2015 ECEB Professor John Dallesasse

Home , Fermi level

M,W,F 12:00-12:50 (X), 2015 ECEB Professor John Dallesasse Department of Electrical and Computer Engineering 2114 Micro and Nanotechnology Laboratory Tel: (217) 333-8416 E-mail: [email protected] Office Hours: Wednesday 13:00 – 14:00 Please Review e-Mail Sent Today Today’s Discussion

• Distribution Functions • Carrier Concentration Calculation • Assignments • Topics for Next Lecture

3 Tentative Schedule [1]

JAN 17 JAN 19 Course overview Intro to semiconductor electronics

JAN 22 JAN 24 JAN 26 Materials and crystal Bonding forces and energy Metals, semiconductors, structures bands in solids insulators, electrons, holes

JAN 29 JAN 31 FEB 2 Intrinsic and extrinsic Distribution functions and Distribution functions and material carrier concentrations carrier concentrations

FEB 5 FEB 7 FEB 9 Temperature dependence, Conductivity and mobility Resistance, temperature, compensation impurity concentration

FEB 12 FEB 14 FEB 16 Invariance of Fermi level at Optical absorption and Generation and equilibrium luminescence recombination

**Subject to Change** 4 Brief Comments Extrinsic Material: Temperature Effects

n n, p i 

n n 0  i

2 n0 p0 = ni

500 K 100 K

Hot Cold Important Definitions

• Intrinsic Material: For a “perfect” crystal in equilibrium, electron and hole densities are equal

– The value of this density ni is called the intrinsic concentration • Extrinsic Material: The crystal is “doped” with – donors, where the material is n-type and the electron density exceeds the hole density, or – acceptors, where the material is p-type and the hole density exceeds the electron density – An amphoteric dopant can act as either a donor or acceptor, depending upon what atom it is replacing in the crystal – In extrinsic material, the carrier with the higher concentration is the majority carrier and the carrier with lower concentration is the minority carrier • We will find out later that the product of the minority carrier density and majority carrier density is constant for a given temperature (in equilibrium), 2 and equal to ni • In p-type material, holes are the majority carrier, electrons the minority carrier • In n-type material, electrons are the majority carrier, holes the minority carrier

7 Determining Conductivity Answer:

• For electrons, we integrate the product of the number (density) of states at a given energy times the probability a state is full at that energy – Conduction band only – Because the valence band is full, electrons in the valence band do not contribute to the conduction process ∞ n0 = f (E)Nc (E)dE ∫Ec • For holes, we integrate the product of the number (density) of states at a given energy times the probability a state is empty at that energy – Valence band only

Ev p0 = ⎡1− f (E)⎤ Nv (E)dE ∫−∞ ⎣ ⎦

But how do we do that? 9 Determining the Probability of State Occupancy Fermions and Bosons

• Fermion: Only a single fermion can occupy any given quantum state – they are subject to the Pauli exclusion principle. Electrons are fermions, and follow Fermi-Dirac statistics. • Boson: There is no limit to the number of bosons that can occupy any given quantum state – they are not subject to the Pauli exclusion principle. Photons are bosons, and follow Bose-Einstein statistics.

11 Fermi-Dirac Distribution

• The probability that an available energy state at E will be occupied by an electron at temperature T f E ( ) T2>T1 (temperature in Kelvin) is given T=0K by the Fermi-Dirac Distribution 1

Function T=T1 • Fermi-Dirac distribution function: 1/2 1 T=T2 f (E) = (E E )/kT 1 e − f + E f E E : Fermi Level f 1 1 (a) E = E f E = E = = k : Boltzmann Constant f ( f ) 1+ e0 2 k = 8.62x10−5 eV / K (b) T = 0K 1 f E < E = = 1 ( f ) 1+ e−∞ 1 f E > E = = 0 ( f ) 1+ e+∞ 12 What About Holes?

Hole Probability: • A state is either empty or full, so the probability a state is empty is f (E) T=0K f (E f − ΔE) 1-P(full) 1 • P(full) is given by the Fermi f (E f + ΔE) distribution f(E) 1/2 • P(empty) = 1 - f(E) T>>0K • f(Ef+ΔE) is the probability of finding an electron at E+ΔE E f E At T = T : • 1-f(Ef-ΔE) is the probability of 1 finding a hole at E-ΔE 1 f (E) = (E E )/kT Note, for a given ΔE : 1+ e − f 1 f E + ΔE = 1− f E − ΔE (E−E f )/kT1 ( f ) ( f ) e 1 1− f (E) = (E E )/kT = (E E)/kT 1+ e − f 1 1+ e f − 1 13 What Do They Mean? Practical Points

• The Fermi Level or Fermi Energy is a number that we can use to calculate the density of free carriers in a semiconductor • The Fermi level has units of energy (typically eV) • The Fermi level, used with the Fermi Distribution, tells us the probability a state is filled if we know the energy of the state relative to the Fermi Level • The Fermi Level is generally referenced to some other energy – make sure you know what it is referenced to when doing your calculations

15 Fermi Level in Metals

• At absolute zero, all states above the Fermi level are empty and all states below the Fermi level are full ✓ Ef • Above absolute zero, the Fermi level defines the energy where there is a 50% probability a state is occupied • For metals, the Fermi level is the energy of the topmost filled level in the ground state of an N electron system (implies the temperature is absolute zero, 0K) – Related to the energy needed to remove an electron from a metal at absolute zero ? – This definition gives useful insight, but is E problematic for a semiconductor where f the Fermi level is typically in the forbidden gap where there are no energy states

16 Metal Fermi Energy

Maximum Kinetic Energy Vacuum Em = hv − qΦ Level

• In a metal, the Fermi Energy referenced to the vacuum level is the metal work function

http://www.sci.ccny.cuny.edu/~rstein/papers/jcmst00/jcmst00.html Fermi Level: Definitions

• There are several equivalent definitions which can be used: – The change in free energy of the crystal when an electron is added or taken away • Effectively, a measure of how difficult it is to add or remove electrons – The sum of the chemical potential and internal electrostatic potential energy, generally equal to the electrochemical potential • Note: most texts fail to mention the electrostatic component • Comment 1: If material with different Fermi levels are brought together, the Fermi levels must align. This has certain implications which will be discussed soon. • Comment 2: Detailed understanding of the Fermi level and Fermi energy requires a background in statistical mechanics, which is beyond the scope of this course

18 Electrochemical Potential

• Electrochemical Potential: a measure of the amount of energy needed to add or remove an incremental amount of a given charged species from a defined locus that may be under the influence of an electric field, incorporating both chemical potential and electrostatics • The Fermi Energy is equal to the electrochemical potential in a semiconductor Other Definitions

• Chemical Potential: The incremental change in the Gibbs Free Energy of the system per particle (fixed temperature, pressure, etc.). ∂G µ = ∂n • Gibbs Free Energy: In a chemical system, the enthalpy of the system minus the product of the temperature and entropy (at constant pressure/volume) G = H − TS = U + PV − TS U = internal energy, S = entropy, H = enthalpy Fermi Level: Experimental Determination

• In practice, it is relatively straight forward to measure carrier concentration • The Fermi level is typically calculated from the measured carrier concentration using the relationships discussed last time (non- degenerate case), or slightly more complex relationships for degenerate material

21 Fermi Level Versus Doping Type

n-Type Intrinsic p-Type Additional Information Conduction Band Density of States

Consider a cube of material having sides of length "L" A free electron in this material has a wavefunction given by:

j(kxx+kyy+kzz) ψ = Ae where 2D 2π 2π 2π k = n , k = n , k = n x x L y y L z z L We also know: p2 2k 2 2π 22 2π 22 E = = = n2 + n2 + n2 = n2 2m* 2m* m*L2 ( x y z ) m*L2 L m*E We can also express n as a function of E: n = π 22 The constant energy surface is therefore the 3D the surface of a sphere of radius "n" where: 1 2 2 2 2 n = (nx + ny + nz ) The total number of states within the sphere is: 4 N′ = πn3 3 Note: Other variants of this derivation get the same result 24 Conduction Band Density of States (2)

Including spin, the total number of states up to a given energy is therefore: 4 8 N′(n) = 2 × πn3 = πn3 3 3 Using the expression for n: L m*E n = so: π 22 3/2 8 ⎛ m*E ⎞ N′(E) = L3 3π 2 ⎝⎜ 22 ⎠⎟ 3/2 dN′(E) L3 ⎛ 2m* ⎞ = E dE 2π 2 ⎝⎜ 2 ⎠⎟ The Density of States is defined as the number of states per incremental unit of energy near E per unit volume: 3/2 dN′(E) 1 1 ⎛ 2m* ⎞ N (E) = = E dE L3 2π 2 ⎝⎜ 2 ⎠⎟ 25 Other Cases

• Valence Band – Need to include both light holes and heavy holes in the density of states • Near the Direct-Indirect Crossover – Need to include both valleys • Indirect Materials – Multiple valleys due to symmetry and energy degeneracy – This is included in the density of states effective http://www.ioffe.rssi.ru/SVA/NSM/Semicond/Ge/Figs/221.gif mass

Carrier Concentration at Thermal Equilibrium

The Fermi Function Multiplied by the Density of States

∞ ni = f (E)Nc (E)dE ∫Ec

EV = pi = ⎡1− f (E)⎤ NV (E)dE ∫−∞ ⎣ ⎦

∞ n0 = f (E)Nc (E)dE ∫Ec

Ev p0 = ⎡1− f (E)⎤ Nv (E)dE ∫−∞ ⎣ ⎦

28 DOS: Appendix IV Approximating the Fermi Function

1 E f (E) = (E−E )/kT 1+ e f

T=T ⎛ −5 eV ⎞ 2 kT = ⎜ 8.62x10 ⎟ (300K )  0.0259eV EC ⎝ K ⎠ T=T 1 In the case where E = E E f C 1 f (EC ) = (E −E )/kT 1+ e C f Note: e1  2.7, e2  7.4, e3  20

EV 1 1 1 1 = = 0.048 ~ 0.05 = = = e−3 1+ e3 21 20 e3 1 (E − E ) f E C f ( ) 1/2 So, for > 3, or EC − E f > 3kT kT ( )

1 −(EC −E f )/kT f (EC ) = (E −E )/kT ~ e 1+ e C f Electron Concentration no in a n-type Semiconductor at Equilibrium

∞ no(K)=Nd, majority carrier n = f E • N(E)dE o ∫ ( ) e e e e e e Ec Ec no = Nc (Ec )• f (Ec ) e e E d f (E) = Fermi Distribution Funtion Ef N(E) = Density of States (cm-3 ) Ei Nc (Ec ) = Effective Density of States (located at conduction band edge) Ev e e e e e e e e e e e e e 1 f (Ec ) = (E −E )/kT 2 c f po(K)=ni /no, minority carrier 1+ e −(Ec −E f )/kT (Ec −E f )/kT f (Ec ) = e if e >> 1 ∞ ∞ * 3/2 1 2 ⎛ m ⎞ 1/2 3/2 no = f (E)• N(E)dE = (E E )/kT • E dE * ∫ ∫ 1+ e − f π 2 ⎝⎜ 2 ⎠⎟ ⎛ 2π • m kT ⎞ Ec Ec n Nc (Ec ) = 2 ∞ * 3/2 * 3/2 ⎜ 2 ⎟ −(E−E )/kT 2 ⎛ m ⎞ ⎛ 2π m kT ⎞ E E /kT ⎝ h ⎠  e f • E1/2dE = 2 n e( F − C ) ∫ π 2 ⎝⎜ 2 ⎠⎟ ⎝⎜ h2 ⎠⎟ Ec −(Ec −E f )/kT no = Nc • e = N T e(EF −EC )/kT = N T e−(EC −EF )/kT C ( ) C ( ) 30

Assignments

• Read info packet – key course policies and schedule are outlined here, including hourly exam dates • Homework assigned every Friday, due following Friday • Reading from Streetman’s book: – Wed 1/24: Review Streetman Chapter 2 – Wed 1/24: §'s 3.1, 3.1.1, 3.1.2 – Fri 1/26: §'s 3.1.3, 3.2.1 (HW1 Due) – Mon 1/29: §'s 3.2.3, 3.2.4 – Wed 1/31: §'s 3.3.1, 3.3.2 • Chapter 1&2 in Pierret covers similar material

32 Assignments

• Read info packet – key course policies and schedule are outlined here, including hourly exam dates • Homework assigned every Friday, due following Friday • Reading from Streetman’s book: – Wed 1/31: §'s 3.3.1, 3.3.2 – Fri 2/2: §'s 3.3.1, 3.3.2 (HW2 Due) – Mon 2/5: §'s 3.3.3, 3.3.4 – Wed 2/7: § 3.4.1 • Chapter 1&2 in Pierret covers similar material

Outline, 2/5/18

• Temperature Effects • Compensation

Instructional Objectives (1)

By the time of exam No. 1 (after 17 lectures), the students should be able to do the following: 1. Outline the classification of solids as metals, semiconductors, and insulators and distinguish direct and indirect semiconductors. 2. Determine relative magnitudes of the effective mass of electrons and holes from an E(k) diagram. 3. Calculate the carrier concentration in intrinsic semiconductors. 4. Apply the Fermi-Dirac distribution function to determine the occupation of electron and hole states in a semiconductor. 5. Calculate the electron and hole concentrations if the Fermi level is given; determine the Fermi level in a semiconductor if the carrier concentration is given. 6. Determine the variation of electron and hole mobility in a semiconductor with temperature, impurity concentration, and electrical field. 7. Apply the concept of compensation and space charge neutrality to calculate the electron and hole concentrations in compensated semiconductor samples. 8. Determine the current density and resistivity from given carrier densities and mobilities. 9. Calculate the recombination characteristics and excess carrier concentrations as a function of time for both low level and high level injection conditions in a semiconductor. 10. Use quasi-Fermi levels to calculate the non-equilibrium concentrations of electrons and holes in a semiconductor under uniform photoexcitation. 11. Calculate the drift and diffusion components of electron and hole currents. 12. Calculate the diffusion coefficients from given values of carrier mobility through the Einstein’s relationship and determine the built-in field in a non-uniformly doped sample.

https://my.ece.illinois.edu/courses/description.asp?ECE340 37 Instructional Objectives (2)

By the time of Exam No.2 (after 32 lectures), the students should be able to do all of the items listed under A, plus the following: 13. Calculate the contact potential of a p-n junction. 14. Estimate the actual carrier concentration in the depletion region of a p-n junction in equilibrium. 15. Calculate the maximum electrical field in a p-n junction in equilibrium. 16. Distinguish between the current conduction mechanisms in forward and reverse biased diodes. 17. Calculate the minority and majority carrier currents in a forward or reverse biased p-n junction diode. 18. Predict the breakdown voltage of a p+-n junction and distinguish whether it is due to avalanche breakdown or Zener tunneling. 19. Calculate the charge storage delay time in switching p-n junction diodes. 20. Calculate the capacitance of a reverse biased p-n junction diode. 21. Calculate the capacitance of a forward biased p-n junction diode. 22. Predict whether a metal-semiconductor contact will be a rectifying contact or an ohmic contact based on the metal work function and the semiconductor electron affinity and doping. 23. Calculate the electrical field and potential drop across the neutral regions of wide base, forward biased p+-n junction diode. 24. Calculate the voltage drop across the quasi-neutral base of a forward biased narrow base p+-n junction diode. 25. Calculate the excess carrier concentrations at the boundaries between the space-charge region and the neutral n- and p-type regions of a p-n junction for either forward or reverse bias.

https://my.ece.illinois.edu/courses/description.asp?ECE340 38 Instructional Objectives (3)

By the time of the Final Exam, after 44 class periods, the students should be able to do all of the items listed under A and B, plus the following: 26. Calculate the terminal parameters of a BJT in terms of the material properties and device structure. 27. Estimate the base transport factor “B” of a BJT and rank-order the internal currents which limit the gain of the transistor. 28. Determine the rank order of the electrical fields in the different regions of a BJT in forward active bias. 29. Calculate the threshold voltage of an ideal MOS capacitor. 30. Predict the C-V characteristics of an MOS capacitor. 31. Calculate the inversion charge in an MOS capacitor as a function of gate and drain bias voltage. 32. Estimate the drain current of an MOS transistor above threshold for low drain voltage. 33. Estimate the drain current of an MOS transistor at pinch-off. 34. Distinguish whether a MOSFET with a particular structure will operate as an enhancement or depletion mode device. 35. Determine the short-circuit current and open-circuit voltage for an illuminated p/n junction solar cell.

https://my.ece.illinois.edu/courses/description.asp?ECE340 39 Course Purpose & Objectives

• Introduce key concepts in semiconductor materials • Provide a basic understanding of p-n junctions • Provide a basic understanding of light-emitting diodes and photodetectors • Provide a basic understanding of field effect transistors • Provide a basic understanding

of bipolar junction transistors Forward Reverse Bias Bias n-type emitter n-type collector injected hole fow holes leakage current injected electron fow electrons p-type base

40 Tentative Schedule [2]

FEB 19 FEB 21 FEB 23 Quasi-Fermi levels and Carrier diffusion Built-in fields, diffusion and photoconductive devices recombination

Feb 26 FEB 28 MAR 2 Review, discussion, Steady state carrier p-n junctions in equilibrium problems (2/27 exam) injection, diffusion length & contact potential

MAR 5 MAR 7 MAR 9 p-n junction Fermi levels Continue p-n junction NO CLASS (EOH) and space charge space charge

MAR 12 MAR 14 MAR 16 p-n junction current flow Carrier injection and the Minority and majority diode equation carrier currents

3/19-3/23 Spring Break MAR 28 MAR 30 MAR 26 Stored charge, diffusion Photodiodes, I-V under Reverse-bias breakdown and junction capacitance illumination

**Subject to Change** 41 Tentative Schedule [3]

APR 2 APR 4 APR 6 LEDs and Diode Lasers Metal-semiconductor MIS-FETs: Basic operation, junctions ideal MOS capacitor

APR 9 APR 11 APR 13 MOS capacitors: flatband & Review, discussion, MOS capacitors: C-V threshold voltage problems (4/12 exam) analysis

APR 16 APR 18 APR 20 MOSFETs: Output & MOSFETs: small signal Narrow-base diode transfer characteristics analysis, amps, inverters

APR 23 APR 25 APR 27 BJT fundamentals BJT specifics BJT normal mode operation

APR 30 MAY 2 (LAST LECTURE) FINAL EXAM BJT common emitter Review, discussion, **Date & time to be amplifier and current gain problem solving announced**

**Subject to Change** 42

Important Information

• Course Website: – http://courses.engr.illinois.edu/ece340/ • Download and Review Syllabus / Course Information from Website! • Course Coordinator: Prof. John Dallesasse – [email protected] – Coordinates schedule, policies, absence issues, homework, quizzes, exams, etc. • Contact Information and Office Hours for All ECE340 Professors & TAs in Syllabus • Lecture Slides: Click on “(Sec. X)” next to my name in instructor list • DRES Students: Contact Prof. Dallesasse ASAP • Textbook: – “Solid State Electronic Devices,” Streetman & Banerjee, 7th Edition – Supplemental: “Semiconductor Device Fundamentals,” Pierret – Additional reference texts listed in syllabus

44 Key Points

• Attend Class! – 3 unannounced quizzes, each worth 5% of your grade – You must take the quiz in your section – Excused absences must be pre-arranged with the course director – Absences for illness, etc. need a note from the Dean • See policy on absences in the syllabus • No Late Homework – Homework due on the date of an excused absence must be turned in ahead of time – You must turn in homework in your section – No excused absences for homework assignments – Top 10 of 11 homework assignments used in calculation of course grade • Do all of them to best prepare for the exams! • No Cheating – Penalties are severe and will be enforced • Turn Off Your Phone – No video recording, audio recording, or photography

45 Homework

• Assigned Friday, Due Following Friday – Due dates shown in syllabus • Due at Start of Class • Follow Guidelines in Syllabus • Peer Discussions Related to Homework are Acceptable and Encouraged • Directly Copying Someone Else’s Homework is Not Acceptable – Graders have been instructed to watch for evidence of plagiarism – Both parties will receive a “0” on the problem or assignment

46 Absences

• The absence policy in the syllabus will be strictly enforced • To receive an excused absence (quiz), you must: – Pre-arrange the absence with the course director (valid reason and proof required) – Complete an Excused Absence Form at the Undergraduate College Office, Room 207 Engineering Hall (333-0050) • The form must be signed by a physician, medical official, or the Emergency Dean (Office of the Dean of Students) • The Dean’s Office has recently put a strict policy in place (3 documented days of illness) – Excused quiz score will be prorated based upon average of completed scores – No excused absences are given for homework, but only the best 10 of 11 are used to calculate your final grade – Excused absences are not given for exams, except in accordance with the UIUC Student Code – Unexcused work will receive a “0” • Failure to take the final will result in an “incomplete” grade (if excused) or a “0” (if unexcused)

47 Exams

• Exam I: Tuesday February 27th, 7:30-8:30 pm • Exam II: Thursday April 12th, 7:30-8:30 pm • Final Exam: Date/Time To Be Announced – Determined by University F&S

48 Grading

Grading Criterion Historical Grade Trends*

Homework 10 % Spring Fall Spring Quizzes 15 % 2016 2016 2017 Hour Exam I 20 % A’s 27 % 28 % 27 % Hour Exam II 20 % B’s 37 % 26 % 38 % Final Exam 35 % C’s 27 % 25 % 27% Total 100 % D’s 6 % 16 % 4 % F’s 3 % 5 % 4 %

*Past performance is not necessarily indicative of future results

49 My Recommendations

• Read the syllabus and information posted on the course website • Attend class & participate • Attend office hours (TA and Professors) • Read the book • Re-read the book • Look at and read selected portions of the supplemental texts • Form study groups to review concepts and discuss high- level approaches for solving homework problems – Don’t form study groups to copy homework solutions • Don’t miss any homework, quizzes, or exams • It’s hard to overcome a zero • Ask questions in class!