ECE 4070: Physics of Semiconductors and Nanostructures K M K’ ECE 4070 – Spring 2010 – Farhan Rana – Cornell University ECE 4070: Physics of Semiconductors and Nanostructures Instructor: Farhan Rana Office: PH316 Email: fr37@cornell.edu Syllabus: The course covers fundamentals of solid state physics relevant to semiconductors, electronic and photonic devices, and nanostructures. Crystal lattices and the reciprocal lattice; Electron states and energy bands in molecules and solids; Metals, insulators, and semiconductors; Graphene, 2D atomic materials, and carbon nanotubes; Lattice dynamics and phonons in 1D, 2D, and 3D materials; Electron statistics and dynamics in energy bands; Effective mass theorem; Electron transport and Boltzmann equation; Optical transitions and optical interband and intraband processes; Optical loss, optical gain, and Kramers-Kronig relations; Excitons and polaritons; Semiconductor heterostructures; Electron states in zero, one, and two dimensional nanostructures; Quantum wells, wires, and dots; Quantum transport in nanostructures and ballistic transport; ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 1 Course Website and Homeworks • All course documents, including: - Lecture notes - Homeworks and solutions - Exam solutions - Extra course related material will appear on the course website: http://courses.cit.cornell.edu/ece407/ Homeworks • Homeworks will be due on Tuesdays at 5:00 PM • New homeworks and old homework solutions will appear on the course website by Tuesday night • Homework 1 will be due next Tuesday and will be available on the course website by tomorrow night ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Course Grading and Textbooks • Course grading will be done as follows: - Homeworks (25%) - 2 Evening Prelims (20% each) – dates TBD - Final exam (35%) – date TBD • No in-class quizzes, no pop-quizzes • Final exam will be comprehensive Textbooks • There are no required textbooks. Highly recommended textbooks are: - Introduction to Solid State Physics, by Charles Kittel (8th edition) - Electronic and Optoelectronic Properties of Semiconductor Structures, by Jasprit Singh - Solid State Physics, by Ashcroft and Mermin ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 2 Handout 1 Drude Model for Metals In this lecture you will learn: • Metals, insulators, and semiconductors • Drude model for electrons in metals • Linear response functions of materials Paul Drude (1863-1906) ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Inorganic Crystalline Materials Ionic solids Covalent solids Mostly insulators Example: NaCl, KCl Semiconductors Si, C, GaAs, InP, GaN PbSe, CdTe, ZnO Insulators SiO2, Si3N4 Metals Au, Ag, Al, Ga, In Metals 1- Metals are usually very conductive 2- Metals have a large number of “free electrons” that can move in response to an applied electric field and contribute to electrical current 3- Metals have a shiny reflective surface ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 3 Properties of Metals: Drude Model Before ~1900 it was known that most conductive materials obeyed Ohm’s law (i.e. I =V/R). In 1897 J. J. Thompson discovers the electron as the smallest charge carrying constituent of matter with a charge equal to “-e” e 1.6 10 19 C sea of electrons ions In 1900 P. Drude formulated a theory for conduction in metals using the electron concept. The theory assumed: 1) Metals have a large density of “free electrons” that can move about freely from atom to atom (“sea of electrons”) 2) The electrons move according to Newton’s laws until they scatter from ions, defects, etc. 3) After a scattering event the momentum of the electron is completely random (i.e. has no relation to its momentum before scattering) + + + + + + + + + + + + + + electron path + + ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Drude Model - I + Applied Electric Field: + electron path In the presence of an applied external electric field E the electron motion, on average, can be described as follows: + + Let be the scattering time and 1/ be the scattering rate This means that the probability of scattering in small time interval time dt is: The probability of not scattering in time dt is then: 1 dt dt Let pt be the average electron momentum at time t , then we have: dt dt pt dt 1 pt e E t dt 0 If no scattering happens then Newton’s law If scattering happens then average momentum after scattering is zero dp t p t e E t dt ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 4 Drude Model - II Case I: No Electric Field dp t p t dt p t 0 Steady state solution: Case II: Constant Uniform Electric Field Steady state solution is: Electron path Electron path p t e E E Electron “drift” velocity is defined as: = e/m = electron mobility p t e v E E (units: cm2/V-sec) m m Electron current density J (units: Amps/cm2) is: J n e v n e E E 3 Where: n electron density (units : # /cm ) electron conductivity (units : Siemens/cm ) ne ne 2 m ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Drude Model - III Case III: Time Dependent Sinusoidal Electric Field dp t p t e E t dt There is no steady state solution in this case. Assume the E-field, average momentum, and currents are all sinusoidal with phasors given as follows: E t Re E e i t pt Re p e i t J t Re J e i t dpt pt p e E t ip eE dt e p e m p E v E 1 i 1 i m Electron current density: J n e v E Where: ne 2 0 m 1 i 1 i Drude’s famous result !! ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 5 Linear Response Functions - I The relationship: J E is an example of a relationship between an applied stimulus (the electric field in this case) and the resulting system/material response (the current density in this case). Other examples include: P o e E electric polarization density electric susceptibility electric field M m H magnetic polarization magnetic density susceptibility magnetic field The response function (conductivity or susceptibility) must satisfy some fundamental conditions …. (see next few pages) ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Linear Response Functions - II Case III: Time Dependent Non-Sinusoidal Electric Field For general time-dependent (not necessarily sinusoidal) e-field one can always use Fourier transforms: d E t E e i t 2 E dt E t e i t (1) Then employ the already obtained result in frequency domain: J E And convert back to time domain: d d J t J e i t E e i t 2 2 Now substitute from (1) into the above equation to get: d d J t E e i t dt ' e i t t ' E t ' 2 2 J t dt ' t t ' E t ' ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 6 Linear Response Functions - III J t dt ' t t ' E t ' Where: d e i t t ' 2 t t ' The current at time t is a convolution of the conductivity response function and the applied time-dependent E-field Drude Model: 0 1 i d 0 d e i t t ' e i t t ' 2 2 1 i t t ' 0 t t ' e t t ' t t ' step function t t ' t t ' ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Linear Response Functions - IV The linear response functions in time and frequency domain must satisfy the following two conditions: 1) Real inputs must yield real outputs: Since we had: d J t dt ' e i t t ' E t ' 2 This condition can only hold if: * 2) Output must be causal (i.e. output at any time cannot depend on future input): Since we had: J t dt ' t t ' E t ' This condition can only hold if: t t ' 0 for t t ' Both these conditions are satisfied by the Drude model ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 7 Drude Model and Metal Reflectivity - I When E&M waves are incident on a air-metal interface there is a reflected wave: o Ei o o Et Hi Er Hr Ht The reflection coefficient is: Er Ei o o Question: what is for metals? ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Drude Model and Metal Reflectivity - II From Maxwell’s equation: Ampere’s law: Phasor form: E r , t H r , t J r , t o t H r J r i o E r E r i o E r i eff E r Effective dielectric constant of metals eff o 1 i o Metal reflection coefficient becomes: Er Ei o eff o eff Using the Drude expression: 0 1 i the frequency dependence of the reflection coefficient of metals can be explained adequately all the way from RF frequencies to optical frequencies ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 8 Drude Model and Plasma Frequency of Metals eff o 1 i For metals: o and ne 2 m 0 1 i 1 i For small frequencies 1 : ne 2 m 0 eff o 1 i 0 o For large frequencies 1 (collision-less plasma regime): 0 ne 2 i m i eff o 1 p where the plasma frequency is: ne 2 om p2 2 For most good metals this frequency is in the UV to visible range Electrons behave like a collision-less plasma Note that for p 1 the dielectric constant is real and negative ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Plasma Oscillations in Metals Consider a metal with electron density n Now assume that all the electrons in a certain region got displaced by distance u -ve charge +ve charge left accumulated behind + + + + + + + + + + + + + + + + + + + + + + + + + The electric field generated E neu o Force on the electrons F eE + + + + + + + + + + + E u u 2 ne u o As a results of this force electron displacement u will obey Newton’s second law: m d 2u t dt 2 F eE n e 2 u t o d 2u t Solution is: u t A cos pt B sin pt dt 2 p2 u t second order system Plasma oscillations are charge density oscillations ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 9 Plasma Oscillations in Metals – with Scattering From Drude model, we know that in the presence of scattering we have: dpt pt e E t dt m d 2u t dt 2 e E t As before, the electric field generated E t m du t dt (1) n e u t (2) o Combining (2) with (1) we get the differential equation: d 2u t dt 2 p2 u t 1 du t dt p Or: d 2u t dt 2 1 du t dt p2 u t 0 ne 2 om + + + + + + + second order system with damping + + + + + + + + + + + E u u ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Plasma Oscillations in Metals – with Scattering Case I (underdamped case): p 1 2 Solution is: u t e t A cos pt B sin pt Damped plasma oscillations Where: 1 2 p p2 2 Case II (overdamped case): p 1 2 Solution is: u t A e 1 t B e 2 t No oscillations Where: 1 1 1 p2 2 4 2 1 1 1 p2 2 4 2 ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 10 Appendix: Fourier Transforms in Time OR Space Fourier transform in time: f dt f t e i t Inverse Fourier transform: d f e i t 2 f t Fourier transform in space: g k dx g x e i k Inverse Fourier transform: x dk g k e i k 2 gx x ECE 4070 – Spring 2010 – Farhan Rana – Cornell University Appendix: Fourier Transforms in Time AND Space Fourier transform in time and space: hk , dx dt h x , t e i k x e i t Inverse Fourier transform: dk d 2 2 h x , t h k , e i k x e i t ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 11 Appendix: Fourier Transforms in Multiple Space Dimensions Fourier transform in space: h k x , k y , k z dx dy dz h x , y , z e i k x x e i ky y i kz z e Need a better notation! Let: k k x xˆ k y yˆ k z zˆ r x xˆ y yˆ z zˆ 3 d r dx dy dz h k d 3r hr e i k . r Inverse Fourier transform: hr d 3k 2 3 h k ei k . r ECE 4070 – Spring 2010 – Farhan Rana – Cornell University 12