Energy Bands in Solids 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/cmp2013
Electrons in Solids Considering electrons in metals as free particles, electron gas in a box explains many experimental results which? Still, why are some solids metals and others insulators? For metals we assumed that electrons are free We know, however, that there are no free electrons in insulators QM give us the answer! Need a more realistic potential for electrons reflecting periodic ionic structure of solids
Splitting of Energy levels If the atoms are far apart- atoms can be considered as isolated Under the influence of neighbouring atoms- splitting of energy levels occurs The width of K-shell is less as compared to L-shell and so on Effect of distance on splitting of energy levels
Energy E3 (M shell) E2 (L shell) E1 (K shell)
Band Energy Energy Band Forbidden Energy Gap Energy Band
ENERGY BANDS IN SOLIDS Explains and distinguish between conductors, insulators and semi conductors Conduction Band small energy can remove an electron from an atom Valence Band Below the conduction band Filled Band- Near the nucleus Insulators- Conduction band is empty but valence is almost completely filled+ wide energy gap between the two bands (Poor conductivity) Conductors- Overlapping of both conduction and valence bands Conduction es move almost twice as fast as holes
Realistic Potential in Solids n i are integers Example: 2D Lattice c n b n a n T T U r U r 3 2 1 ) ( ) ( 3 ; 2 2 1 2 1 n n b n n a T
Realistic Potential in Solids For one dimensional case where atoms (ions) are separated by distance d, we can write condition of periodicity as U( x) U( x dn)
Realistic Potential in Solids Multi-electron atomic potentials are complex Even for hydrogen atom with a simple Coulomb potential solutions are quite complex So we use a model one-dimensional periodic potential to get insight into the problem
Bloch Functions Felix Bloch showed that the actual solutions to the Schrödinger equation for electrons in a periodic potential must have the special form: k (r) uk(r) i(kr) where u has the period of the lattice, that is u r) u k ( k (r e T)
Bloch s Theorem Bloch s Theorem states that for a particle moving in the periodic potential, the wavefunctions ψ(x) are of the form ( x) u k ( x) e ikx, whereu u k k ( x) u ( x) is a pe riodicfunction k ( x d) u k (x) is a periodic function with the periodicity of the potential The exact form depends on the potential associated with atoms (ions) that form the solid
Bloch s Theorem Bloch s Theorem imposes very special conditions on any solution of the Schrödinger equation, independent of the form of the periodic potential The wave vector k has a two-fold role: 1. It is still a wave vector in the plane wave part of the solution 2. It is also an index to u k (x) because it contains all the quantum numbers, which enumerate the wavefunction
Bloch s Theorem What is probability density of finding particle at coordinate x? 2 * * * * 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ] ) ( [ ] ) ( [ ) ( ) ( ) ( ) ( ) ( x u x P x u x u e e x u x u x P e x u e x u x P x x x x P k k k ikx ikx k k ikx k ikx k But uk(x) 2 is periodic, so P(x) is as well 2 (x) u k
Bloch s Theorem P( x) P( x d) The probability of finding an electron at any atom in the solid is the same!!! Each electron in a crystalline solid belongs to each and every atom forming the solid
Kronig-Penney Model U(x) U0 (a+b) b 0 a a+b x The wave equation can be solved when the potential is simple... such as a periodic square well.
Kronig-Penney Model 2 2 d U( x) 2m 2 dx Region I - where 0 < x < a and U = 0 The eigenfunction is a linear combination of plane waves traveling both left and right: Ae ikx Be ikx The energy eignevalue is: 2 K 2m 2
Kronig-Penney Model 2 2 d U( x) 2m 2 dx Region II - where b < x < 0 and U = U0 Within the barrier, the eigenfunction looks like this Ce Qx De Qx and U 0 2 Q 2m 2
Kronig-Penney Model U(x) U0 II I III (a+b) b 0 a a+b x To satisfy Mr. Bloch, the solution in region III must also be related to the solution in region II. III II e ik ( ab)
Kronig-Penney Model A,B,C, and D are chosen so that both the wavefunction and its derivative with respect to x are continuous at the x = 0 and a. At x = 0... A B C D ik( A B) Q( C D) At x = a... III II e ik ( ab) Ae ika Be ika ( Ce Qb De Qb ) e ik( ab) ika ika Qb Qb ik( a b) Be Q Ce De e ik Ae
Result for E < U 0 : Kronig-Penney Model 2 Q K 2QK 2 sinhqb sin Ka coshqb cos Ka cos k( a b) To obtain a more convenient form Kronig and Penney considered the case where the potential barrier becomes a delta function, that is, the case where U 0 is infinitely large, over an infinitesimal distance b, but the product U0b remains finite and constant. 2 Q U and also goes to infinity as U 0. Therefore: 0 Lim U0 ( Q 2 K 2 ) Q 2
Kronig-Penney Model What happens to the product Qb as U0 goes to infinity? b becomes infinitesimal as U 0 becomes infinite. However, since Q is only proportional to the square root of U 0, it does not go to infinity as fast as b goes to zero. So, the product Qb goes to zero as U 0 becomes infinite. As a results of all of this... sinqb Qb cos Qb 1 Q K 1
Kronig-Penney Model P Q 2 ba 2 P Ka sin Ka cos Ka cos ka
Kronig-Penney Model Plot of energy versus wavenumber for the Kronig-Penney Potential, with P = 3/2. 0 2 3 ka 0 ka
Covalent Bonding Revisited When atoms are covalently bonded electrons supplied by atoms are shared by these atoms since pull of each atom is the same or nearly so H 2, F 2, CO, Example: the ground state of the hydrogen atoms forming a molecule If the atoms are far apart there is very little overlap between their wavefunctions If atoms are brought together the wavefunctions overlap and form the compound wavefunction, ψ 1 (r)+ψ 2 (r), increasing the probability for electrons to exist between the atoms 1s 1 r / a a 3 0 e 0
Schrödinger Equation Revisited If a wavefunctions ψ 1 (x) and ψ 2 (x) are solutions for the Schrödinger equation for energy E, then functions -ψ 1 (x), -ψ 2 (x), and ψ 1 (x)±ψ 2 (x) are also solutions of this equations the probability density of -ψ 1 (x) is the same as for ψ 1 (x) 2 2m d 2 2 2 d 2m dx 1,2 2 ( x) U( x)[ 1,2 ( x)] E [ 1( x) 2( x)] U( x)[ 1( x) 2( x)] 2 dx 1,2 ( x) E[ ( x) ( x)] 1 2
Band Theory of Solids Consider an atom with only one electron in s-state outside of a closed shell Both of the wavefunctions below are valid and the choice of each is equivalent s s ( r) ( r) Af Af ( r) e ( r) e Zr/ na Zr/ na 0 0 If the atoms are far apart, as before, the wavefunctions are the same as for the isolated atoms
Band Theory of Solids Once the atoms are brought together the wavefunctions begin to overlap There are two possibilities 1. Overlapping wavefunctions are the same (e.g., ψ s+ (r)) 2. Overlapping wavefunctions are different The sum of them is shown in the figure These two possible combinations represent two possible states of two atoms system with different energies
Symmetric and Anti-symmetric Combinations of Ground State Eigenfunctions
Six States for Six Atom Solid
Splitting of 1s State of Six Atoms
Atoms and Band Structure Consider multi-electron atoms: 1. The outer electrons (large n and l) are closer to each other than the inner electrons Thus, the overlap of the wavefunctions of the outer electrons is stronger than overlap of those of inner electrons Therefore, the bands formed from outer electrons are wider than the bands formed from inner electrons Bands with higher energies are therefore wider!
Splitting of Atomic Levels in Sodium
Occupation of Bands Sodium
Splitting of Atomic Levels in Carbon
Occupation in Carbon at Large Atomic Separation
Actual Occupation of Energy bands in Diamond
Insulators, Semiconductors, Metals The last completely filled (at least at T = 0 K) band is called the Valence Band The next band with higher energy is the Conduction Band The Conduction Band can be empty or partially filed The energy difference between the bottom of the CB and the top of the VB is called the Band Gap (or Forbidden Gap)
Insulators, Semiconductors, Metals Consider a solid with the empty Conduction Band If apply electric field to this solid, the electrons in the valence band (VB) cannot participate in transport (no current)
Insulators, Semiconductors, Metals The electrons in the VB do not participate in the current, since Classically, electrons in the electric field accelerate, so they acquire [kinetic] energy In QM this means they must acquire slightly higher energy and jump to another quantum state Such states must be available, i.e. empty allowed states But no such state are available in the VB! This solid would behave as an insulator
Insulators, Semiconductors, Metals Consider a solid with the half filled Conduction Band (T = 0K) If an electric field is applied to this solid, electrons in the CB do participate in transport, since there are plenty of empty allowed states with energies just above the Fermi energy This solid would behave as a conductor (metal)
Band Overlap Many materials are conductors (metals) due to the band overlap phenomenon Often the higher energy bands become so wide that they overlap with the lower bands additional electron energy levels are then available
Band Overlap Example: Magnesium (Mg; Z =12): 1s 2 2s 2 2p 6 3s 2 Might expect to be insulator; however, it is a metal 3s-band overlaps the 3p-band, so now the conduction band contains 8N energy levels, while only have 2N electrons Other examples: Zn, Be, Ca, Bi
Band Hybridization In some cases the opposite occurs Due to the overlap, electrons from different shells form hybrid bands, which can be separated in energy Depending on the magnitude of the gap, solids can be insulators (Diamond); semiconductors (Si, Ge, Sn; metals (Pb)
Insulators, Semiconductors, Metals There is a qualitative difference between metals and insulators (semiconductors) the highest energy band containing electrons is only partially filled for Metals (sometimes due to the overlap) Thus they are good conductors even at very low temperatures The resistivity arises from the electron scattering from lattice vibrations and lattice defects Vibrations increases with temperature higher resistivity The concentration of carriers does not change appreciably with temperature
Insulators, Semiconductors, Metals The difference between Insulators and Semiconductors is quantitative The difference in the magnitude of the band gap Semiconductors are Insulators with a relatively small band gap At high enough temperatures a fraction of electrons can be found in the conduction band and therefore participate in transport. Hence resistivity decrease with increase of temperature in semiconductor.
Insulators vs Semiconductors There is no difference between Insulators and Semiconductors at very low temperatures In neither material are there any electrons in the conduction band and so conductivity vanishes in the low temperature limit
Insulators vs Semiconductors Differences arises at high temperatures A small fraction of the electrons is thermally excited into the conduction band. These electrons carry current just as in metals The smaller the gap the more electrons in the conduction band at a given temperature Resistivity decreases with temperature due to higher concentration of electrons in the conduction band in semiconductore. 1 q 2 n m *
Holes Consider an insulator (or semiconductor) with a few electrons excited from the valence band into the conduction band Apply an electric field Now electrons in the valence band have some energy sates into which they can move The movement is complicated since it involves ~ 10 23 electrons
Concept of Holes Consider a semiconductor with a small number of electrons excited from the valence band into the conduction band If an electric field is applied, the conduction band electrons will participate in the electrical current the valence band electrons can move into the empty states, and thus can also contribute to the current
Holes from the Band Structure Point of View If we describe such changes via movement of the empty states the picture can be significantly simplified This empty space is a Hole Deficiency of negative charge holes are positively charged Holes often have a larger effective mass (heavier) than electrons since they represent collective behavior of many electrons in the VB (i.e. low lying energy band)
Holes We can replace electrons at the top of e th band which have negative mass (and travel in opposite to the normal direction) by positively charged particles with a positive mass, and consider all phenomena using such particles Such particles are called Holes Holes are positively charged and move in the opposite direction as electrons they replace
Hole Conduction To understand hole motion, one requires another view of the holes, which represent them as electrons with negative effective mass To imagine the movement of the hole think of a row of chairs occupied by people with one chair empty To move all people rise all together and move in one direction, so the empty spot moves in the opposite direction
Concept of Holes If we describe such changes via movement of the empty states the picture will be significantly simplified This empty space is called a Hole Deficiency of negative charge can be treated as a positive charge Holes act as charge carriers in the sense that electrons from nearby sites can move into the hole Holes are usually heavier than electrons since they depict collective behavior of many electrons
Conduction Electrical current for holes and electrons in the opposite direction
What are quantum numbers? Quantum numbers label eigenenergies and eigenfunctions of a Hamiltonian Sommerfeld: k-vector ( k is momentum) Bloch: -vector ( is the crystal momentum) and n k k (the band index). The Crystal Momentum is not the Momentum of a Bloch electron: the rate of change of an electron momentum is given by the total forces on the electron, but the rate of change of electronic crystal momentum is: dr dt 1 n( k) d k v ( ) ˆ n k n k v n k ; ee( r, t) vn( k) B( r, t) k dt where forces are exerted only by the external fields, and not by the periodic field of the lattice.
Semiclassical dynamics of Bloch electrons Bloch states have the property that their expectation values of and, follow classical dynamics. The only change is that now ( n k ) (band structure) must be used: A perfectly periodic ionic arrangement has zero resistance. Resistivity comes from imperfections (example: a barrier induces a reflected and transmitted Bloch wave), which control the mean-free path. This can be much larger than the lattice spacing. A fully occupied band does not contribute to the current since the electrons cannot be promoted to other empty states with higher. The current is induced by rearrangement of states near the Fermi energy k in a partially occupied band. Limits of validity: e Hclassical n k ea( r) e( r) BL k 2m dp d( k) H dr H, v dt dt r dt p p eea n const (no interband transitions) ( k) ( k),, ( k) ( k) ( k) 2 2 gap eb gap c gap n n F m F, gap a r k
What is the range of quantum numbers? k Sommerfeld: runs through all of k-space consistent with the Born-von Karman periodic boundary conditions: ( x L, y, z) ( x, y, z) 2 ( x L, y, z) ( x, y L, z) k nx, ny, nz nx, ny, nz 0, 1, 2, L ( x, y, z) ( x, y, z L) ( x L, y, z) ( x, y, z) 0 ( x L, y, z) ( x, y L, z) 0 k nx, ny, nz nx, ny, nz 1,2, L ( x, y, z) ( x, y, z L) 0 n k Bloch: For each, runs through all wave vectors in a single primitive cell of the reciprocal lattice consistent with the Bornvon Karman periodic boundary conditions; n runs through an infinite set of discrete values.
What are the energy levels? Sommerfeld: ( k) 2 2 k 2m ( ) n k Bloch: For a given band index n, has no simple explicit form. The only general property is periodicity in the reciprocal space: n ( k G) ( k) n
What is the velocity of electron? Sommerfeld: The mean velocity of an electron in a level with wave vector is: k v k 1 m k Bloch: The mean velocity of an electron in a level with band index and wave vector is: n Conductivity of a perfect crystal: 1 ( k) v ( k) n n k NOTE: Quantum mechanical definition of a mean velocity v v r r mi k * ˆ dr ( ) ( )
What is the Wave function Sommerfeld: The wave function of an electron with wave vector k is: 1 () e ikr k r Bloch: The wave function of an electron with band index and wave vector is: k where the function n has no simple explicit form. The only general property is its periodicity in the direct lattice (i.e., real space): V ( r) e kr ( r) k k nk () r ( r R) ( r) i nk nk n
Sommerfeld vs. Bloch: Density of States 2 D( ) d ( ) d 2 k k Sommerfeld Bloch 2 D( ) dk n( k) d 2 n BZ..
Bloch: van Hove singularities in the DOS 2 2 D( ) dk ( k) dk d n 2 2 d n n B. Z. n S ( E) k n 1 D( ) d dsk d 2 ds ( k) d ( k) Δk ( k) k k n k n D( ) 63
Sommerfeld vs. Bloch: Fermi surface ( T 0) Fermi energy represents the sharp occupancy F cut-off at T=0 for particles described by the Fermi-Dirac statitics. Fermi surface is the locus of points in reciprocal space where ( k) F F No Fermi surface for insulators! k F kf Points of Fermi Surface in 1D
Sommerfeld vs. Bloch: Fermi surface in 3D Sommerfeld: Fermi Sphere Bloch: Sometimes sphere, but more likely anything else For each partially filled band there will be a surface reciprocal space separating occupied from the unoccupied levels the set of all such surfaces is known as the Fermi surface and represents the generalization to Bloch electrons of the free electron Fermi sphere. The parts of the Fermi surface arising from individual partially filled bands are branches of the Fermi surface: for each n solve the equation in variable. n( k) k F
Is there a Fermi energy of intrinsic Semiconductors? F If is defined as the energy separating the highest occupied from the lowest unoccupied level, then it is not uniquely specified in a solid with an energy gap, since any energy in the gap meets this test. People nevertheless speak of the Fermi energy on an intrinsic semiconductor. What they mean is the chemical potential, which is well defined at any non-zero temperature. As T 0, the chemical potential of a solid with an energy gap approaches the energy of the middle of the gap and one sometimes finds it asserted that this is the Fermi energy. With either the correct of colloquial definition, does not have a solution in a n( k) F solid with a gap, which therefore has no Fermi surface 66
DOS of real materials: Silicon, Aluminum, Silver
Colloquial Semiconductor Terminology in Pictures PURE DOPPED
Measuring DOS: Photoemission spectroscopy Fermi Golden Rule: Probability per unit time of an electron being ejected is proportional to the DOS of occupied electronic states times the probability (Fermi function) that the state is occupied: 1 I( kin ) D( bin) f ( bin) D( kin ) f ( kin ) ( ) kin
Measuring DOS: Photoemission spectroscopy Once the background is subtracted off, the subtracted data is proportional to electronic density of states convolved with a Fermi functions. We can also learn about DOS above the Fermi surface using Inverse Photoemission where electron beam is focused on the surface and the outgoing flux of photons is measured.
Band Theory of Solids In isolated atoms the electrons are arranged in energy levels
In solids the outer electron energy levels become smeared out to form bands The highest occupied band is called the VALENCE band. This is full. For conduction of electrical energy there must be electrons in the CONDUCTION band. Electrons are free to move in this band.
Insulators : There is a big energy gap between the valence and conduction band. Examples are plastics, paper.. Conductors : There is an overlap between the valence and conduction band hence electrons are free to move about. Examples are copper, lead. Semiconductors : There is a small energy gap between the two bands. Thermal excitation is sufficient to move electrons from the valence to conduction band. Examples are silicon,germanium.
When a conductor is heated its resistance increases ; The atoms vibrate more and the electrons find it more difficult to move through the conductor. R But in a semiconductor the resistance decreases with an increase in temperature. Electrons can be excited up to the conduction band. Conductivity increases R T T
Doping This is the replacement of a silicon atom in the crystal structure with a different atom. Typically 1 atom per 100 million silicon atoms is replaced. This changes the electrical properties. Silicon is in group 4 of the periodic table. It has 4 valence electrons.
P type silicon. Replace a silicon atom with a group 3 atom such as aluminium.( 3 valence electrons) This lowers the number of electrons in the valence band which creates a hole in the structure. Electrons can move through the structure. This is called p type silicon. P is for positive as there is a deficiency of electrons.
N type Replace a silicon atom with a group 5 atom such as phosphorous ( 5 valence electrons ) The extra electron introduced here goes into the conduction band as the valence band is full. This is n type silicon. N for negative as additional electrons have been introduced into the structure.
Energy Band View of Semiconductors Conductors, semiconductors, insulators: Why is it that when individual atoms get close together to form a solid such as copper, silicon, or quartz they form materials that have a high, variable, or low ability to conduct current? Understand in terms of allowed, empty, and occupied electronic energy levels and electronic energy bands.
Fig. 1. Calculated energy levels in the diamond structure as a function of assumed atomic spacing at T = 0 o K. (From Introduction to Semiconductor Physics, Wiley, 1964)
In Fig. 1, at right atoms are essentially isolated; at left atomic separations are just a few tenths of a nanometer, characteristic of atoms in a silicon crystal. If we start with N atoms of silicon at the right, which have 14 electrons each, there must be 14N allowed energy levels for the electrons. (You learned about this in physics in connection with the Bohr atom, the Pauli Exclusion principle, etc.) If the atoms are pushed together to form a solid chunk of silicon, the electrons of neighboring atoms will interact and the allowed energy levels will broaden into energy bands.
When the actual spacing is reached, the quantummechanical calculation results are that: At lowest energies very narrow ranges of energy are allowed for inner electrons (these are core electrons, near the nuclei); A higher band of 4N allowed states exists that, at 0 o K, is filled with 4N electrons; Then an energy gap, E G, appears with no allowed states (no electrons permitted!); and At highest energies a band of allowed states appears that is entirely empty at 0 o K. Can this crystal conduct electricity?
NO, it cannot conductor electricity at 0 o K because that involves moving charges and therefore an increase of electron energy but we have only two bands of states separated by a forbidden energy gap, E G. The (lower) valence band is entirely filled, and the (upper) conduction band states are entirely empty. To conduct electricity we need to have a band that has some filled states (some electrons!) and some empty states that can be occupied by electrons whose energies increase.
Fig. 2 shows the situation at 0 o K for (left) a metallic solid such as copper, and (right) a semiconductor such as silicon. The metal can conduct at 0 o K because the uppermost band contains some electrons and some empty available energy states. The semiconductor cannot conduct it is an insulator. If we raise the temperature of the semiconductor, some electrons in the filled valence band may pick up enough energy to jump up into an unoccupied state in the conduction band. Thus, at a finite temperature, a pure (intrinsic) semiconductor has a finite electrical conductivity.
Fig. 2. Electronic energy bands for (a) metallic conductor at T = 0 o K; (b) insulator or intrinsic semiconductor at 0 o K.
How much conductivity can a pure (intrinsic) semiconductor exhibit? This depends on how much thermal energy there is and the size of the energy gap, E G : Mean thermal energy is kt, where k = Boltzmann s constant = 1.38 x 10-23 J/K and T is the absolute temperature. In electron volts this is kt/qe, or 26 millivolts for room temperature (300 o K) For silicon, E G = 1.12 ev at 300 o K This leads in pure (intrinsic) Si to a carrier concentration n i = 10 10 carriers/cm 3 at 300 o K
Adding Impurities (Doping) to Adjust Carrier Concentrations Adjust carrier concentrations locally in semiconductor by adding easily ionized impurities to produce mobile electrons and/or holes To make silicon N-type: Add valence 5 phosphorous (P) atoms to valence 4 silicon. Fifth electron is easily freed from the atom by a little thermal energy (0.045 ev for phosphorous) to create (donate) a mobile electron. Fig. 3a shows the donor energy level just below bottom of conduction band. To make silicon P-type: Add valence 3 boron (B) to silicon. An electron at the top of the valence band can pick up enough thermal energy to release it from the silicon so it attaches to a boron atom, completing its outer ring of electrons. In the band picture, Fig. 3b, this is represented by an acceptor. energy level 0.045 ev above the top of the valence band.
Fig. 3a. Electronic energy band for n-type semiconductor (Ge) with donors only.
Fig. 3b. Electronic energy band for p-type semiconductor (Ge) with acceptors only.
Energy Bands Bloch Functions Nearly Free Electron Model Kronig-Penney Model Wave Equation of Electron in a Periodic Potential Number of Orbitals in a Band
Some successes of the free electron model: C, κ, σ, χ, Some failures of the free electron model: Distinction between metals, semimetals, semiconductors & insulators. Positive values of Hall coefficent. Relation between conduction & valence electrons. Magnetotransport. Band model New concepts: Effective mass Holes finite T impuritie s
Nearly Free Electron Model Bragg reflection no wave-like solutions energy gap Bragg condition: 2 2 k G k 2k G G 0 k 1 2 G / / e e i x a i x a 2cos x/ a 2i sin x / a k a
Origin of the Energy Gap cos 2 / U x U x a 2cos x/ a 2isin x / a 4co 4sin 2 s / 2 x a x/ a a 2 2 a 2 4U 0 dx cos 2 x / a U 0 P.Ravindran, PHY075- Condensed Matter Physics, Spring 02013 : Energy Bands in Solids Eg dx U x E E
Bloch Functions Periodic potential Translational symmetry Abelian group T = {T(Rl)} Rr r R T 1 T R f r f T R r f r R k-representation of T(Rl) is i e kr l Basis = e i kr Corresponding basis function for the Schrodinger equation must satisfy T 1 i kr l Rl k r k T Rl r r R e k r k l i l or r R e kr r k l k This can be satisfied by the Bloch function k i r e kr ukr where u r R u r k l k i G l e R 1 representative values of k are contained inside the Brillouin zone.
Kronig-Penney Model 2 2 d U 2 x 2m dx U na b x na b a 1 0 U 0 n a b a x n a b 0 xa U 0 Ae ik x Be ik x K 2m 2 2 b x 0 U U 0 Ce Qx De Qx Bloch theorem: ψ(0) continuous: A B C D a x a b b x e 0 ik a b U 0 Q 2m 2 2 ika ika ψ(a) continuous: Ae Be b e ik ab Qb Qb ikab Ce De e ψ (0) continuous: ika ikb QC QD ika ika ψ (a) continuous: ikae ikbe b e i k ab Qb Qb ik ab QCe QDe e
1 1 1 1 A 0 ika ika Qbik ab Qbi k ab e e e e B 0 ik ik Q Q C 0 ika ika Qbi k ab Qbik ab ike ike Qe Qe D 0 Q K 2QK 2 2 sinh Qb sin Ka cosh Qb cos Ka cos k a b 1 2 Delta function potential: b 0, U0 such that Q ba P finite 2 Thus Q K, Qb 1 so that P Ka sin Ka cos Ka cos ka
2 2 2 2 2 K Ka in units of 2 2m 2ma P Ka sin Ka cos Ka cos ka P 3 2
H j j Matrix Mechanics E Ansatz r ci ir r c H E c j j j j j r H c E c i j j j i j j H i j E Si j c j 0 Hi j i H j i j i j j i 3 * A d r A S r r Matrix equation H E S c 0 Secular equation: HE S 0 Eigen-problem Orthonormal basis: Si j i j i j S I
Fourier Series of the Periodic Potential U r R Ur m l U r U Gm e m i G r R 3 l li ai i1 3 G m b a b 2 m i i i1 1 3 i 1 m n 3 i m n d x e G G r d x e mn G G r V V i j i j V = Volume of crystal volume of unit cell 1 m 3 i G r G r U m d x e U For a lattice with atomic basis at positions ρα in the unit cell r rρ U r ρ R U rρ U U i m i m r G U U e e m m G ρ G r m U G U G e l U i Gm G e i G ρ m m is the structural factor m m r
Plane Wave Expansion Bloch function r k r k V 1 i k e r V 1 i k r i Gm e m C e m r k Gm k Gn mn Cm r k Gm m V = Volume of crystal 2 k G 2 m k G H k G k G U k G 2m n m nm n m, k G k G G U G G U U n m p n p m p Matrix form of the Schrodinger equation: 2 2 k Gn E nm U Gn Gm Cm 0 m 2m n m n = 0: 2 2 E C0 U G mcm 0 P.Ravindran, PHY075-2mCondensed Matter m Physics, Spring 2013 : Energy Bands in Solids k (central equation)
Crystal Momentum of an Electron Properties of k: i e kr u k ikr l ikr i l r r r R e e u r R e kr r k k k l k l e i U = 0 k r kr Selection rules in collision processes crystal momentum of electron is k. Eq., phonon k q kg absorption:
Solution of the Central Equation m E U C 0 k G G G m nm n m m k k 2m 2 2 1-D lattice, only U g U g U 0 k2g E U 0 0 C 2 0 U kg E U 0 C 1 0 0 U k E U 0 C0 0 0 U E U C 0 kg 1 0 0 U k2g E C2 0
Kronig-Penney Model in Reciprocal Space U x A x sa a/2 a a/2 s 2 U cosg x n n /2 1 a a a/2 n 1 Un dx U x cos Gnx dx A A (only s = 0 term contributes) a/2 1 1 dx cos G x cos G x a 2 a/2 m n mn Eigen-equation: m E A kg C 0 m nm m cot x n C 1 n x n A E k G 2 1 2mA 2mE 2 n k 2 a n m C m 1 n A E k G 1 1 1 2 n 2K 2 n 2 n n k K k K a a n n E 2 A k 2m K 2 n a 2 2mE 2 a cot 1 cot 1 k K a k K a 4K 2 2
2 a 1 1 cot k K a cot k K a 2mA 4K 2 2 K 2mE 2 a 4K sin Ka 1 1 sin k K a sin k K a 2 2 cot a cot b sin a b sin asin b a sin Ka 2K cos ka cos Ka 2sin a sin b cos a b cos a b P Ka sin Ka cos Ka cos ka with P maa 2 2 2 (Kronig-Penney model)
Empty Lattice Approximation Free electron in vacuum: Free electron in empty lattice: k k 2m 2 2 k G 2 2 k n k k kg k1 st BZ 2m Simple cubic
Approximate Solution Near a Zone Boundary m E U C 0 k G G G Weak U, λk2g >> U m nm n m m k k 2m 2 2 kg E U 0 C 1 0 U E U C 0 k 0 0 U k g E C 1 0 k near zone right boundary: k 1 2 k 0 g a kg E U C1 0 U E C 0 2 2 kg k kg k E E U 0 k g U U C C C 1 0 0 E k g for E near λk 1 E 2 2 2 4U kg k kg k
1 E k g k k g k 2 2 2 4U 2 2 2 g 2 g K K 2 2 2m 2m kg k K m 2 2 2 1 k g K 2 K << g/2 g 4 2 2 2 g 2 g K K 2 2 2m 2m kg k 2 Kg m 2U 2 2 2 2 1 Kg 4U 2U 1 8 2 U m kg k 2 2 g/2 K 2U 1 2 2 U 2m E 2m 2 2m U 2 2 2 2 g K g/2 U 1 2
C C 1 0 C k g C C k C 0 1 x C e C e ik x 0 1 i kg x
Number of Orbitals in a Band Linear crystal of length L composed of of N cells of lattice L Na constant a. ik x Periodic boundary x L x e u x condition: 2 n 2 n k L 2 n k N inequivalent values L Na of k Generalization to 3-D crystals: Number of k points in 1st BZ = Number of primitive cells Each primitive cell contributes one k point to each band. Crystals with odd numbers of electrons in primitive cell must be metals, e.g., alkali & noble metals metal semi-metal insulator
Braggs Law of Diffraction 2dsin(0)=nl d=spacing of parallel atomic planes 0=angle of incidence l=wavelength of incident beam Requires l<=2d Bragg reflection of electron waves in crystals cause energy gap. At locations of Bragg reflection solution to the Schrödinger equation do not exist.
Brillouin Zone Defined as the primitive zone in reciprocal lattice Made up of all the wave vectors that satisfy the diffraction condition Wave vectors outside of Brillouin zone do not lead to solution of the Schrödinger Equation
Origin of Energy Gap In one dimension, solution to the Schrödinger equation at the boundaries of Brillouin are standing waves. Standing waves made up of equal parts of left and right moving waves Waves traveling in different directions have different energy, corresponding to energy gap
Band Gap zone boundary
+ Origin of the Band Gap To get a standing wave at the boundaries, you can take a linear combination of two plane waves: a x i ikx a x i ikx e e e e / 2 / 1 a x e e a x i a x i 2cos / / 2 1 / / 1 2 2 sin i x a i x a x e e i a
Origin of the Band Gap Electron Density
Origin of the Band Gap
Summary Band Structures What is being plotted? Energy vs. k, where k is the wavevector that gives the phase of the AO s as well as the wavelength of the electron wavefunction (crystal momentum). How many lines are there in a band structure diagram? As many as there are orbitals in the unit cell. How is the center of gravity energy level of each band determined? Usually follows from the MO diagram. How do we determine whether a band runs uphill or downhill? By comparing the orbital overlap at k=0 and k=/a. How do we distinguish metals from semiconductors and insulators? The Fermi level cuts a band in a metal, whereas there is a gap between the filled and empty states in a semiconductor. Why are some bands flat and others steep? This depends on the degree of orbital overlap between building units. Wide bands Large intermolecular overlap delocalized e - Narrow bands Weak intermolecular overlap localized e -
Energy Bands in Solids:
Bandstructure - Nomenclature For most purposes, it is sufficient to know the En(k) curves - the dispersion relations - along the major directions of the reciprocal lattice. This is exactly what is done when real band diagrams of crystals are shown. Directions are chosen that lead from the center of the Wigner- Seitz unit cell - or the Brillouin zones - to special symmetry points. These points are labeled according to the following rules: Points (and lines) inside the Brillouin zone are denoted with Greek letters. Points on the surface of the Brillouin zone with Roman letters. The center of the Wigner-Seitz cell is always denoted by a G
Bandstructure - Nomenclature For cubic reciprocal lattices, the points with a high symmetry on the Wigner- Seitz cell are the intersections of the Wigner Seitz cell with the low-indexed directions in the cubic elementary cell. simple cubic
BZ for fcc - Nomenclature We use the following nomenclature: (fcc): The intersection point with the [100] direction is called X (H) The line G X is called. The intersection point with the [110] direction is called K (N) The line G K is called S. The intersection point with the [111] direction is called L (P) The line G L is called L. Brillouin Zone for fcc is wigner seitz cell for bcc and vice versa.
Nomenclature BZ for bcc lattice We use the following nomenclature: (red for fcc, blue for bcc): The intersection point with the [100] direction is called X (H) The line G X is called. The intersection point with the [110] direction is called K (N) The line G K is called S. The intersection point with the [111] direction is called L (P) The line G L is called L.
Electron Energy Bands in 3D Real crystals are three-dimensional and we must consider their band structure in three dimensions, too. Of course, we must consider the reciprocal lattice, and, as always if we look at electronic properties, use the Wigner-Seitz cell (identical to the 1st Brillouin zone) as the unit cell. There is no way to express quantities that change as a function of three coordinates graphically, so we look at a two dimensional crystal first (which do exist in semiconductor and nanoscale physics). The qualitative recipe for obtaining the band structure of a two-dimensional lattice using the slightly adjusted parabolas of the free electron gas model is simple:
Electron Energy Bands in 3D The lower part (the "cup") is contained in the 1st Brillouin zone, the upper part (the "top") comes from the second BZ, but it is folded back into the first one. It thus would carry a different band index. This could be continued ad infinitum; but Brillouin zones with energies well above the Fermi energy are of no real interest. These are tracings along major directions. Evidently, they contain most of the relevant information in condensed form. It is clear that this structure has no band gap.
Band Structure: KCl We first depict the band structure of an ionic crystal, KCl. The bands are very narrow, almost like atomic ones. The band gap is large around 9 ev. For alkali halides they are generally in the range 7-14 ev.
Band Structure: silver (fcc)
Electron Density of States The electron density of states is a key parameter in the determination of the physical phenomena of solids. Knowing the energy levels, we can count how many energy levels are contained in an interval E at the energy E. This is best done in k - space. In phase space, a surface of constant energy is a sphere as schematically shown in the picture. Any "state", i.e. solution of the Schrodinger equation with a specific k, occupies the volume given by one of the little cubes in phase space. The number of cubes fitting inside the sphere at energy E thus is the number of all energy levels up to E.
Electron Density of States: Free Electrons Counting the number of cells (each containing one possible state of ) in an energy interval E, E + E thus correspond to taking the difference of the numbers of cubes contained in a sphere with "radius" E + E and of radius E. We thus obtain the density of states D(E) as DE ( ) 1 N( E, E E) N( E) V E 1 dn V de where N(E) is the number of states between E = 0 and E per volume unit; and V is the volume of the crystal.
Electron Density of States: Free Electrons The volume of the sphere in k-space is V 4 k 3 3 The volume Vk of one unit cell, containing two electron states is V k 2 L 3 The total number of states is then N V 4 k L k L 2 2 3 3 3 3 3 V 3 k 38 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Energy Bands in Solids 2
Electron Density of States: Free Electrons E 2 2 k 2mE k 2 2m N 3 3 3 k L L 2mE 2 2 2 3 3 32 / D( E) 1 dn 1 2m L de 2 3 2 2 3/2 E
Electron Density of States: Free Electrons D(E)
DOS in real Materials
Electron Density of States: If we know the band structure at every point in the Brillouin zone, then the DOS is given by the formula ds D( ) k 3 n 4 n 1 where the integral is over the surface Sn() is the surface in k space at which the n th eigenvalue has the value n. Obviously we can not evaluate this integral directly, since we don't know n(k) at all points; and we can only guess at the properties of its gradient. One common approximation is to use the tetrahedron method, which divides the Brillouin zone into (surprise) tetrahedra, and then linearly interpolate within the tetrahedra to determine the gradient. This method is an approximation, but its accuracy obviously improves as we increase the number of k-points.
Electron Density of States: ds D( ) k 3 n 4 n 1 When the denominator in the integral is zero, peaks due to van Hove singularities occur. Flat bands give rise to a high density of states. It is also higher close to the zone boundaries as illustrated for a two dimensional lattice below. Leon van Hove
Electron Density of States: Real Solids For the case of metals, the bands are very free electronlike (remember we compared with the empty lattice) and the conduction bands are partly filled. The figure shows the DOS for the cases of a metal, Cu, and a semiconductor Ge. Copper has a free electron-like s-band, upon which d-bands are superimposed. The peaks are due to the d-bands. For Ge the valence and conduction bands are clearly seen.
Electron Density of States The basic shape of the density of states versus energy is determined by an overlap of orbitals. In this case s and d orbitals fcc
Electron Density of States: bcc tungsten