大二上固态电子学复习chapter4lecture35band gap density of states fermi statistics

1、1 1Origin of the Energy Gap in Solids2 2Advanced band theoryForward waveBAAReflected wavesCBCkBackward wave-kxaAn electron wave propagation through a linear lattice. For certain k valuesthe reflected waves at successive atomic planes reinforce each other togive rise to a reflected wave travelling in

2、 the backward direction. Theelectron then cannot propagate through the crystal.p Combination of forward wave and reflective wave results in standing wave- wave does not propagate 3 3Advanced band theory4 4Advanced band theoryWave cannot propagate-standing wave5 5In the case of 3D diffractionElectron

3、kk1k2Diffracted electron(11) Planes(01) Planes(10) Planesk30110yx11454545aa2a6 6Metal, semiconductor and insulatorEnergy gap1110Overlappedenergy gapsEnergy gap = Eg1st BZband2nd BZband(b) Semiconductor and insulatorEnergy gap1110Bands overlapenergy gapsEnergy gap1st BZband2nd BZ&1st BZoverlapped

4、band2nd BZband(a) Metal(a) Metal: For the electron in a metal there is no apparent energy gapbecause the 2nd BZ (Brillouin Zone) along 10 overlaps the 1st BZ along11. Bands overlap the energy gaps. Thus the electron can always find anyenergy by changing its direction.(b) Semiconductor or insulator:

5、For the electron in a semiconductor there isan energy gap arising from the overlap of the energy gaps along 10 and11 directions. The electron can never have an energy within this energygap, Eg.7 7E-K diagram : Velocity and effective massp Velocity is proportional to the slope of the E-k curvep Mass

6、is reverse proportional to the curvature of the E-k curveSlope approach 0 = Velocity approach 0Curvature is small=effective mass is larger8 8At k=0, electron has a constant positive value and it rises rapidly as k value increases. After experiencing a singularity (infinite mass) the effective mass b

7、ecomes negative up to the top Therefore: - m* is positive near the bottoms of all bands, - m* is negative near the tops of all bands.dE/dkEd2E/dk2E-K diagram : Velocity and effective mass9 9Density of state and statistics of electrons1010IntroductionEcEvCBVBEgThermalexcitationElectron energyQuestion

8、s:p The CB and VB can theoretically accommodate how many electrons? A: related to density of statesp Is doping concentration the larger the better? A: related to density of statesp How many electrons are actually in the bands? 1111Density of states1(a)NEnergy Band(b)(c)Eg(E)1212Density of statesn12+

9、 n22= n2n1= 1n2= 3012345123456n1= 2, n2= 2Each state, electron wavefunction in the crystal, can be representedby a box at n1,n2.n2-n2n1-n1n2In here n12+n22+n32n2nVol. =1/8(4/3n3)n3n11313Density of states1414Electron distribution statisticsp Number of electrons decreases exponentially with energyp At

10、 the bottom of the band, E small = more electrons1515Fermi-Dirac statistics1616Fermi level for intrinsic semiconductor1717Fermi level for n-type semiconductor1818Fermi level for p-type semiconductor1919Insulator, semiconductor and conductor2020Number of electronsp So far we know g(E) and f(E), how t

11、o obtain n(E)?p N(E): number of electrons per unit energy per unit volume, is simply the product of g(E)*f(E) Number of electrons per unit volume is 2121Carriers in semiconductor2222Conductivity and Fermi level3s3p3d4sEFCu: 1s2 2s2 2p6 3s2 3p6 3d10 4s12323Summaryp E-K diagram of solids. p Origin of

12、band gap: standing wave in the bottom and the top of the energy band. p Effective mass and velocity from E-K diagram.p Density of states.p Fermi-Dirac distribution for semiconductors. p Fermi level for semiconductors. p Carrier concentration 2424Homeworkp 4.9 Fermi energy and electron concentrationp

13、 4.10 Temperature dependence of the Fermi energyp Plot the Fermi-Dirac distribution curves in one diagram for T1, T2, T3 respectively with T1T2T3.p Plot the electron Fermi-Dirac distribution curves in one diagram for p-doped, intrinsic, n-doped Si respectively. p Plot the hole Fermi-Dirac distributi

14、on curves in one diagram for p-doped, intrinsic, n-doped Si respectively. Submission deadline 20DEC2013 (next Fir.)2525Applications: thermo couple, vacuum tube, field emission2626Metal-Metal contact2727Seebeck effectnegative: free path L increases with TPositive: free path L decreases with TL affect

15、ed by scattering such as lattice vibrations, impurities and crystal defects2828Seebeck effectp X: Takes into account how various charge transport parameters such as L depend on the energyp kt/EF0: only those electrons about a kT around the Fermi level are involved in the transport and scattering pro

16、cess2929Thermo coupleAt 300 K3030Thermo couple3131Thermo coupleThermocouples are widely used to measure the temperature.LEFT: A thermocouple pair embedded in a stainless steel sheath-probe. The thermocouple junction inside the probe is in thermal contact with the probe tip, and, electrically insulat

17、ed from the probe metal.|SOURCE: Courtesy of Omega3232Thermionic emission3333Thermionic emissionp Neglect the effect of the applied field and image charge.p To obtain an efficient thermionic cathode, we need high temperature and low work functionsp However, high melting temperature materials such as

18、 W and Ta, have high work functions, vise versa. p Combination of high melting temperature materials (base) and low work function materials to make the cathode. E.g., Ni+Th, or W+BaO/ThO/Cao etc. 3434Thermionic emission3535Vacuum tubesp Phase out as appearance of transistorp Switching, amplifier, re

19、ctifier, high power and high frequency applicationsp CRT3636Vacuum tubes3737World first computer: ENIAC第一台计算机(ENIAC)于1946年2月,在美国诞生。重30 吨，用了18000 个电子管，(2.4 m 0.9 m 30 m), took up 1800 square feet (167 m2), and consumed 150 kW of power.3838Effect of image charges-q+qinterfaceImage chargex-x3939Effect

20、of applied filedECathodeGrid or AnodeHV V(c)4040Field emission4141Field emissionJ increase with low work function and larger E4242Field emission4343Field emission4444Field emission display (FED)Cross-section of a field emission display showing a Spindt tip cathode, (b) Sony portable DVD player using a field emission display. 4545Field emission display (FED)Advantages:Thinner, lighter, vivid color, fast response, wide viewing angle etc.Hot cathodeCold cathode4646Phonon4747Harmonic osciallator4848Lattice wavePhonon: lattice vibrationPhoton: electromagnetic radiation4949