Chapter 6. the Free-Electron Model Free Electron Model for Electrons in Metals

Chapter 6. The Free-Electron Model Free electron model for electrons in metals

• It is sometimes found that the outmost valence Positively charged ions are shielded by other electrons in a solid can be treated as if they are electrons. 𝑉~0 inside solid essentially free electrons, particularly in metals where outer-most valence electrons are not “Particle in a box” model involved in chemical bonding. ex) metallic sodium has 1s22s22p63s1, and the 𝑉𝑖𝑛 0

outermost 3s electron can be considered 𝑉𝑜𝑢𝑡~∞ to be essentially free. • Almost-free electrons confined within the metal → The potential barrier to overcome for escape is the work function.

[Main questions in chapter 6] 1. What energies are allowed for the valence electrons in a metal? 2. What is the density of allowed states? How many states lie between E and E+dE? 3. How are these states occupied by electrons at various temperatures? What energies are allowed?

ℏ Outside metal; 𝜓~0, inside metal; ∇𝜓𝐸𝜓0 (Schrodinger eq.) (separation of variables) 𝜓𝑥, 𝑦, 𝑧 𝑋𝑥𝑌𝑦𝑍𝑧 𝐸𝐸 𝐸 𝐸 𝑑𝑋 2𝑚 𝑒. 𝑔. 𝐸 𝑋0 𝑑𝑥 ℏ 𝑉 0 2 𝑛𝜋𝑥 𝑖𝑛 𝑋 sin 𝑉 ~∞ 𝐿 𝐿 𝑜𝑢𝑡 ℏ 𝐸 𝑛 (𝑛 :integer) 2 𝑛 𝜋𝑥 𝑛𝜋𝑦 𝑛 𝜋𝑧 𝜓 sin sin sin 𝐿 𝐿 𝐿 𝐿

ℏ𝜋 ℏ 𝐸 𝑛 𝑛 𝑛 𝐤 2𝑚𝐿 2m 𝑛 𝜋 𝑛𝜋 𝑛 𝜋 with 𝐤𝐞 𝐞 𝐞 𝐿 𝐿 𝐿 What energies are allowed?

3.42 10 The ground state energy is 𝐸 𝑛 𝑛 𝑛 𝑒𝑉 𝐿 effectively zero, and the distribution of energy states can be treated as ℏ Ground state: E quasi−continous since the differences between discrete states 𝐿~𝑂 𝑐𝑚 , 𝐸~0, Δ𝐸~0 are so small

Degeneracy : Different state with different spatial distribution of probability density, but the same energy (single energy level)

Degenerate state 𝑛,𝑛,𝑛 1,1,1, 2,2,2 : not degenerate

ℏ𝜋 Degeneracy: 𝐸41 2𝑚𝐿 1,2,6, 1,6,2, 2,1,6, 2,6,1, 6,1,2, 6,2,1 3,4,4, 4,3,4, 4,4,3 9 fold degenerate state The density of the allowed states as a function of E

Number of states within ℏ the interval of 20 The density of states as a function of E

How many states lie within a particular energy range, e.g., the energy range between E and E+dE? Let n(E) is the density of electron at energy E, Then n(E)=N(E)ƒ(E) where, N(E): density of allowed state, # of states between E and E+dE ƒ(E) : probability of occupation of a state n N(E): the total number of states with energy less than E z

N N𝐸 𝑁𝐸𝑑𝐸, 𝑁𝐸 (1,1, 2) (1,1,1)

How to count N(E)? (2,1,1) 2𝑚𝐿 ny 𝑛 𝑛 𝑛 𝐸 ℏ𝜋 (1, 2,1) 2𝑚𝐿 n :equation of a sphere with radius of 𝐸 x ℏ𝜋 [n-space]

: one state (nx, ny, nz) per unit volume The density of states as a function of E 2𝑚𝐿 N𝐸: volume of the positive octant of the sphere with radius 𝑅 𝐸 ℏ𝜋 1 4𝜋 2𝑚𝐿 𝐿 2𝑚 NEV () N𝐸 𝐸 𝐸 8 3 ℏ𝜋 6𝜋 ℏ 𝑉 2𝑚 𝐸 𝑉 𝐿 ) 6𝜋 ℏ 𝑑N𝐸 𝑉 2𝑚 𝑁𝐸 𝐸 𝑑𝐸 4𝜋 ℏ 𝐸 E ∶ Density of orbital states without considering spin F

𝑁𝐸: the number of states per unit volume "density of states" 𝑁𝐸 2 due to spin degeneracy

1 2𝑚 𝑁 𝐸 𝐸 2𝜋 ℏ Occupation probability: ƒ(E)

At 0 K, we can expect that electrons will fill up the states, starting from the lowest energy first, until the last electron to be added has an energy EF which marks the occupied states from the higher-lying unoccupied states.

NEV ()

Fermi energy Ef : The highest occupied electron state

E EF - Density of free electrons (n) 1 2𝑚 𝑛 𝑁𝐸𝑑𝐸 𝐸 3𝜋 ℏ ℏ ℏ 𝑘 𝐸 3𝜋𝑛 𝑘 3𝜋𝑛 2𝑚 2𝑚 Occupation probability: ƒ(E)

What is 𝑓𝐸 for T > 0 K? 𝐸 is a very slow function of temperature.

1 fE() F 2

Requirements on 𝑓𝐸 1 For E ≪E, 𝑓𝐸=1 2 For E ≫E, 𝑓𝐸~0 1 𝑓𝐸 [Fermi−Dirac distribution function] 𝑒/ 1 Occupation probability: distribution

𝑓𝐸 Bose Einstein 𝐸 0 for Boltzman & B−E 𝐸 𝐸 for F−D Fermi Dirac 1.0 Boltzman

Eref 𝐸

At large energies, three distributions are the same.

For Fermi function For B-E distribution

𝑓𝐸=1 for E ≪E Infinite occupancy for phonons and photons with 𝑓𝐸= for E E zero energy 𝑓𝐸=0 for E ≫E Density of occupied states : every physically distinct distribution of N particles among various energy states in equally likely to occur. 1 2𝑚 1 𝑁 𝐸 𝐸 𝑓𝐸 2𝜋 ℏ 𝑒/ 1

1 2𝑚 𝐸 𝑛 𝐸𝑓E 𝑁 E 2𝜋 ℏ 𝑒/ 1

𝑓𝐸 𝑁𝐸 ∝𝐸 1.0

𝑛𝐸

𝐸 Photoemission

KE.. Vacuum

 work function 𝑞𝜙  e E detector F −measure the kinetic metal energy or electron current 0 metal surface Vacuum

• To be emitted from the metal, electrons need to overcome the energy barrier, work function.

𝐾. 𝐸. ℏ𝜔 𝑉 • At low temperature, the minimum energy to produce photoemission is hc q min max q

: Method for measurement of the work function of a metal surface Note work function varies depending on the crystal face. Photoemission (a) Rate of electron excitation by light 𝐶exp𝛼𝑥 𝛼: absorption coefficient

(b) Absorption coefficient is a function of the photon energy, ℏ𝜔. The photoemission current depends on the photon energy.

(d) If electrons are scattered during excitation, there would be no emission. (probability of not being scattered during traveling a distance x) = exp 𝑥: mean free path for scattering

(e) Electrons approaching the surface and to be photoemitted undergo reflection due to wave-like properties and change in kinetic energy at the surface. Photoemission

ex) 𝑞𝜙 ≡𝑉 𝐸 𝑞𝜙 of metals: 1.8 𝑒𝑉 cesium ~ 4.8 𝑒𝑉 nickel Cs Ni 700 nm 260 nm

Internal photoemission : In a contact between metal and another material, there is a barrier for photoemission. Photoemission yield ~ (photon E – barrier height)2

control of work function in intermetallic compounds Thermionic emission

• Emission of high energy electron corresponding to the high energy tail of the Fermi-distribution at high temperature. 1 For thermionic emission, 𝐸 𝑚𝑣 𝐸 𝑞𝜙 2 𝐽 density of electrons with sufficient energy to pass into vacuum velocity component in the x directon  J 𝐽 𝑞𝑣𝑑𝑛 𝑞𝑣𝑛 𝐸𝑑𝐸 x

𝑞𝜙

𝑛𝐸𝑑𝐸 𝑓𝐸𝑁 𝐸𝑑𝐸 E 8𝜋𝑚 𝑣𝑑𝑣 F ℎ 𝑒 / 1 𝑛𝐸 1 0 x ∵ 𝐸 𝑚𝑣,𝑑𝐸𝑚𝑣𝑑𝑣 2 metal surface Vacuum Thermionic emission

4𝜋𝑣 𝑑𝑣in spherical coordinate~𝑑𝑣𝑑𝑣𝑑𝑣 in courtesian coordinate

Since 𝐸𝐸 ≫𝑘𝑇 1 →𝑒/ 𝑒/𝑒 𝑒/ 1

2𝑚 𝐽 𝑑𝑣 𝑑𝑣 𝑞𝑣 𝑒/𝑒 𝑑𝑣 ℎ 2𝜋𝑘𝑇 𝑒 𝑑𝑣 𝑒 𝑑𝑣 𝑚 𝑘𝑇 𝑣 𝑒 𝑑𝑥 𝑒/ 𝑚 4𝜋𝑞𝑚 4𝜋𝑞𝑚 𝑞𝜙 𝐴 𝑘𝑇 ∴𝐽 𝑘𝑇 exp [Richardson Eq.] ℎ ℎ 𝑘𝑇 (Richardson constant) Thermionic emission

𝐽 1 From a plot of ln vs. from 𝐽 𝑘𝑇 exp 𝑞𝜙/𝑘𝑇 𝑇 𝑇 Straight line with a slope of 𝑞𝜙/𝑘

ex) contact between Pd and a-Si The slope gives an effective barrier height of 0.97 eV, which is similar to the barrier measured by photoemission. Field emission

• Field-induced electron emission from metals • The mechanism: quantum mechanical tunneling • The rate of emission ∝ n(E)✕T(E) n(E): the density of occupied states at a given E T(E): tunneling transmission coefficient at the same E Field emission • Rate of emission ~ (density of occupied states at given E) (tunneling transmission coeff.) • Total emission = (the rate of emission)dE E

• Most of the emission, however, comes from state close to EF For high energy, n(E) ↓, T(E) ↑ For low energy, n(E) ↑, T(E) ↓

𝜀∇𝜙 (𝜙: potential, 𝜀: electric field)

𝜙𝑞𝜀 for const. 𝜀 : In electric field, potential energy varies with distance x from the surface • A traveling wave moving in the +x direction approaching the barrier from the left would be attenuated in passing “through” the region of the barrier Field emission

• Approximate tunneling probability T is given by Wentzel−Kramers−Brillouin 2 tungsten 𝑇exp 2𝑚𝑉 𝐸 𝑑𝑥 [WKB model] ℏ 𝑉 𝐸 : ffectivee barrier height, 𝑑: barrier width Considering the tunneling at the Fermi level, 𝑥 𝑉 𝐸𝑞𝜙 1 , 𝑑𝑙 𝑙 4 𝜙 ∴𝑇 𝐸 exp 2𝑚𝑞 3ℏ 𝜀

• From integration over the whole distribution of n(E) T(E)

4 𝜙 𝐽~𝜀 exp 2𝑚𝑞 Plot of ln vs. ⟹ straight line with slope of 𝜙 3ℏ 𝜀

note: J is independent of temperature Scanning tunneling microscope (STM) STM tip (tungsten)

Tunneling distance

Low High Graphene current current

Leakage current in nanoscale transistor

The size of the transistor constantly scales down. At the same time, the thickness of gate oxide - an insulator separating the gate electrode and channel layer - also becomes thinner. When the thickness of the oxide is only a few nanometers, the tunneling current through the gate oxide becomes sizeable, which induces failure of the devices. To resolve this issue, the industry (pioneered by

Intel) changed the material for the gate oxide from SiO2 to HfO2 that has high dielectric constant. So, we can decrease the oxide thickness while maintaining the capacitance. Heat capacity

• A significant contribution to the heat capacity comes from the free electrons present in the metals in addition to the lattice vibration. ex) semiconductor and insulator – lattice vibration metal – free electrons significantly contribute to the heat capacity • The contribution of lattice vibrations:

𝐶 𝜕𝑈/𝜕𝑇 Phonon E Density of lattice vibrational states 𝑈 𝑈𝜔𝑑𝜔 ℏ𝜔exp ℏ𝜔/𝑘𝑇 1 𝑁𝜔𝑑𝜔 𝐶 ∝ 𝑇 • The contribution of free electrons

𝐶 𝜕𝑈/𝜕𝑇 𝑈 𝑈𝐸𝑑𝐸 𝐸𝑓𝐸𝑁𝐸 𝑑𝐸 𝐶 ∝𝑇for low T • At low temperature, 𝐶𝐶 𝐶 𝐴𝑇 𝐵𝑇 Plot of vs. 𝑇 ⟹ slope = A and intercept = B