 200 300

Line W

idth (e

V)

0.1

0.2

0.3

(N, n')

(3, 1)(7, 2)(12, 3)(13, 3)

(2, 1)

1

3

2

4

Periodic Fermi level crossing of quantum well states with

increasing thickness

Counting Quantum Well States

Number of monolayers N

n

Thickness N (ML)0510152025Wo

rk F

un

ctio

n (

eV)

4.34.4

Bin

din

g E

ner

gy

(eV

) 01212345678(a) Quantum Well States for Ag/Fe(100)

(b)n

Kawakami et al.Nature 398, 132 (1999)

HimpselScience 283, 1655 (1999)

Quantum Well Oscillations in Electron Interferometers

Fabry-Perot interferometer model: Interfaces act like mirrors for electrons. Since electrons have so short wavelengths, the interfaces need to be atomically precise.

n

12

34

56

The Important Electrons in a Metal

Energy EFermi

Energy Spread 3.5 kBT

Transport (conductivity, magnetoresistance, screening length, ...)

Width of the Fermi function:

FWHM 3.5 kBT

Phase transitions (superconductivity, magnetism, ...)

Superconducting gap:

Eg 3.5 kBTc (Tc= critical temperature)

Energy Bands of Ferromagnets

States near the Fermi level cause

the energy splitting between

majority and minority spin bands

in a ferromagnet (red and green).-10

-8

-6

-4

-2

0

2

4

XK

Ni

En

erg

y R

ela

tiv

e t

o E

F [

eV

]

0.7 0.9 1.1

k|| along [011] [Å-1 ]

Calculation Photoemission data

(Qiu, et al. PR B ‘92)

Quantum Well States and Magnetic Coupling

The magnetic coupling between layers plays a key role in giant magnetoresistance (GMR), the Nobel prize winning technology used for reading heads of hard disks.

This coupling oscillates in sync with the density of states at the Fermi level.

Minority spins discrete,Majority spins continuous

Magnetic interfaces reflect the two spins differently, causing a spin polarization.

Spin-Polarized Quantum Well States

Filtering mechanisms

• Interface: Spin-dependent Reflectivity Quantum Well States

• Bulk: Spin-dependent Mean Free Path Magnetic “Doping”

Parallel Spin Filters Resistance Low

Opposing Spin Filters Resistance High

Giant Magnetoresistance and Spin - Dependent Scattering

Giant Magnetoresistance (GMR): (Metal spacer, here Cu)

Tunnel Magnetoresistance (TMR): (Insulating spacer, MgO)

Magnetoelectronics

Spin currents instead of charge currents

Magnetoresistance = Change of the resistance in a magnetic field