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 ⋅ π/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.

λn = 2d / n

kn = 2π / λn = n ⋅ π/d

d

E

0 π/aπ/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 ⋅ π/d

Quantization in Thin Graphite FilmsE

0 π/aπ/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)

101616

161616

16

13

1414

11.5

13

1413

14

hν (eV)Ag/Fe(100)

Binding Energy (eV)012

Pho

toem

issi

on In

tens

ity (a

rb. u

nits

)

1

2

3

45

6

7

8

9

10

11

13

14

15

12

N

1

3

2

4

101616

161616

16

13

1414

11.5

13

1413

14

hν (eV)Ag/Fe(100)

Binding Energy (eV)012

Pho

toem

issi

on In

tens

ity (a

rb. u

nits

)

1

2

3

45

6

7

8

9

10

11

13

14

15

12

N

1

3

2

4

n

Periodic Fermi level crossing of quantum well states with

increasing thickness

Counting Quantum Well States

Number of monolayers N

Bin

ding

Ene

rgy

(eV)

0

1

2

1 2 3 4 5

6

7

8

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

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

Γ XK

Ni

Ener

gy R

elat

ive

to E

F[e

V]

0.7 0.9 1.1k|| along [011] [Å-1 ]

Calculation Photoemission data

(Qiu, et al.PR B ‘92)

Quantum Well States and Magnetic CouplingThe 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)

MagnetoelectronicsSpin currents instead of charge currents

Magnetoresistance = Change of the resistance in a magnetic field