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2/2/14 Things you should know when you leave… ECE 340 Lecture 6 : Intrinsic and Extrinsic Material I Class Outline: • Effective Mass • Intrinsic Material • Extrinsic Material Key Questions • What is the physical meaning of the effective mass • What does a negative effective mass mean? • What is intrinsic material? • What is thermal equilibrium? • What is extrinsic material? • How does doping work? M.J. Gilbert Effective Mass Effective Mass At the end of lecture 5, we talked about effective mass… We even defined the effective mass… • In a vacuum, we can apply Newton’s second law: Electric Field Electric Field F = −qE = m0 We can define the effective mass as: dv dt m* = • In a semiconductor, we cannot. ECE 340 – Lecture 6 dv dt !2 d 2 E / dk 2 Nevertheless, two questions remain: – For overall motion – NO! – For motion in-between scattering – NO! • We defined a new “effective” mass which incorporated all of the complicated interactions. F = −qE = mn* M.J. Gilbert ECE 340 – Lecture 6 1. Where does this definition come from? 2. What does it mean physically? M.J. Gilbert ECE 340 – Lecture 6 1 2/2/14 Effective Mass Effective Mass Let’s begin to think about where effective mass comes from… What are the forces that the electron is experiencing? Start with the energy-wavevector (dispersion) relation for free electrons: ! 2k 2 Ek = 2m (6.1) k’ Now look at the equation of motion for how electrons move in an energy band in an electric field. k ⎛ eE field ⎝ ! δk = −⎜⎜ (6.2) ECE 340 – Lecture 6 M.J. Gilbert Effective Mass approximation to the band: – W (Band Width) ~ 5 eV – a (lattice spacing) ~ 0.5 nm dt d 2 E 1 ⎛ d 2 E dk ⎞ ⎟ = ⎜ !dkdt ! ⎜⎝ dk 2 dt ⎟⎠ ⎛ 1 d 2 E ⎞ ⎟ F = ⎜⎜ 2 2 ⎟ ⎝ ! dk ⎠ = Newton’s 2nd law! 1 m* ECE 340 – Lecture 6 The group velocity goes to zero!! What about the effective mass? 1 ⎛ ka ⎞ E (k ) = W (1 − cos ka ) = W sin 2 ⎜ ⎟ 2 ⎝ 2 ⎠ 0.3 E (k ) eV 5 Effective mass: 0 π a −π a -0.3 • The effective mass becomes negative! 1 dE (k ) aW Group velocity: vg (k ) = = sin (ka ) − π ! dk 2! a ECE 340 – Lecture 6 m* (k ) = 2! 2 sec(ka ) m0 a 2W – States of positive mass occur near the bottom of the bands due to positive band curvature. – States of negative mass occur at the top of bands. What are the group velocity and the effective mass? M.J. Gilbert dv g Effective Mass Simple Example… Consider a simple cosine • Sample parameters ⎞ ⎟⎟δt ⎠ where, dk ! = −eE field = F dt All of the information of the effects of the crystal on the motion of the electron are in the dispersion relation. M.J. Gilbert (6.4) Combine eqns. 6.3 and 6.4 to arrive at an external force that is exerted on the electrons by the applied electric field. The wavepacket is moving with some group velocity, vg: k’ We observe that by using eq. 6.2… ⎛ dE ⎞ ⎟δk = !v g δk ⎝ dk ⎠ E 1 dE vg = ! dk k δE = ⎜ Suppose that the wavepacket is made of wavefunctions near a particular k. Ψ(x) How much work is the field doing on the vg electron? δE = −eE field vg δt (6.3) E Ψ(x) vg(k) π a • Physically, it means that on going from k to k+Δk the momentum transfer to the lattice from the electron is larger than that of the momentum transfer from the applied force to the electron. – As we approach Bragg reflection at the edge, when we increase the wavevector we can get an overall decrease in the forward momentum. M.J. Gilbert ECE 340 – Lecture 6 2 2/2/14 Intrinsic Material Intrinsic Material Intrinsic Material is pure with no additional contaminants… But there are more processes at work… T=0K Generation Rate: ⎛ 1 ⎞ G = Gth + Gopt + Gmech + ...⎜ 3 ⎟ ⎝ cm ⋅ s ⎠ T = 300 K • At T = 0 K, there is no energy in the system. – All of the covalent bonds are satisfied. – Valence band is full and conduction band is empty. • At T > 0 K, thermal energy breaks bonds apart – Crystal lattice begins to vibrate and exchange energy with carriers. – Electrons leave the valence band to populate the conduction band. M.J. Gilbert ECE 340 – Lecture 6 • Generation – Break up of a covalent bond to form an electron and a hole. – Requires energy from thermal, optical, mechanical or other external sources. – Supply of bonds to break is virtually inexhaustible. • Atomic density >> # of electrons or # of holes. M.J. Gilbert Intrinsic Material Intrinsic Material Since we are in thermal equilibrium, there must be an opposite process… In the steady state… = Recombination Rate: ⎛ 1 ⎞ R ∝ n • p⎜ 3 ⎟ ⎝ cm ⋅ s ⎠ • N – number of electrons • P – number of holes • Recombination – Formation of a bond by bringing together and electron and a hole. – Releases energy in the form of thermal or optical energy. – Recombination events require the presence of 1 electron and 1 hole. – These events are most likely to occur at the surfaces of semiconductors where the crystal periodicity is broken. M.J. Gilbert ECE 340 – Lecture 6 ECE 340 – Lecture 6 • The generation rate must be balanced by the recombination rate. G0 = R0 ⇒ n0 p0 = ni2 n0 = p0 ⇒ n0 = p0 = ni • Important consequence is that for a given semiconductor the np product depends only on the temperature. M.J. Gilbert ECE 340 – Lecture 6 3 2/2/14 Intrinsic Material Extrinsic Semiconductors Putting numbers to the intrinsic concentrations… The great strength of semiconductors… Silicon ni ~ 1010 cm-3 Germanium ni ~ 2 x 1013 cm-3 • For silicon GaAs ni ~ 2 x 106 cm-3 M.J. Gilbert – – – – – 5 x 1022 atoms/cm3 4 bonds per atom 2 x 1023 bonds/cm3 ni (300 K) ~ 1010 cm-3 1 broken bond per 1013 bonds. ECE 340 – Lecture 6 Extrinsic Materials How does a donor work? • We can change their properties many orders of magnitude by introducing the proper impurity atoms. • Which columns add – Electrons? – Holes? • What about impurities? M.J. Gilbert ECE 340 – Lecture 6 Extrinsic Materials Phosphorous (P) 5 valence electrons Silicon (Si) 4 valence electrons How does an acceptor work? Silicon (Si) 4 valence electrons Boron (B) 3 valence electrons Si! B M.J. Gilbert ECE 340 – Lecture 6 M.J. Gilbert ECE 340 – Lecture 6 4 2/2/14 Extrinsic Materials Extrinsic Materials In general, we can modify the materials properties with the introduction of immobile impurity atoms… How tightly bound is the extra electron or hole? • We can – Selectively create regions of n and p. • Needed for CMOS. • Acceptor in Si Binding energy (eV) P 0.045 B 0.045 M.J. Gilbert Extrinsic Materials EB = − As 0.054 Al 0.067 Acceptor * n 4 mq 2 32π 2 (ε 0ε r ) ! 2 Sb 0.039 Ga 0.072 r In 0.16 ECE 340 – Lecture 6 Extrinsic Material Visualizing donors on the band diagram… Ec Remember the intrinsic concentrations… Ec Ed Ev Ea Ev Silicon ni ~ 1010 cm-3 Let’s take a look at Silicon with Phosphorus impurity atoms: Ec Ed Eg = 1.12 eV GaAs ni ~ 2 x 106 cm-3 ECE 340 – Lecture 6 Germanium ni ~ 2 x 1013 cm-3 • For silicon 0.045 eV Ev M.J. Gilbert Donor The electron mass must be represented by the effective mass Donor in Si Binding energy (eV) ECE 340 – Lecture 6 Δx e- – Different relative permittivity. – Modify the conductivity over several orders of magnitude. – Manipulate the number of conduction electrons over 5 orders of magnitude. M.J. Gilbert • We can use the Bohr’s hydrogen model to get an idea. • Electrons move in Si and not in a vacuum. h+ M.J. Gilbert – – – – – 5 x 1023 atoms/cm3 4 bonds per atom 2 x 1023 bonds/cm3 ni (300 K) ~ 1010 cm-3 1 broken bond per 1013 bonds. ECE 340 – Lecture 6 5 2/2/14 Extrinsic Materials Extrinsic Material Revisiting the effect of temperature… Commonly used terms: T = 0 K M.J. Gilbert T = 50 K ECE 340 – Lecture 6 T = 300 K • Dopants – specific impurity atoms that are added to semiconductors in controlled amounts for the express purpose of increasing either the electron or hole concentrations. • Intrinsic semiconductor – undoped semiconductor; extremely pure semiconductor sample containing an insignificant amount of impurity atoms; a semiconductor whose properties are native to the material. • Extrinsic semiconductor – doped semiconductor; a semiconductor whose properties are controlled by added impurity atoms. • Donor – impurity atom that increases the electron concentration; n-type dopant. • Acceptor – impurity atom that increases the hole concentration; p-type dopant. • N-type material – a donor doped material; a semiconductor containing more electrons than holes. • P-type material – an acceptor doped material; a semiconductor containing more holes than electrons. • Majority carrier – the most abundant carrier in a given semiconductor sample; electrons in ntype and holes in p-type. • Minority carrier – the least abundant carrier in a given semiconductor sample; electrons in ptype and holes in n-type. M.J. Gilbert ECE 340 – Lecture 6 6