03 Introduction to quantum theory of solids
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03 Introduction to quantum theory of solids

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03 Introduction to quantum theory of solids 03 Introduction to quantum theory of solids Presentation Transcript

  • Chapter 3Electron in a finite space  energy is quantized.Pauli exclusion principle  one quantum state canbe occupied by only one electronCan we extend these concepts to electron in acrystal lattice? September/Oktober 2007 1 Department of Microelectronics & Computer Engineering
  • Formation of Energy BandsWhen two hydrogen atoms are brought close enough for the wave functionsof n=1 electrons to start interacting, the n=1 state splits into two differentenergies, in accordance with Pauli exclusion principle. (a) Probability density function of n=1 electron in an isolated hydrogen atom. (b) Overlapping probability density functions in two adjacent hydrogen atoms. (c) splitting of n=1 state. September/Oktober 2007 1 Department of Microelectronics & Computer Engineering
  • Hypothetically, if we have a periodic arrangement of manyhydrogen atoms and they are brought close enough  initialquantized energy level will split into band of discrete levels. Quasi-continuous when the number of atoms in the system is large. The splitting of an energy state into a band of allowed energies (r0 is the equilibrium inter-atomic distance in the crystal) September/Oktober 2007 3 Department of Microelectronics & Computer Engineering
  • What happens in an atom containing manymore electrons? As atoms come closer, the states in the outermost shell split first. Based on r0, Forbidden energy other energy bands states may or may not splitSplitting of energy states into allowed bands of energies in an atomcontaining electrons up to n=3. September/Oktober 2007 4 Department of Microelectronics & Computer Engineering
  • Actual splitting can be more T=0K lower (valence)complex as indicated for n=3 band full, uppershell in silicon. (conduction) band empty 2N 3s and 6N 3p states merge and form 8N new Isolated silicon atom. states, in which lower 4N states are occupied and upper 4N states are empty. September/Oktober 2007 5 Department of Microelectronics & Computer Engineering
  • KRONIG-PENNEY MODELTo develop the concept of allowed and forbidden energy levelsusing Schrodinger’s wave equation  we consider Kronig-Penny model.First we need to understand how potential and electron energyvary inside an atom September/Oktober 2007 9 Department of Microelectronics & Computer Engineering
  • KRONIG-PENNEY MODEL• potential (V) is inversely Vproportional to distance frompositively charged nucleus.• electron is negatively charged. EE=-eV  energy is negative(which means the electron isattracted to the nucleus) Potential and electron energy functions of a single, non-•At infinite distance from nucleus interacting, one-electron atomboth V and E are zero (freeelectron) September/Oktober 2007 9 Department of Microelectronics & Computer Engineering
  • KRONIG-PENNEY MODEL (a) (b)(a) Overlapping Energy functions of adjacent atoms (b) Net Energy function of a one-dimensional single crystal. September/Oktober 2007 11 Department of Microelectronics & Computer Engineering
  • KRONIG-PENNEY MODEL Simplified potential function  Kronig-Penny modelOne-dimensional periodic potential function of the Kronig-Penny model September/Oktober 2007 12 Department of Microelectronics & Computer Engineering
  • Two regions: I. V = 0  a “free” particle in a potential well.II. V=V0  ∞ an infinite number of potential barriers. • Electron is bound in the crystal when E<V0 • Electrons are contained in the potential wells. •There is possibility of tunneling between potential wells. September/Oktober 2007 12 Department of Microelectronics & Computer Engineering
  • KRONIG-PENNEY MODELBloch Theorem – All one-electron wave functions forproblems involving periodically varying potentialfunctions, must be of the form – ψ ( x) = u ( x )e jkx Bloch functionk  constant of motion (explained later)u(x)  periodic function with period (a+b) September/Oktober 2007 13 Department of Microelectronics & Computer Engineering
  • Referring to chapter 2, total solution to Schrodinger’s equation = Time-independent solution x Time-dependent solution (  )t − j E Ψ ( x, t ) = ψ ( x)ϕ (t ) = u ( x)e .e jkx  E  j  kx − t  Ψ ( x, t ) = u ( x )e    This travelling wave solution represents the motion of an electron in a single-crystal material. k  also referred to as wave number. Department of Microelectronics & Computer Engineering
  • Time-independent Schrodinger wave equation -Region I : V(x)=0. Substituting ψ ( x) = u1 ( x)e jkx 2mE Where α 2 = 2 September/Oktober 2007 15 Department of Microelectronics & Computer Engineering
  • 2 September/Oktober 2007 16Department of Microelectronics & Computer Engineering
  • 2mENow, α 2 = 2 Let September/Oktober 2007 17 Department of Microelectronics & Computer Engineering
  • There are two differential equations for regions I and II September/Oktober 2007 18Department of Microelectronics & Computer Engineering
  • There are two differential equations which give for region I and II the following solutions: September/Oktober 2007 18Department of Microelectronics & Computer Engineering
  • The wave function must be finite everywhere. Also the wave function and its first derivative must be continuous everywhere. September/Oktober 2007 19Department of Microelectronics & Computer Engineering
  • Due to periodicity, ψ (a ) = ψ (− b) Therefore Also September/Oktober 2007 20Department of Microelectronics & Computer Engineering
  • our homogeneous equations in the unknowns A,B,C and D. Only a solution if: determinant (coefficients of A,B,C & D) = 0 September/Oktober 2007 22 Department of Microelectronics & Computer Engineering
  • Only a solution if: det = 0 For det=0, Now : β is an imaginary quantity. So γ = real Substituting for β, September/Oktober 2007 23 Department of Microelectronics & Computer Engineering
  • To simplify further let barrier width b  0 and V0  ∞, suchthat bV0 remains constant. Then - Let September/Oktober 2007 24 Department of Microelectronics & Computer Engineering
  • • This is a relation between parameter k, total Energy E(through parameter α ) and potential barrier bVo.•This is not a solution to Schrodinger’s equation but acondition for a solution to exist.•Crystal infinitely large  k can assume a continuum ofvalues and must be real September/Oktober 2007 24 Department of Microelectronics & Computer Engineering
  • 1st case, free electron:V0=0 hence: α = k and k= p Constant of motion parameter k is related to the  particle momentum for the free electron ( A result which we already know. This validates Kronig-Penny model) September/Oktober 2007 25 Department of Microelectronics & Computer Engineering
  • Parabolic E versus k curve for free electron September/Oktober 2007 26Department of Microelectronics & Computer Engineering
  • Coming back toCrystal-lattice LetThis function can only have values between -1and +1, because it is equal to cos(ka)Hence α a can only have certain allowed valuesHence also the energy can have certain allowedvalues. September/Oktober 2007 27 Department of Microelectronics & Computer Engineering
  • Allowed values of α a September/Oktober 2007 28Department of Microelectronics & Computer Engineering
  • Since E is related to k, we can make a plot of E vs. k. E versus k diagram generated from the figure in the previous slide.Note that E is discontinuous  there are allowed and forbidden energy bands. September/Oktober 2007 29 Department of Microelectronics & Computer Engineering
  • cos ka = cos(ka + 2π ) = cos(ka − 2π ) Various sections of the E versus k diagram can be shifted by n 2aπ 2π E versus k diagram showing displacements by n a of several sections of the allowed energy bands. September/Oktober 2007 30 Department of Microelectronics & Computer Engineering
  • allowed allowed E versus k diagram in the reduced k-space representation September/Oktober 2007 31Department of Microelectronics & Computer Engineering
  • T=0K Empty Conduction band Eg Full Valence band (b) line representation of energy(a) Covalent bonding in Si band diagram 31 Department of Microelectronics & Computer Engineering
  • Band-gap energy in some covalent elements Element Eg (eV) C (diamond) 5.48 Si 1.1 Ge 0.7 Sn (gray) 0.08 32Department of Microelectronics & Computer Engineering
  • T>0KTwo dimensional representation At T>0K, some covalent bonds of breaking of a covalent break giving rise to positively bond. charged empty states and electrons September/Oktober 2007 33 Department of Microelectronics & Computer Engineering
  • E vs. K diagrams at T=0K and T>0K in the absence of external electric field. Electron and empty states are symmetric with respect to k. Even within a band allowed energies are still quantized. This is because electrons are still confined to potential wells. September/Oktober 2007 34Department of Microelectronics & Computer Engineering
  • Drift currentq  electron chargeN  charge density (C/cm3)Vd  drift velocity of electron Considering individual electron velocities (summation taken over unit volume) September/Oktober 2007 35 Department of Microelectronics & Computer Engineering
  • Drift current Vd in +x direction Vd in -x direction K +ve K -ve No of electrons with k related toNo external +|k| value = No of momentum.force electrons with -|k| So drift value. current J = 0 September/Oktober 2007 35 Department of Microelectronics & Computer Engineering
  • When ext force (electric field) applied  Electrons move intoempty energy states in conduction band, gain net energyand a net momentum. EDrift current density due tomotion of electrons - (summation taken Asymmetric distribution of over unit volume) electrons in E vs. k diagram when ext force is applied. September/Oktober 2007 36 Department of Microelectronics & Computer Engineering
  • Electron effective massTotal force acting on electrons in crystal – FTotal = Fext + Fint = maFint  internal forces in the crystal due to +vely charged ionsand –vely charged electronsFext  externally applied electric fieldm  rest mass of electrona  acceleration. September/Oktober 2007 36 Department of Microelectronics & Computer Engineering
  • Electron effective massIt is difficult to account for all of the internal force. So wedefine – Fext = m*am*  effective mass of the electron (which takes intoaccount internal forces)Acceleration a  directly related only to ext applied force. 39Department of Microelectronics & Computer Engineering
  • Free electron: Taking derivative of E w.r.t k, p 2  2k 2 hE= = , since p = = k 2 m 2m λThis implies that from E vs. k diagram we can get thevelocity of the electron in real space. September/Oktober 2007 37 Department of Microelectronics & Computer Engineering
  • Again taking derivative of E w.r.t k,This implies that from E vs. k diagram we can also get the effective massof the electron in real space. For a free electron, the second derivativeand hence mass is constant.(why do we need to extract the electron mass, when we already knowit ??? - read further) September/Oktober 2007 37 Department of Microelectronics & Computer Engineering
  • Free electronMotion of electron is in opposite direction of the appliedelectric field due to the negative charge.We now apply these concepts to the electrons in thecrystal September/Oktober 2007 38 Department of Microelectronics & Computer Engineering
  • Electron in conduction bandThe energy of electrons at thebottom of the conduction bandcan be approximated as - C1  +ve Conduction and valence bands in reduced k space with their parabolic approximations to compare with free electron September/Oktober 2007 39 Department of Microelectronics & Computer Engineering
  • Electrons in conduction band Effective mass: • connects quantum mechanics and classical mechanics • effective mass varies with k, but almost constant at the bottom of the conduction band. • positive value since C1 is +ve September/Oktober 2007 40Department of Microelectronics & Computer Engineering
  • Concept of the hole• When valence electron goes to conduction band positively charged emptystate is created.• If a valence electron gets a small amount of energy, it can occupy this emptystate.• Movement of valence electron movement of +vely charged empty statein opposite direction.• The charge carrier in the form of +vely charged empty state is called the hole. September/Oktober 2007 41 Department of Microelectronics & Computer Engineering
  • September/Oktober 2007 42Department of Microelectronics & Computer Engineering
  • Drift current density due to electrons in valence band is - ∑ vi = i ( filled ) ∑ i ( total ) vi − ∑ vi i ( empty )Now,Also in a band that is completely full,electron distribution is symmetric withrespect to k. The net drift current densitygenerated from a completely full band isthen 0. September/Oktober 2007 43 Department of Microelectronics & Computer Engineering
  • and ThereforeDrift current due to electronsin the filled state can also belooked at as that due toplacing positively chargedparticles in the empty states. September/Oktober 2007 43Department of Microelectronics & Computer Engineering
  • Valence band with conventional Concept of positive chargeselectron filled states and empty occupying the original empty states. states September/Oktober 2007 43 Department of Microelectronics & Computer Engineering
  • Energy at the top of allowed energyband can be written as - C2  +ve Compare with free electron: September/Oktober 2007 44 Department of Microelectronics & Computer Engineering
  • Negative effective mass Bit strange!• negative mass is a result of our attempt to relate quantum and classicalmechanics. It is due to the inclusion of the effect of internal forces due to ionsand other electrons.• The net motion of electrons in the nearly full valence band can also be describedby considering the empty states, provided +ve electronic charge and +veeffective mass is associated with them•This new particle with +ve electronic charge and +ve effective mass denoted bymp* is called the hole September/Oktober 2007 45 Department of Microelectronics & Computer Engineering
  • Electrons: Holes:- in almost empty band - in almost full band- negative charge - positive charge- positive effective mass - positive effective mass Both: * qE September/Oktober 2007 46 Department of Microelectronics & Computer Engineering
  • Electron and Hole effective mass in different semiconductors. mn*/m0 mp*/m0 Silicon 1.08 0.56 Gallium Arsenide 0.067 0.48 Germanium 0.55 0.37 September/Oktober 2007 47Department of Microelectronics & Computer Engineering
  • Metals, semiconductors and insulatorsInsulators: Empty Conduction band Eg Full Valence band• Eg 3.5 to 6 eV or higher• valence band full, conduction band empty• no charged particles that contribute to drift current. September/Oktober 2007 49 Department of Microelectronics & Computer Engineering
  • Semiconductors: • Eg in semiconductors is small. Eg ~ 1eV • At T>0K, there will be some electrons in conduction band and holes in valence band, which can contribute to current. T > 0K • resistivity can be varied over many orders of magnitude by doping September/Oktober 2007 49 Department of Microelectronics & Computer Engineering
  • Metals: • Two possibilities - half filled conduction band or conduction and valence bands overlap. • plenty of electrons and empty energy states that these electrons can move into. •Very high electrical conductivity. September/Oktober 2007 49 Department of Microelectronics & Computer Engineering
  • Extending energy band theory to three dimensions • Extend concepts of energy band and effective mass to 3-dimensions • distance between atoms not the same in different directions •E vs. k diagram  as seen before depends on ‘a’ and ‘a’ varies with direction.Different potentials in different directions September/Oktober 2007 51 Department of Microelectronics & Computer Engineering
  • E vs. k diagrams of Si and GaAs Indirect direct “band gap” “band gap” transition transition• [100] and [111] directions plotted along +k and –k axes. September/Oktober 2007 52 Department of Microelectronics & Computer Engineering
  • E vs. k space diagrams of Si and GaAs•In GaAs the minimum conduction band energy and maximumvalence band energy occur at same k value. Suchsemiconductors are called direct band-gap semiconductors.Others are called indirect band-gap semiconductors. Eg. Si,Ge.• Direct bandgap semiconductors are used for making opticaldevices. In indirect bandgap semiconductors there is loss ofmomentum during transition from conduction band to valenceband, making them unsuitable for optical devices. 59 Department of Microelectronics & Computer Engineering
  • E vs. k space diagrams of Si and GaAs d 2E dk 2 is larger in GaAs than in silicon So m* in GaAs < m* in silicon• effective mass can be different along 3 different k-vectors.We consider average effective mass hence forth. 60 Department of Microelectronics & Computer Engineering
  • Density of states function• we want to know the number of charge carriers and theirtemperature dependence• hence the question is: how many energy levels do we haveand what is the chance that they are populated independence of the temperature. September/Oktober 2007 55 Department of Microelectronics & Computer Engineering
  • Mathematical model for density of states• we consider a free electron confined to an 3-dimensional cubical infinitepotential well of side ‘a’.One dimensional andpotential well September/Oktober 2007 57 Department of Microelectronics & Computer Engineering
  • Extending to 3-dimensions, 2mE π 2 = k 2 = k x2 + k y + k z2 = (nx + n y + nz2 )( 2 ) 2 2 2 2 aWhere nx, ny and nz are positive integers (negative valueslead to the same orbitals and hence do not represent adifferent quantum state) Two-dimensional array of allowed quantum states in k-space September/Oktober 2007 57 Department of Microelectronics & Computer Engineering
  • In 3-dimensional k-space, only1/8th of the spherical k-spaceneeds to considered fordetermining the density of states-ve value of (kx,ky,kz) result insame k 2 = k x2 + k y + k z2 and hence 2same energy. Since these do notresult in separate energy state,they are not considered. Positive 1/8th of spherical k-space.Spherical surface is consideredbecause value of k2 and henceenergy is same along this surface. September/Oktober 2007 58 Department of Microelectronics & Computer Engineering
  • Distance between quantum statesin any one axial direction is - Positive 1/8th of spherical k-space.Therefore Volume of a k-point: September/Oktober 2007 58 Department of Microelectronics & Computer Engineering
  • Number of energy statesbetween k, k+Δk Where: 2 two spin states allowed for each quantum number  differential volume in k-space between shells of radius k and K+ Δk Simplifying, September/Oktober 2007 59 Department of Microelectronics & Computer Engineering
  • Use relationship between k & E  to move from k-space toreal space.Substituting for dk in previous equation, πa 3 π 3 September/Oktober 2007 60 Department of Microelectronics & Computer Engineering
  • πa 3 π 3 h = 2πgT(E)dE is the density of states in volume a3. Dividing by a3we can get the volume density of states as September/Oktober 2007 61 Department of Microelectronics & Computer Engineering
  • Extension to semiconductors.We have derived expression for density of allowed electronquantum states using model of free electron in an infinitepotential well.Electrons and holes are also confined within a semiconductorcrystalSo we can extend this model to derive density of quantumstates in conduction and valence bands. September/Oktober 2007 68 Department of Microelectronics & Computer Engineering
  • Extension to semiconductors.Conduction band: Parabolic E vs. k for free electron September/Oktober 2007 68 Department of Microelectronics & Computer Engineering
  • Valence band: September/Oktober 2007 69 Department of Microelectronics & Computer Engineering
  • Density of states in the conduction band For free electron, E =  2 k 2 and 2mFor conduction band, September/Oktober 2007 70 Department of Microelectronics & Computer Engineering
  • Density of states in the valence band September/Oktober 2007 71Department of Microelectronics & Computer Engineering
  • Density of states functionIn general: September/Oktober 2007 73Department of Microelectronics & Computer Engineering
  • Statistical Mechanics.• While studying, large number of particles, we are interestedonly in statistical behavior of the group as a whole.• In a semiconductor crystal, we are not interested in thebehavior of each individual electron.• The electrical characteristics will be determined by statisticalbehavior of large number of electrons.• While studying statistical behavior, we must consider thelaws that the particles obey. 75 Department of Microelectronics & Computer Engineering
  • Distribution Laws:Distribution of particles among energy states  3 laws. Particles distinguishable Particles Identical Unlimited number of Maxwell-Boltzmann Bose-Einstein function. particles allowed in probability function. each energy state E.g. Behavior of photons or E.g. Behavior of gas at low black body radiation. pressure. Limited number of Fermi-Dirac probability particles allowed in function each energy state E.g. Electrons is a semiconductor crystal. September/Oktober 2007 74 Department of Microelectronics & Computer Engineering
  • Fermi-Dirac distribution function: N (E) 1 = f F (E) = g (E)  E − EF  1 + exp   kT N(E)  number of electrons per unit volume per unit energy.g(E)  number of quantum states per unit vol per unit energy.EF  Fermi energy (explained in more detail later) September/Oktober 2007 75 Department of Microelectronics & Computer Engineering
  • At T = 0 K >EF F(E < E F )=1 <EF F(E > E F )=0 The Fermi probability function versus energy for T=0K Discrete energy states and quantum states for a particular system. September/Oktober 2007 77 Department of Microelectronics & Computer Engineering
  • At E=EFSo Fermi Energy is the energy level where probability is ½.At T>0 The fermi probability function versus energy for different temperatures. Applet At T>0K, there is non-zero probability that some states above EF are occupied and some states below are empty  some electrons have jumped to higher energy levels with increasing thermal energy. September/Oktober 2007 78 Department of Microelectronics & Computer Engineering
  • At T > 0 K >EF <EF Discrete energy states and quantum states for the same system at T>0K September/Oktober 2007 79 Department of Microelectronics & Computer Engineering
  • At a certain T The function fF(E) is symmetrical with the function 1-fF(E) about the Fermi energy EF September/Oktober 2007 80Department of Microelectronics & Computer Engineering