1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 13Properties of Electrons in Energy BandsIn this lecture you will learn:• Properties of Bloch functions• Average momentum and velocity of electrons in energy bands• Energy band dispersion near band extrema• Effective mass tensor• Crystal momentumECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Functions: A Review1) The quantum states of an electron in a crystal are given by Bloch functions that obey the Schrodinger equation:rkErHknnkn,,ˆ2) Under a lattice translation, Bloch functions obey the relation:reRrknRkikn,.,3) Bloch functions can be written as the product of a plane wave times a lattice periodic function: ruVerknrkikn,.,where the wavevector is confined to the FBZ and “n” is the band indexkjrGkijknknjeVGcr.,,14) Bloch function of wavevector can be written as a superposition of plane waves with wavevectors that differ from by reciprocal lattice vectors:kk2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityOrthogonality:Bloch functions are eigenstates of a Hermitian operator and therefore must be orthogonal. In “d ” dimensions:mnddmnkkkmkndkkVrrrd,,',',,*'2 Bloch Functions: Orthogonality and CompletenessBoth expression valid depending upon contextCompleteness:Bloch functions for ALL wavevectors in the FBZ and for ALL energy band satisfy the following completeness relation in “d ” dimensions: ''2'FBZ,*,FBZ in ,*,rrrrkdVrrdknknddknknknnECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityAnother Schrodinger-like Equation for Bloch FunctionsThe periodic part of a Bloch function satisfies a Schrodinger-like equation:rukErurVmkkmPmPrukErurVmkPruekErurVmkPerkErrVmPrkErHknnknknnknknrkinknrkiknnknknnkn,,222,,2,.,2.,,2,,ˆ2.ˆ2ˆˆ2ˆˆ2ˆˆ2ˆˆWhere the following two relations have been used:rfkPerfePrfkPerfePrkirkirkirki2..2..ˆˆˆˆResult:3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBloch Functions and Electron Momentum• For an electron with wavefunction given by a plane wave:the quantity is the momentum of the electronrkikeVr.1k• A plane wave is an eigenfunction of the momentum operator with eigenvalue :• A Bloch function is a superposition of plane waves of different wavevectors:So clearly it is not an eigenfunction of the momentum operator (i.e. it has no well defined momentum). So what exactly is the significance of the wavevector that labels a Bloch function?• As you will see, even the average momentum of an electron in a Bloch state is NOT given by : rkrirPkkkˆkjrGkijknknjeVGcr.,,1kk krirrdPknkndknkn,,*,,ˆECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityAverage Momentum and Velocity of Bloch StatesWe need to find the average momentum and average velocity of an electron in a Bloch state:?ˆ?ˆ,,,,knknnknknmPkvPSuppose we have solved the Schrodinger-like equation for a particular wavevector :Start from a very different point:krukErurVmkkmPmPknnkn,,222ˆ2.ˆ2ˆSuppose now we want to solve it again for a neighboring wavevector :kkrukkErurVmkkkkmPmPkknnkkn,,222ˆ2.ˆ2ˆThe “Hamiltonian” is:rVmkkmPmPHkˆ2.ˆ2ˆˆ2224ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityAverage Momentum and Velocity of Bloch StatesThe new “Hamiltonian” is:kkkkkHHmkkkkmPHrVmkkkkmPmPHˆˆ2.2.ˆˆ ˆ2.ˆ2ˆˆ22222Treat this part as a perturbation to the old “Hamiltonian”Using concepts from time-independent perturbation theory, the first order correction to the energy eigenvalue would be:knkknnnuHukEkkE,,As written, the above expression is approximate but becomes exact in the limit0kknknnkknknnkknkknnnumkPukEukmkkmPukEkuHukEkkEk,,,,,,ˆ1 ..ˆ. :0limECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityAverage Momentum and Velocity of Bloch States knknnkknkndnkknrkiknrkidnkknknrkirkidnkknkndknknnkmPkErmPrrdkEruemPruerdkErumkPrueerdkErumkPrurdumkPukE,,,,*,.,*.,,*..,,*,,ˆ1ˆ1ˆ1ˆ1ˆˆ1 (Contd…) The average momentum of an electron in a Bloch state is:kEmPnkknkn,,ˆ The average velocity of an electron in a Bloch state is: kEmPkvnkknknn1ˆ,,5ECE 407 – Spring 2009
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