Chapter XXIV Crystalline Solids

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Chapter XXIV Crystalline Solids

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To have a quantum-mechanical treatment we model a crystalline solid as matter in which the atoms have long-range order, that is a recurring (periodical) pattern of atomic positions that extends over many atoms. We will describe the wavefunctions and energy levels of electrons in such periodical atomic structures.  We want to answer the question: Why do some solids conduct curr We want to answer the question: Why do some solids conduct current ent and others don and others don’ ’t? t?...

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Nội dung Text: Chapter XXIV Crystalline Solids

GENERAL PHYSICS III Optics & Quantum Physics
Chapter XXIV Crystalline Solids §1. Wavefunctions and energy band of electrons §2. Electronic conduction in metals §3. Semiconductors
To have a quantum-mechanical treatment we model a crystalline solid as matter in which the atoms have long-range order, that is a recurring (periodical) pattern of atomic positions that extends over many atoms. We will describe the wavefunctions and energy levels of electrons in such periodical atomic structures.  We want to answer the question: Why do some solids conduct current current and others don’t? don’
§1. Wavefunctions and energy band of electrons: 1.1 Potential energy of electrons: Atom H Two atoms (molecule H2 ) +e +e +e r r n=3 n=2 n=1 2 e 2 e 2 e U ( r )  U ( r )   r r 1 r  2 r r
• A simple (1-D) model of the recurring structure is shown below. • An electron interacts not with one, but with many nuclei  every energy level splits into a band of energies Atom Solid +e r n = 3 n = 2 n = 1 Discrete atomic states  band of crystal states  Fill according to Pauli Principle
1.2 Wavefunctions and Energies: +e r 2 e • Atomic ground state: U ( r )  A r n=1 ( r ) e / a0 r • Molecular states: +e +e +e +e r r  odd even Bonding state Antibonding state Now we fill these orbitals with the 2 available electrons (one from each hydrogen atom). Both can go into ‘bonding’, thanks to spin
• 1-D periodic lattice: +e r Again start with simple atomic state: A n=1 Bring N atoms together together forming a 1-D crystal (a periodic lattice) (N atomic states N crystal states): Energy band What do these crystal states look like? -- approximately linear combinations of atomic orbitals.
• Simple model of a crystal with covalent bonding: “in between” states N Highest energy orbital (N-1 nodes) 1 Lowest energy orbital (zero nodes)
• The “in between" states: Length of crystal, L n Lattice spacing, a Real Envelope: eikx The wavevector k has N possible values from k =  to k =  /L /a. n kn  n 1,2, ..... L/a N = L/a states L Bloch Wavefunction for u(x) depends on the electron in a solid: n (x) u(x)eik n x atomic states involved: 2s-states u(x) x 1s-states x
• An example of Bloch Wavefunctions and the Energy Band: For N = 6 there are six different n (x) u(x)eik n x superpositions of the atomic states that form the crystal states Highest energy wavefunction  Energy 6 5 4 3 2 Closely spaced  energy levels in 1 this “1s-band” Lowest energy wavefunction
1.3 Bloch Wavefunctions and the Free Electron Model: n (x) u(x)e ikn x 2 Envelope: Re eikx  Bloch wavefunction acts almost like free electron wavefunction: k 2 2  p2  Free electron: (x) Ae ikx Energy E      2m  2m   k n 2 2 Bloch wave: n (x) u(x)e ikn x Energy E  m*  effective mass" " 2m* e-  In a perfectly periodic lattice, an electron moves freely without scattering from the atomic cores !! It’s in a packet of stationary states –- the Bloch wavefunctions.
• Electron in a periodic potential has a well defined wavevector and momentum:  p k (  h/ ) even though it is traveling in a complicated potential.  If there is a defect in the crystal, the electron may scatter to another Bloch state. I.e. a Bloch state is not stationary in the full potential, with defect. e-  Also lattice vibrations break the periodicity – electrons in metals scatter more at higher temperatures. Here the electrons see a time-varying potential.
1.4 Insulators, Semiconductors, Metals: Why do some solids conduct current and others don ’t?  Energy bands and the gaps between them determine the conductivity and other properties of solids.  Insulators have a valence band which is full and a large energy gap (few eV) E  apply an electric field - no states of higher energy available for electron  Semiconductors are insulators at T = 0. They have a small energy gap (~1 eV) between valence and conduction bands, so they become conducting at higher T.  Metals have an upper band which is only insulators semi- metals partly full conductors  apply an electric field - lots of states of higher energy available for electron
§2. Electronic conduction in metals: 2.1 Semi-classical picture of conduction: n = # free electrons/volume e-  time between scattering events = E Wire with J = current density = I/A cross section A F = force = -eE F eE J  drift nev vdrift  a    a = acceleration = F/m m m ne 2 ne 2 J  EE  conductivity m m Resistivity Metal: scattering time gets 1 m shorter with increasing T   2  ne  Temperature, T
2.2 Quantum-mechanical description of conduction: • Example: Na Z = 11 1s22s22p63s1 3s N states 2s, 2p 4N states 1s N states The 2s and 2p bands overlap. Basis Number of functions Since there is only one electron in the n = 3 shell, Bloch (atomic we don’t need to consider the 3p or 3d bands of states states) states, which partially overlap the 3s band. N = total number of Fill the crystal states atoms with 11N electrons
Fill the Bloch states Total number of atoms = N according to Pauli Principle: Total number of electrons Z = 11 1s22s22p63s1 = 11N 3s N states 2s, 2p 4N states 1s N states 2N electrons fill 8N electrons fill these states. these states. These electrons are easily The 3s band is only half promoted to higher states filled (N orbital states in the band. Therefore, and N electrons) Na is a good conductor. Partially filled band  good conductor
2.3 Fermi-Dirac distribution: The probability distribution for occupation of electrons in energy states depends on temperature. This distribution is represented by f(E): 1 f(E)  ( E  F ) / kT Fermi-Dirac distribution e E 1 (EF: the Fermi level ) T=0 T>0
§3. Semiconductors: Consider the structure of energy band for Si : Fill the Bloch states Total number of atoms = N Z = 14 1s2 2s2 2p6 3s23p2 Total number of electrons =14N according to Pauli Principle 3s, 3p 4N states 2s, 2p 4N states 1s N states 2N electrons fill 8N electrons fill these states. these states. It appears that, like Na, By this analysis, Si should be Si will also have a half a good metal, just like Na. filled band: The 3s3p band has 4N orbital But something special states and 4N electrons. happens for Group IV elements.
Fill the Bloch states Total number of atoms = N according to Pauli Principle Z = 14 1s22s22p6 3s23p2 Total number of electrons = 14N 3s, 3p 4N states 2s, 2p 4N states 1s N states 2N electrons fill 8N electrons fill these states. these states. In Si, the eigenstates There are 2 types of are made of hybrid superpositions between The s-p band splits into two: combinations of s and p neighbors: states: Antibonding states Antibonding Bonding sp hybrid state states Bonding
• The dependence of electronic conductivity of a semiconductor on temperature: The example of Si Z = 14 1s22s22p63s23p2 Empty band at T = 0. (Conduction Band) valence electrons 3s/3p Energy Gap  Poor conductor at T = 0 band Egap ≈1 eV Filled band at T = 0 (Valence band) The electrons in a filled band cannot contribute to conduction, because with reasonable E fields they cannot be promoted to a higher kinetic energy. Therefore, at T = 0, Si is an insulator. At higher temperatures, however, electrons are thermally promoted into the conduction band: Metal: scattering time  gets Resistivity shorter with increasing T 1 m   2 Semiconductor: number n of free  ne  electrons increases rapidly with T (much faster than  decreases) Temperature, T (This graph only shows trends. A semiconductor has much higher resistance than a metal.)

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