The Free Electron Theory of Metals

1
The Free Electron Theory of Metals

• Chapter 6 of Solymar

2
Free Electrons

• The valence electrons in the metal are free to
  move around, but they cant escape the metal
• They are in a potential well
• In 1-dimensional case
• If a cube of side L

3
Quantized Energy Levels?

• The allowed energy is an integral multiple of
  h2/8mL2
• If L 1 cm
• Yes! The energy is quatized!
• The difference between adjacent energy levels
  increases at higher energy
• If the maximum energy E 3 eV
• If
• Then nX 1.64?107
• Then energy level just below the maximum energy
  is nX 1, nY, nZ

4
Density of States

• How many states between energy levels E and E
  dE?
• Consider a 3-dimensional space formed by nX, nY,
  and nZ
• Each point with integer coordinates represents a
  state, i.e. a unit cube contains exactly one
  state
• The number of states in a sphere of radius n is
  equal to the numerical value of the volume
• The number of states having energy less than E

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Density of States (Cont.)

• Number of states with energy less than E dE
• Number of states with energy between E and E dE
• Z(E) density of states
• Only positive values for nX, nY, and nZ , so it
  needs to multiply 1/8
• Each state allows two values for spin, so it
  needs to multiply 2

6
Fermi-Dirac Distribution

• What is the probability of occupancy for a state?
• Based on Paulis exclusion principle
• EF Fermi level
• At T 0 (Fig. 6.1)
• F(E) 1 when E lt EF Electrons fill up states to
  EF
• F(E) 0 when E gt EF All states above EF are
  empty

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Fermi Level

• Therefore
• N number of electrons/volume
• Table 1 calculated EF

8
Fermi-Dirac Distribution (Cont.)

• When E EF gtgt kT (low temperature or high
  energy)
• This is Maxwell-Boltzmann distribution
• When EF E gtgt kT (low energy)
• The probability of non-occupancy is 1 F(E), so
  the probability of non-occupancy varies
  exponentially
• In the range E EF, F(E) changes sharply
• If we define the transition region as between
  F(E) 0.9 and F(E) 0.1, the width of the
  region is 4.4kT

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Specific Heat

• The specific heat per electron
• ltEgt is the average energy of electrons
• In classical physics
• In quantum mechanics, ltEgt is finite when
  temperature goes to zero
• So
• Electrons way below EF have nowhere to go when
  they receive a small amount of energy, so they
  dont contribute to specific heat

10
Work Function

• It takes energy to remove electrons from a metal
• Heat, light, etc.
• There is a certain threshold energy the electrons
  must possess in order to escape a given metal
• This threshold energy is called work function f
• At T 0, f is the energy barrier for electrons
  to escape (Fig. 6.2)

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Thermionic Emission

• Emission of electrons from a metal at high
  temperatures is called thermionic emission
• They must have energy more than EF f
• For free electrons
• If x is normal to the surface of the metal,
  electrons must travel in the x-direction to
  escape
• If the reflection coefficient is g(px), the
  probability of escape is 1 - g(px)

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Thermionic Emission (Cont.)

• The number of electrons arriving at the surface
  per second per unit area
• N(px) number of electrons having momentum
  between px and px dpx
• Number of electrons with a chance to escape
• The emission current density

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How to Find N(px)

• First find density of states Z(px)
• For a cube of metal of side 1
• The number of states in a volume dnxdnydnz is
  dnxdnydnz or
• Divide it by 8 and multiply by 2

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Mathematics

• The number of electrons in the momentum range px,
  px dpx, py, py dpy, pz, pz dpz
• Integrate

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Thermionic Current

• Assume g(px) g is independent of px
• Where
• J varies exponentially, which agrees with
  experiments
• Work functions of several metals (Table 6.2)
• Measurement setup (Fig. 6.3)
• Current cant sustain due to charge build-up
• (a) thermionic electrons scattered by air
  molecules
• (b) thermionic electrons accumulates in the
  vicinity of the metal
• (c) a dc voltage sweeps electrons to anode

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Image Force

• Thermionic emission with image force and electric
  field
• Image Force (Fig. 6.4)
• An electron in front of an infinitely conducting
  sheet experiences a force which is equivalent to
  replacing the sheet by the mirror charge
• The potential energy
• The reference point is at x ?
• Previously the reference point was the bottom of
  the potential well, i.e. a valence electron at
  rest

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Electric Field

• Fig. 6.5
• (a) Metal-vacuum interface without image force or
  electric field
• (b) Effect of image force
• (c) Effect of a constant electric field
• (d) Combined effect of image force and electric
  field
• The maximum of the potential energy
• The energy required to escape is reduced by -Vmax

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The Schottky Effect

• Due to image force and electric field, the
  effective work function is
• The thermionic current
• This is known as the Schottky effect or barrier
  lowering effect
• Fig. 6.6 Log(J) vs. 1/T or Log(J/T2) vs. 1/T
  gives a straight line

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Field Emission

• The presence of an electric field increases the
  emission current due to barrier lowering
• Further increase in electric field (109 V/m)
  leads to electrons tunneling through the
  potential barrier
• Fig. 6.7 The barrier becomes thin under the
  electric field and the electrons can tunnel
  through field emission
• Higher energy electrons have thinner barrier to
  tunnel through, but they are few of them
• The main contribution to the tunneling current
  comes from electrons around the Fermi level
• The width of the barrier
• The height of the barrier is feff

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Field Emission Current

• Equivalent barrier (Fig. 6.8)
• Tunneling current can be derived
• J is an exponential function of xF
• This current is temperature independent
• Tunneling current is temperature independent, but
  thermionic current is temperature dependent

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Photoelectric Effect

• Photons can excite electrons in a metal to escape
  (Fig. 6.11)
• Experimental set-up
• When an electromagnetic wave is incident, there
  is no current unless the frequency of the wave is
  high enough
• Then an electric current flows between the
  electrodes. The magnitude of the current is
  proportional to incident power
• Photons are the particle equivalents of
  electromagnetic waves
• Each photon has energy
• If incident power is P and frequency is w, the
  number of photons incident is

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Metallic Junction

• When two metals of different work functions are
  in contact
• Fig. 6.12 Electrons flow from left to right
  particle equivalents of electromagnetic waves
• An excess of positive charge on left and negative
  charge on right
• An electric field is built in the metallic
  junction
• The electric field drives electrons from right to
  left
• At equilibrium, equal numbers of electrons move
  left to right and right to left
• Fig. 6.13
• At equilibrium, the Fermi level is leveled
• The potential difference between two metals, the
  contact potential, is equal to the work function
  difference
• If we put an electron in the contact region, it
  will be pulled to left

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HW Assignment

• 6.3, 6.5, 6.7, 6.8