PHYS 342/555 Introduction to solid state physics Instructor: Dr. Pengcheng Dai Professor of Physics The University of Tennessee (Room 407A 407A, Nielsen Nielsen, 974 974-1509) 1509) Chapter 7: Energy Bands Lecture in pdf format will be available at: http://www.phys.utk.edu
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Bloch’s Bloch s theorem The eigenstates of the one-electron Hamiltonian 2 2 H U (r ) , where U (r R) U (r ) for all R in 2m a Bravias lattice, can be chosen to have the form of a plane wave times a function with the periodicityy of the Bravias lattice: ik r nk (r ) e unk (r ), where unk (r R) unk (r ) f all for ll R in i the th Bravias B i lattice. l tti or ik R (r R) e (r )
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Chapter 7-2
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Chapter 7-3
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Chapter 7-4
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Chapter 7-5
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Chapter 7-6
Also, for small k , U1 effectively averages to zero so it still looks like the free electron case where ik r (r ) Ae and 2 k 2 / 2m A k approaches As h /a (k 2 / / a ), ) we expect a Bragg B reflection or a standing wave. We can form two different standing waves from f the h two traveling li waves exp((i x / a ), ) or
(+) exp(i x / a ) exp(i x / a ) 2 cos( x / a ); () exp((i x / a) exp((i x / a) 2i sin( i ( x / a ) Both standing waves are composed of equal parts of right- and leftdi directed d traveling li waves.
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Chapter 7-7
Origin of the energy gap The two standing waves (+) and () pile up electrons at different regions, and therefore the two waves have different values of the potential energy. This is the origin of the energy gap. The probability density of a particle is * . For a standing wave (+), 2
() cos 2 x / a. 2
This function piles up electrons (negative charge) on the positive ions centered at x 0, 0 a, 2a,...where where the potential energy is lowest. lowest For the other standing wave () the probability density is
() sin 2 x / a. 2
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Chapter 7-8
The potential energy of is lower than that of the traveling wave, wave and that of is higher than the traveling wave. The difference is the energ energy gap gap. Dai/PHYS 342/555 Spring 2013
Chapter 7-9
Magnitude of the energy gap Th wavefunction The f i at the h Brillouin B ill i zone boundary b d k / a are 2 cos x / a and 2 sin x / a, normalized over unit length of line. If the potential energy of an electron in the crystal at point x is U ( x) U cos 2 x / a The first-order energy difference between the two standing wave states is 1
Eg dxU ( x)[ ( ) () ] 2
2
0
2 dxU cos(2 x / a )(cos 2 x / a sin 2 x / a ) U
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Chapter 7-10
Kronig-Penney Kronig Penney Model
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Chapter 7-11
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Chapter 7-12
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Chapter 7-13
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Chapter 7-14
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Chapter 7-15
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Chapter 7-16
Chapter 7: Energy Bands
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Chapter 7-17
The periodic potential
2 2 H U (r ) 2m Dai/PHYS 342/555 Spring 2013
Chapter 7-18
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Chapter 7-19
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Chapter 7-20
Metals and insulators
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Chapter 7-21
Consider the free electron energy bands of an fcc crystal lattice in the approximation of an empty lattice, but in the reduced zone scheme in which all k's are transformed to lie in the first Brillouin zone. Plot roughly in the [111] direction the energies of all bands up to six times the lowest band at the zone boundary at 1 1 1 k (2 / a )( , , ). ) 2 2 2
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Chapter 7-22
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Chapter 7-23
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Chapter 7-24
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Chapter 7-25
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Chapter 7-26
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Chapter 7-27
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Chapter 7-28
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Chapter 7-29
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Chapter 7-30
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Chapter 7-31
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Chapter 7-32
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Chapter 7-33
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Chapter 7-34
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Chapter 7-35
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Chapter 7-36
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Chapter 7-37
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Chapter 7-38
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Chapter 7-39
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Chapter 7-40
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Chapter 7-41
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Chapter 7-42
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Chapter 7-43
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Chapter 7-44
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Chapter 7-45
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Chapter 7-46
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Chapter 7-47
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Chapter 7-48
Pentavalent impurities Impurity atom with 5 valence electrons produce n-type semiconductors by contributing extra electrons. Trivalent impurities Impurity atoms with 3 valence electrons produce p-type semiconductors by producing a "hole" hole or electron deficiency deficiency.
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Chapter 7-49
The addition of pentavalent impurities such as antimony, arsenic or phosphorous contributes free electrons, greatly increasing the conductivity of the intrinsic semiconductor. Phosphorous may be added by diffusion of phosphine gas (PH3).
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Chapter 7-50
The addition of trivalent impurities such as boron, aluminum or gallium to an intrinsic semiconductor creates deficiencies of valence electrons, called "holes". It is typical to use B2H6 diborane gas to diffuse boron into the silicon material material.
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Chapter 7-51
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Chapter 7-52
Semiconductor Current Both electrons and holes contribute to current flow in an intrinsic semiconductor.
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Chapter 7-53
Depletion Region When a p-n junction is formed, some of the free electrons in the n-region
diffuse across the junction and combine with holes to form negative ions. In so doing they leave behind positive ions at the donor impurity sites. sites
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Chapter 7-54
P-N Energy Bands
For a p-n junction at equilibrium, the fermi levels match on the two sides of the junctions. Electrons and holes reach an equilibrium at the junction and form a depletion region. The upward direction in the diagram represents increasing electron energy. That implies that you would have to supply enery to get an electron to go up on the diagram, and supply energy to get a hole to go down.
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Chapter 7-55
P-N Energy Bands To reverse-bias the p-n junction, the p side is made more negative, making it "uphill" for electrons moving across the junction. The conduction direction for electrons in the diagram is right to left, and the upward direction represents increasing electron energy.
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Chapter 7-56
-N Energy Bands
To forward bias the p-n junction, the p side is made more positive, so that it is "downhill" for electron motion across the junction junction. An electron can move across the junction and fill a vacancy or "hole" hole near the junction. It can then move from vacancy to vacancy leftward toward the positive terminal, which could be described as the hole moving right. The conduction direction for electrons in the diagram is right to left, and the upward direction represents increasing electron energy.
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Chapter 7-57
When the p-n junction is forward biased, the electrons in the n type material which have been elevated to the conduction n-type band and which have diffused across the junction find themselves at a higher energy than the holes in the p-type material. t i l Th They readily dil combine bi with ith th those h holes, l making ki possible a continuous forward current through the junction.
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Chapter 7-58
Forward Biased Conduction The forward current in a p-n junction when it is forward-biased (illustrated below) involves electrons from the n-type material moving leftward across the junction and combining with holes in the p-type p type material. material Electrons can then proceed further leftward by jumping from hole to hole, so the holes can be said to be moving to the right in this process.
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Chapter 7-59
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Chapter 7-60
