Electrons in Solids Simplest Model Free Electron Gas

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Electrons in Solids Simplest Model: Free Electron Gas Quantum Numbers E, k Fermi “Surfaces”

Electrons in Solids Simplest Model: Free Electron Gas Quantum Numbers E, k Fermi “Surfaces” Beyond Free Electrons: Bloch’s Wave Function E(k) Band Dispersion Angle-Resolved Photoemission

Free Electron Gas Quantum numbers of electrons in a solid: E, kx , ky

Free Electron Gas Quantum numbers of electrons in a solid: E, kx , ky , (kz) Two cuts: Fermi “surface”: I(kx, ky) Band dispersion: I( E , kx) E(k) = ħ 2 k 2/2 m = Paraboloid E Fermi circle Band dispersion kx ky

Fabricating a Two-Dimensional Electron Gas Lattice planes V(z) Inversion Layer (z) Bulk Hamiltonian +

Fabricating a Two-Dimensional Electron Gas Lattice planes V(z) Inversion Layer (z) Bulk Hamiltonian + boundary conditions V(z) Doped Surface State Surface Hamiltonian (z)

Measuring E, kx, ky in a Two-Dimensional Electron Gas Fermi Surface I(ky, kx) 0.

Measuring E, kx, ky in a Two-Dimensional Electron Gas Fermi Surface I(ky, kx) 0. 012 0. 015 0. 022 ky e-/atom: 0. 0015 kx Band Dispersion I(E, kx) 0. 086

Superlattices of Metals on Si(111) 1 monolayer Ag is semiconducting: 3 x 3 Surface

Superlattices of Metals on Si(111) 1 monolayer Ag is semiconducting: 3 x 3 Surface doping: 2 1014 cm-2 Equivalent bulk doping: 3 1021 cm-3 Add extra Ag, Au as dopants: 21 x 21

Fermi Surface of a Superlattice Fermi circles are diffracted by the superlattice. Corresponds to

Fermi Surface of a Superlattice Fermi circles are diffracted by the superlattice. Corresponds to momentum transfer from the lattice. ky Angle-resolved photoemission data ky kx Model using Diffraction kx

Fermi “surfaces” of two- and one-dimensional electrons ky 2 D 2 D + superlattice

Fermi “surfaces” of two- and one-dimensional electrons ky 2 D 2 D + superlattice 1 D kx

One-Dimensional Electrons at Semiconductor Surfaces

One-Dimensional Electrons at Semiconductor Surfaces

Beyond the Free Electron Gas

Beyond the Free Electron Gas

E(k) Band Dispersion

E(k) Band Dispersion

Band Dispersion in a Semiconductor E (e. V) Band gap Empty lattice bands [111]

Band Dispersion in a Semiconductor E (e. V) Band gap Empty lattice bands [111] Density of states [100] [110] Wave vector

Two-dimensional bands of graphene E [e. V] Empty EFermi Occupied Empty * K M

Two-dimensional bands of graphene E [e. V] Empty EFermi Occupied Empty * K M Occupied K =0 M kx, y K

“Dirac cones” in graphene A special feature of the graphene -bands is their linear

“Dirac cones” in graphene A special feature of the graphene -bands is their linear E(k) dispersion near the six corners K of the Brillouin zone (instead of the parabolic relation for free electron bands). In a plot of E versus kx, ky one obtains cone-shaped energy bands at the Fermi level.

Topological Insulators A spin-polarized version of a “Dirac cone” occurs in “topological insulators”. These

Topological Insulators A spin-polarized version of a “Dirac cone” occurs in “topological insulators”. These are insulators in the bulk and metals at the surface, because two surface bands bridge the bulk band gap. It is impossible topologically to remove the surface bands from the gap, because they are tied to valence band on one side and to the conduction band on the other. The metallic surface state bands have been measured by angle- and spin-resolved photoemission (left). the

Photoemission (PES, UPS, ARPES) EFermi • Measures an “occupied state” by creating a hole

Photoemission (PES, UPS, ARPES) EFermi • Measures an “occupied state” by creating a hole • Determines the complete set of quantum numbers • Probes several atomic layers (surface + bulk)

Measuring the quantum numbers E, k of electrons in a solid The quantum numbers

Measuring the quantum numbers E, k of electrons in a solid The quantum numbers E and k can be measured by angle-resolved photoemission. This is an elaborate use of the photoelectric effect, which was explained as quantum phenomenon by Einstein in 1905. Photon in Electron outside (final state) Electron inside (initial state) Energy and momentum of the emitted photoelectron are measured. Energy conservation: Momentum conservation: Efinal = Einitial + h k||final = k||initial + G|| h = Ephoton Only k|| is conserved (surface!) kphoton 0

Photoemission setup: D(E) Einitial Efinal h Photoemission process: Efinal Photoemission spectrum: e counts EF+h

Photoemission setup: D(E) Einitial Efinal h Photoemission process: Efinal Photoemission spectrum: e counts EF+h Core Valence Secondary electrons W = width Efinal

Spectrometer with E, kx - multidetection 50 x 50 = 2500 Spectra in One

Spectrometer with E, kx - multidetection 50 x 50 = 2500 Spectra in One Scan

E, k Multidetection: Energy Bands on a TV Screen E Calculated E(k) Ni (e.

E, k Multidetection: Energy Bands on a TV Screen E Calculated E(k) Ni (e. V) Measured E(k) EF = 0. 7 0. 9 1. 1 (Å-1) Electrons within ± 2 k. BT of the Fermi level EF are not locked in by the Pauli principle. This is the width of the Fermi-Dirac cutoff at EF. k Spin-split Bands in a Ferromagnet These electrons determine magnetism, superconductivity, specific heat in metals, …

http: //uw. physics. wisc. edu/~himpsel/551/Lectures /Lecture Energy Bands. pdf

http: //uw. physics. wisc. edu/~himpsel/551/Lectures /Lecture Energy Bands. pdf