Fundamentals of Semiconductor Physics 万 歆 Zhejiang Institute of Modern Physics Fall 2006

In memory of Prof. Xie Xide ( )

Chapter 1. Fundamentals 1.1 Bonds and bands 1.2 Impurities and defects 1.3 Statistical distribution of charge carriers 1.4 Charge transport Total 12 hours.

Resistivity of Conductors, Semiconductors & Insulators

Semiconductor-forming Elements BCNO AlSiPS ZnGaGeAsSe CdInSnSbTe

Chapter 1. Fundamentals 1.1 Bonds and bands –Crystal structures –Bond picture –Band picture “Nearly free” electron model Tight-binding model (LCMO) k·p perturbation 1.2 Impurities and defects 1.3 Statistical distribution of charge carriers 1.4 Charge transport

An Apparently Easy Problem Solid Nuclei Electrons interaction In principle, by solving Schrödinger's equation

Lattice & Unit Cell Crystal = periodic array of atoms Crystal structure = lattice + basis The choice of lattice as well as its axes a,b,c is not unique. But it is usually convenient to choose with the consideration of symmetry Unit cell: Parallelpiped spanned by a,b,c

Cubic Lattices Q : How many atoms are there in each of the unit cells?

Miller Indices

Elemental Semiconductors Si, Ge: 4 valence electrons 14 Si: 1s 2 2s 2 2p 6 3s 2 3p 232 Ge: [Ar]3d 10 4s 2 4p 2

Covalent Bond

Complex Lattices (a)Diamond: Si, Ge (b)Zinc blende: GaAs, ZnS Interpenetrating fcc latticesfcc lattices

Wurtzite Lattice ZnS

Band: An Alternative Picture From hydrogen atom to molecule HH H2H2 bonding antibonding

Band Formation N degenerate levels evolve into an energy band

Silicon 14 Si: 1s 2 2s 2 2p 6 3s 2 3p 2

Resistivity Conduction band, valence band & band gap

Formal Treatment

Approximation 1 Separation of electrons into valence electrons and core electrons Ion cores = core electrons + nuclei e.g. Si: [Ne] 3s 2 3p 2

Approximation 2 Born-Oppenheimer or adiabatic approximation To electrons, ions are essentially stationary. Ions only see a time-averaged adiabatic electronic potential. electron-phonon interaction ionic motion  Lattice vibration (phonons) electronic motion

Separation of Motion

Approximation 3 Mean-field approximation: Every electron experiences the same average potential V(r) V(r): by first principle (ab initio), or by semi-empirical approach

Can we calculate everything? Yes. –First principle band calculations –Slater, … No. –Disordered & strongly correlated systems –Mott, Anderson, …

Reciprocal Lattice Definition: The set of all wave vectors K that yield plane waves with the periodicity of a given lattice is known as its reciprocal lattice. Try to verify the following b i form the reciprocal lattice of the lattice spanned by a i. In 1D, we have simply K = 2n  / a, where a is the lattice constant.

FCC  BCC

FCC Lattice & its Brillouin Zone Lattice (fcc): real space Reciprocal lattice (bcc) Diamond structure

Construct the Brillouin Zone

A Digression on Group Theory Ref.: Yu & Cardona, Section 2.3

Rotational Symmetry Ref.: Yu & Cardona, Section 2.3

Translational Symmetry Define operator T R Discrete translational symmetry Bloch’s theorem:

Example: 1D Empty Lattice V  0: We assume an imaginary periodicity of a. Define the reciprocal lattice constant G = 2  / a. We can therefore restrict k within the range of [-G/2, G/2].

Free Electrons in 1D V  0:

Comments The wave vector k is not momentum p/ , since Hamiltonian does not have complete translational invariance. Rather,  k is known as crystal momentum (quantum number characteristic of the translational symmetry of a periodic potential). The wave vector k can be confined to the first Brillouin zone. More in A/M Chapter 8.

“Nearly Free” Electrons

Nearly Free & Pseudopotential Si

Overlap of Molecular Orbitals

Tight-binding or LCMO hopping E(k) k 2D: z = 4 nearest neighbors

Si – Ge –  -Sn

Comparison Nearly free e - ’s + pseudopotential Electrons nearly free Wave functions approximated by plane waves Electrons in conduction band are delocalized, so can be approximated well by nearly free electrons Tight-binding or LCMO approach e - ’s tightly bound to nuclei Linear combination of atomic wave functions Valence electrons are concentrated mainly in the bonds and so they retain more of their atomic character.

Band Diagram Conduction band Valence band E k Conduction band Valence band Allowed states Forbidden band gap EcEc EiEi EvEv EgEg

Direct Bandgap GaAs

Band Structure: Si & Ge

Electrons and Holes 0

The k·p Method

Conduction Band (Nondegenerate)

Comment on k·p Method Band structure over the entire BZ can be extrapolated by the zone center energy gaps and optical matrix elements. One can btain analytical expression for band dispersion and effective mass around high-symmetry points. Nondegenerate perturbation is applicable to the conduction band minimum in direct-bandgap semiconductors (zinc-blende, wurtzite); degenerate perturbation to top valence band (diamond, zinc-blende, wurtzite). Ref.: Yu & Cardona, Section 2.6 The trend of m* can be explained.

Chapter 1. Fundamentals 1.1 Bonds and bands 1.2 Impurities and defects –Classification of defects –Point defects –Shallow (hydrogenic) impurities 1.3 Statistical distribution of charge carriers 1.4 Charge transport

Classification of Defects Point defects Line defects Surface states

Point Defects VAVA IAIA CACA

More Classifications Intrinsic vs extrinsic –Intrinsic: native, such as vacancies or antisite defects –Extrinsic: foreign, Si:P Shallow vs deep – “effective mass approximation” Donors, double donors, isovalent center –Examples: Si:P, Si:Se, Si:C

Shallow Impurity States impurity potential BCNO AlSiPS ZnGaGeAsSe CdInSnSbTe Si:P Effective mass approximation Break translational symmetry No Bloch’s theorem!? Screened Coulomb potential

Hydrogenic Wave Function 1s E 2s 2p... Bloch wave Hydrogenic envelope Hydrogenic bound states1s, 2s, 2p,... + continuum (conduction band) “Rydberg” “Bohr radius” 10 meV (Ge) 30 meV (Si) 50 Å (Ge) 20 Å (Si)

Band Diagram Donor (Si:P, Ge:As) C. B. V. B. Acceptor (Si:B) E k Conduction band Valence band Allowed states Forbidden band gap EcEc EdEd EiEi EaEa EvEv

Effective Mass Approximation 1.Introduce Wannier Functions (indexed by lattice vector in real space): Fourier transforms of Bloch functionsBloch functions For very localized electrons, Wannier functions are roughly atomic orbitals.

Effective Mass Approximation 2.Express H in the basis of Wannier functions Assume 3.Parabolic, isotropic, nondegenerate

Envelope Wave Function Approximation valid for large effective Bohr radius a*(small  k).

Comment on EMA The net effect of the crystal potential on the donor electron inside the crystal is to change the electron mass from the value in free space to the effective mass m* and also to contribute a dielectric constant  of the host crystal. Only conduction band states over a small region of reciprocal space around the band minimum contribute to the defect wave function if the effective Bohr radius a* is much larger than the lattice constant a 0.

Heavy Doping Light doping: impurity atoms do not interact with each other  impurity level Heavy doping: perturb the band structure of the host crystal  reduction of bandgap E  (E) EcEc EdEd EvEv EgEg

Metal-Insulator Transition Average impurity-impurity distance = Bohr radius Mott criterion

Chapter 1. Fundamentals 1.1 Bonds and bands 1.2 Impurities and defects 1.3 Statistical distribution of charge carriers –Thermal equilibrium –Mass-action law –Fermi level 1.4 Charge transport

Thermal Equilibrium Thermal equilibrium is a dynamic situation in which every process is balanced by its inverse process. Thermal equilibrium means that time can run toward the past as well as into the future. E1E1 E2E2

Mass-Action Law Electron-hole pairs: generation rate = recombination rate Generation: G = f 1 (T)f 1 : determined by crystal physics and T Recombination: R = npf 2 (T) –Electrons and holes must interact to recombine At equilibrium, G = R Intrinsic case (all carriers result from excitation across the forbidden gap): n = p = n i

Fermi Level Fermi-Dirac distribution Boltzmann distribution Density of electrons (not too heavily doped)

Intrinsic Semiconductors

A Parabolic Band 1

Comment Multivalley, such as Si: –additional factor of number of valleys Anisotropic band –Effective mass: Geometrical average of mass components Valence band: N V –Sum of heavy hole, light hole (neglecting split-off band) –replace m* by hole effective mass Density of states effective mass

Intrinsic Carrier Concentration

Doped Semiconductors Assuming full ionization, charge neutrality With intentional doping, typically for n-type Majority carriers Minority carriers Compensation

Extrinsic Semiconductors Conduction band Valence band EcEc EdEd EiEi EaEa EvEv EFEF

Carrier Concentration vs T

Fermi Energy in Si Eg ↓Eg ↓ Nonmonotonic behavior

Occupation of Impurity level 0EdEd EdEd 2E d + 

Chapter 1. Fundamentals 1.1 Bonds and bands 1.2 Impurities and defects 1.3 Statistical distribution of charge carriers 1.4 Charge transport

Charge Transport Ohm’s law Equipartition of energy Room temperature (300K)

The complete notes for Chapter 1 are expected to be available after Sept. 20, 2006.