Solid State Physics. Lecture 10 Band Theory. Professor Stephen Sweeney

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1 Solid State Physics Lecture 10 Band Theory Professor Stephen Sweeney Advanced Technology Institute and Department of Physics University of Surrey, Guildford, GU2 7XH, UK

2 Recap from Lecture 9 Drude model (classical) assumes that the electrons collide with atoms in the lattice and that this can be used to describe the conductivity (and electronic heat capacity) overestimates role of electrons in heat capacity and C el is temperature independent Free electron model (quantum mechanical) Number of available states (density of states) E 1/2 in a 3D crystal Fermi function describes occupation probability as a function of temperature (applies to Fermions, e.g. electrons) The Fermi level (E F ) corresponds to the energy of highest occupied states at T=0K, and more generally to states with 50% occupation probability at finite T Electrical and thermal properties of metals are dominated by electrons with energies near E F

3 Band Theory In the free electron model we saw that an electron s energy can be simply written in terms of its momentum, k, as 2 2 k E 2m e This corresponds to an electron propagating freely (a plane wave) subject to the boundary conditions set by the size of the crystal 1D: x Asink x x E k where k x 2π n L BUT, this ignores the fact that there are positively charged ions in a real crystal

4 Bloch Theory x x Asin k x x In the free electron model, on average >> a wavefunction is almost constant over scale of the lattice spacing x Bloch postulated that the effect of the periodic potential would modify the electron wave-function with a periodicity of the lattice constant, x lattice constant, a Potential Energy, U ion core potential x

5 Bloch Theory Now SE is x Asin k x ux x where since follows the periodicity of the lattice x u 2 2 2m e 2 d V dx x ux na x E u x where V x Vx na lattice constant, a Potential Energy, U ion core potential x

6 Felix Bloch ( ) Born in Zurich, Switzerland Started studying engineering, then switched to physics Studied under Heisenberg, Debye, and Schrödinger PhD thesis was on the quantum theory of solids (what became known as Bloch theory), supervised by Heisenberg Worked with Pauli, Bohr and Fermi in Rome, then moved to the USA (in 1933) Professor in Stanford and Harvard Nobel Prize in Physics (1952), joint with Purcell for nuclear magnetic resonance (NMR) measurements Returned to Europe and became the first Director- General of CERN When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation F. Bloch

7 Bloch Theory and Probability Densities Like X-rays, electrons may be diffracted within the crystal, and at critical values of k electrons will be scattered by the crystal. From before, Bragg condition for scattering in 1D is: e.g. when k a k 1 2 G n a an electron travelling to the right will scatter to the left and vice-versa since the electron then has equal probability of moving to the left or right, the wave-function forms a standing wave a E a First Brillouin Zone free electron k

8 Bloch Theory and Probability Densities Using exponential notation, the forward and backward travelling wave has the form: x x x exp i cos isin a a a Standing waves can arise from either the sum (+) of difference (-) between the travelling waves, thus x x x x x x exp i exp i 2cos and exp i exp i 2i sin a a a a a a In terms of probability densities: x a 2 x a cos and sin

9 Bloch Theory and Probability Densities Physically, (+) corresponds to electrons piling-up at the positive ions where the potential energy is lower than for the travelling wave this is known as BONDING x a Physically, (-) corresponds to electrons piling-up between the ions for which the potential energy is higher than for the travelling wave this is known as ANTI-BONDING x a 2 2 cos 2 2 sin (from Kittel)

10 The Band Gap (E g ) For k a (at point B) the electron wave-function is (from Kittel) For k a wave-function is (at point A) the electron Band gap (E g ) this represents a region of energy for which no states exist. Electrons are allowed in either the lower or upper bands

11 Zone Schemes Extended Zone Scheme shows how multiple band gaps arise and can be closely related to the free electron picture Reduced Zone Scheme since the E(k) relationship repeats every G (reciprocal lattice vector = 2/a in 1D), this scheme saves space by folding the upper bands back in by multiples of ±G. This is most commonly used (figures from Rosenberg)

12 Metals, Insulators and Semiconductors Total number of k-values in each zone = N (the number of atoms) Each state can contain two electrons (spin ) total number of states/zone = 2N Group I metal (e.g. Potassium) has one conduction electron per atom. Electrons fill only half of the states in the lowest band. Electrons therefore can move into higher states due to thermal excitation of application of an electric field - CONDUCTOR For an element with 2 electrons per atom, the lowest band is full. Therefore energy is needed to excite electrons into next band. If E g is > 3eV this is difficult (kt at 300K is ~26meV). Therefore this is an INSULATOR For an element with 2 electrons per atom for which E g is ~1eV some electrons can be thermally excited at 300K, increasing with T. These are called SEMICONDUCTORS (figures from Rosenberg)

13 Extension to 3 dimensions Recall from Lecture 5 that we can construct Brillouin Zones in 3D by bisecting lattice translation (G) vectors along each axis. Different directions (in reciprocal space) give rise to different behaviour The corresponding E(k) relationship is known as the band structure 1 st Brillouin Zone for a fcc lattice (3D) a Band Structure for Silicon

14 The Fermi Surface The Brillouin Zone shape depends on the type of crystal but tells us nothing about where the electrons are In the free electron model, the boundary between full and empty states (at the Fermi energy) is simply a sphere this is known as the FERMI SURFACE Group I metals, such as Potassium have a spherical Fermi surface totally enclosed with the first Brillouin Zone For other elements with a higher number of outer shell electrons the Fermi surface can touch the Brillouin zone and become distorted (e.g. Copper). This has a strong effect on the electrical and thermal properties. Copper Potassium k y k z See the full set of Fermi surfaces here: k x

15 The Tight Binding Model This is an alternative description of how energy bands form in crystals Consider an individual atom with 2 energy levels E Two isolated atoms can have the same energy levels

16 The Tight Binding Model This is an alternative description of how energy bands form in crystals E Consider an individual atom with 2 energy levels

17 The Tight Binding Model This is an alternative description of how energy bands form in crystals E Consider an individual atom with 2 energy levels E g N High density Small lattice spacing Larger E g Forbidden band gap forms due to Pauli exclusion principle

18 Band gaps of common semiconductors and insulators Diamond Crystals with a small lattice constant tend to have a larger band gap and tend to be strong

19 Semiconductors As stated earlier, semiconductors are intermediate band gap materials which may conduct due to thermal excitation of electrons By increasing the number of conduction electrons with temperature, the resistance decreases with increasing temperature (unlike metals). Silicon is commonly used in thermistors Carrier density still much lower than a good metallic conductor, e.g. Si (300K)* n ~ cm -3 Cu (300K) n ~ cm -3 *doping Si with impurities can increase n up to ~10 20 cm -3