1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 7Properties of Bloch States and Electron Statistics in Energy BandsIn this lecture you will learn:• Properties of Bloch functions• Periodic boundary conditions for Bloch functions• Density of states in k-space• Electron occupation statistics in energy bandsECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Functions - Summary• Electron energies and solutions are written as ( is restricted to the first BZ):• The solutions satisfy the Bloch’s theorem:and can be written as a superposition of plane waves, as shown below for 3D:• Any lattice vector and reciprocal lattice vector can be written as:• Volume of the direct lattice primitive cell and the reciprocal lattice first BZ are:rkn,andkEnreRrknRkikn,.,jrGkijnknjeVGkcr.,1k332211bmbmbmG3213. bbb332211anananR3213. aaa2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Function – Product Form ExpressionA Bloch function corresponding to the wavevector and energy band “n” can always be written as superposition over plane waves in the form:kjrGkijnknjeVGkcr.,1The above expression can be re-written as follows:rueVeGceVeVGkcerknrkijrGijknrkijrGijnrkiknjj,..,...,1 1 1Where the function is lattice periodic:rukn,rueGceGcRruknjrGijknjRrGijknknjj,.,.,, satisfiesBloch’s theoremreRrknRkikn,., rueVrknrkikn,.,1Note that:ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityAllowed Wavevectors for Free-Electrons (Sommerfeld Model)xLyLzLzyxLLLVWe used periodic boundary conditions:zyxLzyxzyxzLyxzyxzyLxzyx,,,,,,,,,,,,The boundary conditions dictate that the allowed values of kx, ky, and kz, are such that:zzLkiyyLkixxLkiLpkeLmkeLnkezzyyxx212121n = 0, ±1, ±2,….m = 0, ±1, ±2,….p = 0, ±1, ±2,….32V There are grid points per unit volume of k-spacexL2yL2zL2xkykzk3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Functions – Periodic Boundary Conditions • Any vector in the first BZ can be written as:k332211bbbkwhere 1, 2, and 3range from -1/2 to +1/2:Reciprocal lattice for a 2D latticeDirect lattice1b1a2b2ak212112121221213ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Functions – Periodic Boundary Conditions • Consider a 3D crystal made up of N1primitive cells in the direction, N2primitive cells in the direction and N3primitive cells in the direction1a2a3aAssuming periodic boundary conditions in all three directions we must have: rreaNraNki11.11 rreaNraNki22.22 rreaNraNki33.33 Volume of the entire crystal is:3321332211.NNNaNaNaNV1a2a22aN11aN2212211 :crystal2DtheForNNaNaNA4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Functions – Periodic Boundary Conditions The periodic boundary condition in the direction implies:1111111111.2. :that recall 22integer an is 2.111NmbamNmmaNkekjkjaNkiSince:2121122111NmN m1can have N1different integral values between –N1/2 and +N1/2Reciprocal lattice for a 2D latticeDirect lattice1b1a2b2a1ak332211bbbkECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Functions – Periodic Boundary Conditions Similarly, the periodic boundary conditions in the directions of and imply:333222333222..&2.&2.1&13322NmNmmaNkmaNkeeaNkiaNki m1can have N1different integral values m2can have N2different integral values m3can have N3different integral valuesReciprocal lattice for a 2D latticeDirect lattice1b1a2b2a2ak332211bbbk3aSince any k-vector in the FBZ is given as: there are N1 N2 N3different allowed k-values in the FBZ There are as many different allowed k-values in the FBZ as the number of primitive cells in the crystal22&22333222NmNNmN5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityDensity of States in k-SpaceReciprocal lattice for a 2D lattice332211bbbk22222222NmNNm22333333NmNNm22111111NmNNm12a22a1b2b112aN222aNQuestion: Since is allowed to have only discrete values, how many allowed k-values are there per unit volume of the k-space? kVolume of the first BZ is:3213. bbb• In this volume, there are N1N2N3allowed k-values • The number of allowed k-values per unit volume in k-space are:333321332122VNNNNNNwhere V is the volume of the crystal3D Case:ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityDensity of States in k-Space1b2b1b1D Case:Length of the first BZ is:1112b• In the first BZ, there are N1allowed k-values • The number of allowed k-values per unit length in k-space are:2211111LNNLength of the crystal:1111 NaNL2D Case:Area of the first BZ is:222122bb• In the first BZ, there are N1N2allowed k-values • The number of allowed k-values per unit area in k-space are: 2222122122ANNNNArea of the crystal:2212211 NNaNaNA6ECE 407 – Spring 2009 – Farhan
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