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# Special Topics EECS 598

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This 21 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 598 at University of Michigan taught by Pc Ku in Fall. Since its upload, it has received 22 views. For similar materials see /class/231527/eecs-598-university-of-michigan in Engineering Computer Science at University of Michigan.

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Date Created: 10/29/15

Lecture 6 Photons electrons and other quanta EECS 598002 Winter 2006 Nanophotonics and Nanoscale Fabrication PCKu I From classical to quantum theory In the beginning of the 20th century experiments showed Particle nature of EM radiation Wave nature of electrons In 1927 Heisenberg proposed the uncertainly principle which later on became the foundation of modern quantum mechanics QM Not all the physical quantities can be measured at the same time with however precision we would like Eg the more precisely we measure the position of a particle the less precisely we will be able to measure its momentum EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu I Example Photoelectric effect In the experiment the light shines on a metal and knocks out the electrons on the metal surface When we measured the stopping voltage V0 to counteract the generated current we found a cutoff frequency fo This can not be explained by the Maxwell s equations G A f f ii EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu I Quantum state and operators In the language of QM the physics of a particle or a system that is comprising of many particles can be described by a state ppm a vector in Hilbert space The physical quantities measurables can be obtained by applying a suitable operator Q to the state QI PU This will yield one of the eigenvalues with probability P given by Pww Pt2 The measurement will also change the system to the new state lea eg ln coordinate basis Position operator r Momentum operator th Total energy operator V A 2m VV gt Hamiltonian EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu I Dynamics of quantum states I The evolution of the quantum state obeys the Schroedinger equation dTtgt h H LP t z dt 0 where H is the Hamiltonian For example an electron in a timeindependent potential Vr is governed by hzvz A d LP t 2 VFLPtgtlh d m eiiEth hZVZ A 2m Vr EEgtO EECS 598002 Nanophotonios and Nanosoale Fabrication by PCKu I Comparison bw QM and EM Equation governing QM and EM have lots of mathematical similarities QM EM Equation V2 Z E VolE0 V20120808r 80 10 Potential 47 80 80 Eigenstate l E E and H Eigenvalue E w2 Wavevector l Tw nm 0W EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 6 I First analogy tunneling Vz 82 396 WWW F 2 gt2 QM EM 1 Electron experiences reflection and tunneling through the potential barrier The wavefunctid is exponential decayed in the barrier 2 The tunneling probability increases when the width of the potential barrier decreases EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu I Slab waveguide vs potential well Vz 82 2 gt2 QM EM At a fixed KE in the xy plane the kinetic energy of an electron in the z direction is quan zed EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu I Electrons in crystals When electron travels in a periodic potential eg in the crystal its eigenstate satisfies the Bloch s theorem i137 lI7u776 The Bloch function ul7 has the same periodicity as the crystal lattice The spatial symmetry of the crystal lattice with respect to a fixed point determines the eigenfunctions of the electron Because of the periodicity the energy bandgap exists at certain k values EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu I Effective mass vs constitutive relation Similar to the macroscopic version of the Maxwell s equations we can also derive an effectivemass equation for electrons traveling in the lattice with the assumption that the electron interacts weakly with the lattice hzvz A A 2m mezU Vmacmcopic 0 h2V2 a 2m Vmacroscopicr O EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 10 I Example of a dispersion relation Electron in periodic Vr EM wave in periodic 8r w was I X K y VMLEY A Z Jig 03 LOWER sq vALL Ev Normai zed frequency ma21m w 30 a i i L iii 139 me x EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 11 I Density of states EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu Density of states gE is the total number of allowedto occupy states with frequencies between E and E5E per unit volume idNE gE V dE where NE is the total number of states from O to E In a potential well NE l gE39fE39dE39 Z k39 E39ltE j Ak39fk39dk39 k39ltk 1 A 1 2gtlt d fk39dk392x d M kite Ak i Moog y E gEquot V My dE 427 AkdE 3 dim Ak 47rk2 2 dim Ak 2m 1 dim Ak 1 0 dim Ak 5k kn d 12 I Density of states cont For electrons E 2 1 1sz 2m For EM waves photons E 2 ha 2 hck Note k can be discrete due to quantization dk m For electrons dgt0 E E dE h2k g ang n 1 2m 32 3 d39 E E 1m g 27r2 hz j 2 dim gE ZQE En 73 n 1 m 12 1 1dirn E g Ens n 0 dirn gE 225a En EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 13 I Density of states for electrons gE Density of states dot W39 2D 1 D OD 903 903 903 903 Eg E Eg E Eg E Eg E EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 14 I Occupation probability In thermal equilibrium fermions eg electrons satisfy the FermiDirac statistics two fermions cannot stay in the same state 1 fE expE EFkBT1 In thermal equilibrium bosons eg photons satisfy the BoseEinstein statistics 1 eXpE EFkBT 1 fE EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 15 I From singleparticle to manyparticle system So far we have considered only the quantization of a single particle namely the electron This procedure namely the uncertainly principle applies to all particles that are governed by Newton s Laws classically Similarly we can also treat a multiparticle system as a whole and quantize the system at once To treat the system classically we can imagine the position and momentum of each particle form a field The moving of a particle wrt its equilibrium position looks like a perturbation of the field ie wave EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 16 I Number operator N m 51 n 1 Creation operator 51 n 1 Annihilation operator F ha5z5z 12 Hamiltonian operator EECS 598002 Nanophotonics and Nanosoale Fabrication by PCKu 17 I Photons Similar to the quantization of a manyparticle system EM wave can be thought as a perturbation of the EM field in spacetime Quantization of the EM field 9 photons 13107 01531073 1 mgyk 507 1739 k Properties of photons Mass O Charge O Energy hv M Momentum hk EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 18 I When do we need concepts of photons When do we need to treat the EM wave as photons When the momentum of each photon is comparable to that ofthe material upon which it impinge When the number of photons involved in the interaction is very small Examples Photoelectric effect Spontaneous emission EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 19 I Density of states for photons in 3D dw c Nw J gw39fw39dw39Zfk39 1 A 1 2gtlt k39dk392gtlt 4 k392 k39dk39 Ak3kL Akf w Ejz 602 g 7 da 7239263 EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 20 I Other quanta Phonon Plasmon Surface plasmon EECS 598002 Nanophotonics and Nanoscale Fabrication by PCKu 21

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