1. Prof. Jang-Ung Park (박장웅)
Chapter 3.
Bonds, Bands,
&
Semiconductors
Reference Book 1 (PDF): Elementary Solid State Physics
(M. Ali Omar, Addison-wesley Publishing Company)
Reference Book 2: Solid State Electronic Devices, Chapter 3
(Ben G. Streetman, Prentice Hall Series)
Prof. Jang-Ung Park (박장웅)

2. Summary of Quantum Mechanics
: predicts average values of position, momentum, and energy of electrons in solid.
: describes the probabilistic nature of events involving atoms and electrons.
We must look for “the probability of finding an electron at a certain position”.
(probability density function)
Schrödinger equation (for 1-dimensional cases)
wave function of electron:
probability density function:
Time-independent Schrödinger equation

3. Summary of Quantum Mechanics
: predicts average values of position, momentum, and energy of electrons in solid.
: describes the probabilistic nature of events involving atoms and electrons.
We must look for “the probability of finding an electron at a certain position”.
(probability density function)
Schrödinger equation (for 1-dimensional cases)
wave function of electron:
probability density function:
Comparison to classic equation for particle’s energy
Kinetic Energy + Potential Energy = Total Energy

4. Variables in spherical coordinate
Atomic Structure
Hydrogen Atom model
proton
electron
 Coulomb potential :
 time-independent wave function :
 Separation solutions can be obtained for the r-dependent, the -dependent, and -dependent equations.
[spherical coordinate system]
 Three quantum numbers need to be associated with the three equations.
principal quantum number (specifies a shell)
Variables in spherical coordinate
Quantum numbers r n  l  m
n = 1,2,3,…
l = 0,1,2,…, (n-1)
m = - l, …,-2,-1,0,1,2,…,+l
s =
s (spin)

5. The Periodic Table Relation with Pauli exclusion principle
: no two electrons can have the same set of quantum numbers.
(only two electrons can have the same numbers of n, l, m ,with opposite spin.)
 In the first electronic shell (n=1); l = n-1 = 0, m = 0,
: allowable states become 2 for  (100).
s = 1s 2s 2p
l = 0, 1, 2, 3, 4
s, p, d, f, g 3s 3p 3d

6. The Periodic Table
Si atom :
10 core electrons
4 valence electrons

7. Energy Levels in a Si Atom
 Why does Coulomb potential energy increase with distance ?

8. Interaction Between Two Atoms
[two atoms without interactions]
[two atoms with interactions]
bonding state
probability density of electron:
wave function of electron:
anti-bonding state
For the hydrogen molecule ion (H2+) which has only one electron,
potential energy:
By solving Schrödinger equation (linear combination of the two 1s orbitals),
(bonding state): electron is shared by two protons.
(anti-bonding state)

9. Interaction Between Two Atoms [two atoms with interactions]
bonding state (even)
nucleus
nucleus
anti-bonding state (odd)
anti-bonding state (odd)
nucleus
bonding state (even)
nucleus
anti-bonding state
interaction
no interaction
bonding state
 Pauli exclusion principle dictates that no two electrons in a given interacting system may have the same quantum state.
Energy of bonding state is lower than energy of anti-bonding state.
: Two nuclei shares an electron, and hence the energy level becomes lower (stable).
(The electron occupies the bonding orbital.)
binding energy: 2.65 eV, binding distance: 1.06 Å

10. Interaction Between Two Atoms
[two atoms without interactions]
[two atoms with interactions]
energy band
 Each of the atomic levels is split into N closely spaced sublevels, where N is the number of atoms in a solid.
 Since N is very large, the sublevels are so extremely close to each other that they coalesce, and form an energy band.

11. Energy Band Formation in Solids
 In a solid, many atoms are brought together, so that the split energy levels form essentially continuous bands of energies. 1s 2s 2p 3s 3p
 A high energy state corresponds to a large atomic radius, and hence a strong perturbation, which is the cause of the level broadening in the first place.
 For an atom, electrons are localized around nucleus. For a solid, however, electrons are delocalized from nuclei.

12. Metals, Semiconductors, and Insulators
A band which is completely full carries no electric current, even in the presence of an electric field. It follows therefore that a solid behaves as a metal only when some of the bands are partially occupied. EC
(conduction
band)
Semimetals: Bi, As, Sb, Sn
metals: Alkalis (Na, Li, K)
Cu, Ag, Au, etc. EV
(valence
band)
Eg of Si: 1.1 eV, Eg of Ge: ~0.7 eV : semiconductors when Eg is less than 2eV at room temperature.
Eg of diamond: ~7 eV : insulator

13. Charge Carriers in Semiconductors
[ electrons and holes ]
As the temperature is raised from 0K, some electrons in the valence band receive enough thermal energy to be excited across the band gap to the conduction band.
 Electron-hole pairs (EHPs) are created.
The equilibrium number of EHPs in pure Si at room temperature is only 1.51010 EHP/cm3, compared to the Si atom density of 51022 atoms/cm3.

14. Charge Carriers in Semiconductors
[ electrons and holes ]
In a filled valence band,
 net current is zero.
However, current flows by creation of holes (positively charged particles).
 Both of holes (in the valence band) and electrons (in the conduction band) are charge carriers.

15. Charge Carriers in Semiconductors
[ effective mass ]
The electrons in crystal are not completely free, but instead interact with the periodic potential of the lattice.
electron’s momentum:
electron’s kinetic energy:
: electron energy is parabolic with wave vector k.
: The curvature of the band determines the electron effective mass.
Effective mass of electron:
Effective mass of hole:
electron’s total energy:

16. Semiconductors [ Direct and Indirect Semiconductors ]
A single electron is assumed to travel through a periodic lattice.
The wave function of the electron is assumed to be in the form of a plane wave moving, for example, in the x-direction with propagation constant k, also called a wave vector.
 Energy can be plotted vs. wave vector (k).
 Energy (kinetic + potential) can be plotted vs. momentum (p).
GaAs Si

17. Charge Carriers in Semiconductors
[ electrons and holes ]
When electric field is applied,
An electron at A gains kinetic energy by electric field.  Moves to B. (The electron loose kinetic energy to heat by scattering.)
: The slopes of the energy band edges reflect the local electric field.

18. Charge Carriers in Semiconductors
[ Intrinsic Material ]
Intrinsic material: perfect crystal with no additional impurities.
(no charge carriers at 0K).
Electron concentration :
(at room temperature for Si)
If a steady state carrier concentration is maintained, there must be recombination of EHPs at the same rate at which they are generated.
[ Extrinsic Material ]
: it is possible to create carriers in semiconductors by doping impurities.
n-type dopants : atoms from column V of the periodic table (P, As, Sb, etc)
donates additional electrons (donor) : no » ni.
p-type dopants : atoms from column III of the periodic table (B, Al, Ga, In, etc)
accepts electrons from valence band of Si (acceptor) : po » ni.
If we dope Si with 1015 P atoms/cm3, the conduction electron concentration changes to 1015 /cm3.

19. Charge Carriers in Semiconductors
[ Extrinsic Material ]
Donor binding energy (from Bohr model):
In Si, the usual donor and acceptor levels lie about eV from a band edge.

20. Effects of Doping on Mobility
Intrinsic Si at Room temperature (300k):
=1350 cm2/Vs
Doped Si (1017cm-3) at Room temperature:
=700 cm2/Vs

21. Intrinsic Mobilities &Eg of Various Semiconductors
Source: Intel

22. Approach to Increase Mobility of Si
Strained Si (IBM, 2001 June)
: Chip speeds (including mobilities) can be enhanced up to 35% faster.

23. Carrier Concentrations in Semiconductors
Fermi Energy Level (EF):
: The electron’s energy of the highest occupied level.
EF=1
Density of states
Distribution function for electrons [f(E)]:
: Probability that the level E is occupied by an electron.
At T = 0K,
At T > 0K, thermal energy (kT=0.025 eV) excites electrons.
: only those electrons close to the Fermi level can be excited, because the levels above EF are empty.
: Fermi-Dirac distribution
(k: Boltzmann constant=8.6210-5eV/K)
[1-f(E)] : probability that the level E is empty. probability that the level E is occupied by hole.

24. Charge Concentrations in Semiconductors
[ intrinsic ]
: electron concentration (ni) = hole concentration (pi)
ni = pi = 1.5  1010 /cm3 for Si at room temperature.
: EF is located at the middle of the band gap.
[ extrinsic ]
In n-type, n0 » p0
: EF is located above its intrinsic position.
: As EF moves closer EC, the value of f(E) for each energy level in the conduction band (and therefore total electron concentration n0) increases.
: The energy difference (EC-EF) gives a measure of n.

25. Electron & Hole Concentrations at Equilibrium
Concentration of electrons in the conduction band :
N(E)dE: the density of states in the energy range dE.
n0: electron concentration at equilibrium conditions.

26. Electron & Hole Concentrations at Equilibrium
Concentration of electrons in the conduction band :
 This result is the same as that obtained if we represent all of the distributed electron states in the conduction band by an effective density of states (NC) located at the conduction band edge EC.
: NC is dependent on Temperature.
mn*: effective mass for electrons
For holes,
mp*: effective mass for holes

27. Electron & Hole Concentrations at Equilibrium
For intrinsic material ( EF  Ei ),
Since ni = pi ,
[Problem] A Si is doped with 1017 As atoms/cm3. What is the equilibrium hole concentration p0 at 300K?
Where is EF relative to Ei?
[Solution] Since Nd » ni , n0  Nd EC EV Ei EF
1.1eV

28. Temperature Dependence of Carrier Concentration

29. Temperature Dependence of Carrier Concentration
ni » Nd
Donor atoms are ionized, and electrons are donated to the conduction band.
All donor atoms are ionized (n0=Nd=1015 cm-3).

30. Compensation and Space Charge Neutrality
A semiconductor contains both donors and acceptors.
Due to Nd > Na, EF > Ea : Ea level is filled with electrons.
The resultant concentration of electrons in the conduction band: Nd - Na
Space charge neutrality : the material is to remain electrostatically neutral.

31. Changes in Energy Gap with Temperature and Pressure
The energy gap EG changes with temperature and pressure.
[ Temperature Effect ]
For Si, EG(0)=1.17eV, A: 4.7310-4, B: 636.
[ Pressure Effect ]
In Si, the gap decreases with pressure at a rate of about 210-3 eV/kbar.
s: strain, l: length, s: stress, Y: Young’s modulus
Maximum elastic strain without breaking Si: about 0.01.
 Maximum change in the band gap : ΔEG = -3.810-2 eV.