1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 15Dynamics of Electrons in Energy BandsIn this lecture you will learn:• The behavior of electrons in energy bands subjected to uniform electric fields• The dynamical equation for the crystal momentum• The effective mass tensor and inertia of electrons in energy bands • Examples• Magnetic fields• Appendix: Electron dynamics using gauge invariance arguments, Berry’s phase, and Berry’s curvatureECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityElectron Dynamics in Energy Bands1) The quantum states of an electron in a crystal are given by Bloch functions that obey the Schrodinger equation:rkErHknnkn,,ˆ2) Under a lattice translation, Bloch functions obey the relation:reRrknRkikn,.,where the wavevector is confined to the FBZ and “n” is the band indexkNow we ask the following question: if an external potential is added to the crystal Hamiltonian,then what happens? How do the electrons behave? How do we find the new energies and eigenstates?The external potential could represent, for example, an applied E-field or an applied B-field, or an electromagnetic wave (like light)trUH ,ˆˆ2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityRecall from homework that the energy bands are lattice-periodic in the reciprocal space,When a function in real space is lattice-periodic, we can expand it in a Fourier series, When a function is lattice-periodic in reciprocal space, we can also expand it in a Fourier series of the form, kEGkEnn rGijjjeGVrVrVRrV.jkRijnnnnjeREkEkEGkE.Periodicity of Energy BandsFourier representation of energy bandsECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityConsider the following mathematical identity (Taylor expansion): xfeaxfaxfxfaxfdxda ...........''21'2Generalize to 3 dimensions: rfearfa.Now go back to the relation: and consider the operator:jkRijnnnnjeREkEkEGkE.jRjnnjeREiE.ˆWe apply this operator to a Bloch function from the same band (i.e. the n-th band) and see what happens:?ˆ,.,reREriEknjRjnknnjA New Operator - I3ECE 407 – Spring 2009 – Farhan Rana – Cornell University  rkEreRERrREreREriEknnjknRkijnjknjjnknjRjnknnjj,,.,,., ˆThe result above implies that the action of the operator on a Bloch function belonging to the same band (i.e. n-th band) is that of the Hamiltonian! iEnˆrkErHriEknnknknn,,,ˆˆA New Operator - IIThis also implies that if we have a superposition of Bloch functions from a single band then: FBZin,FBZ in ,FBZ in , ˆˆkknnkknnkknrkEkcrkciErkcHECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe Case of Uniform Electric FieldttritrrEeH,,ˆ.ˆStatement of problem: Need to solve,given that at time t = 0 the state of the electron is a Bloch function with wavevector ,rtrkn,0,Assumption: Assume that the state at any later time is going to be a Bloch function or a linear combination of Bloch functions belonging to the same band (valid for weak E-fields) Then one can replace the Hamiltonian with , ttritrrEeiEttritrrEeHn,,ˆ.,,ˆ.ˆ iEnˆxkaaEnergy0tk4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe Case of Uniform Electric Field tkndtrEetEirtr0,'.'exp,Try the following time-dependent solution with a time-dependent energy:ttritrrEeiEn,,ˆ.First see how the assumed solution behaves under a lattice translation:tredtRrEetEiRrtRrRkitkntEe, '.'exp,.0,So the assumed solution looks like a Bloch function with a time dependent k-vector:tEektk But we still don’t know what is the time-dependent energy E(t)ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe Case of Uniform Electric FieldLHS (first term): trtkEtreREtRrREtreREtriEEenRtkijjnjjjnjRjnnjEej, , , ,,..RHS:trrEetEttri ,.,Take the trial solution and plug it into the equation:ttritrrEeiEn,,ˆ.5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitytEentEennkEtEtrrEetEtrrEekEttritrrEeiE,ˆ.,ˆ.,,ˆ.Putting it together:The Case of Uniform Electric FieldThe time-dependent energy is consistent with our solution being a Bloch function with a time-dependent k-vector,tEektk tntkndttkEirtr0,''exp,rtrkn,0,So the solution for the initial condition:is approximately a Bloch function with a time-dependent k-vector:ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe Case of Uniform Electric FieldFinal result: In the presence of a uniform