材料物理化学 第三篇习题

Chap 11、Semiconductors

11.1 (a) For a semiconductor, show that np product obtained from

Eq.(11.27) is proportional to exp(-βE g ) and thus is independent of the position of the chemical potential μ in the bandgap.

Eq.(11.27):

(b) The law of mass action in semiconductors for reaction creating

pairs of electrons and holes [e.g., Eq.(11.28)] has the form n(T)P(T)∝exp(-βE g ). Explain the significance of this law. (Hint: The law of mass action is described in Section 4.6)

Eq.(11.28): (c) Evaluate the np product at T=300K for Si with E g =1.11ev and

m eds ﹡=1.05m and m hds ﹡=0.58m.

11.2 Using Eq.(11.30) and m eds ﹡=1.05m and m hds ﹡=0.58m for Si,

calculate the change in the position of the chemical potential μ in the energy gap of intrinsic Si between T=0 and 300K.

Eq.(11.30): 11.3 Calculate the values of N c and N v as defined in Eq.(11.27) for Si

at T=300K. The appropriate of density-of-states effective masses for Si are m eds ﹡=1.05m and m hds ﹡=0.58m.

11.4 Consider a semiconductor with a bulk energy gap E g =1.5ev and

g

E v c i i e T N T N T p T n β-=)()()()()()(2/32)()2(2)()(μβμβπ----*===c c E c E B eds i e T N e T k T n T n m )()(2/32)()2(2)()(v v E v E B h ds i e T N e T k T p T p m ----*===μβμβπ **

+=eds hds

B g m m T k E T ln 432)(μ

with m e﹡=m h﹡=0.1m. Calculate the increase in the energy gap of

this semiconductor when it is incorporated into the following

structures:

(a) A quantum well (d=2) with L x=10nm.

(b) A quantum wire (d=1) with L x=L y=10nm

(c) A quantum dot (d=0) with L x=L y=L z=10nm

11.5 A Hall effect measurement is carried out on a rectangular bar

of a semiconductor with dimensions L x=0.04m (the direction

of current flow ) and L y=L z=0.002m. When a current I x=5mA

flows in the +x direction and a magnetic field B z=0.2T is

applied in the +z direction, the following voltages are

measured: V x=6V and V y=+0.3mV (i.e., increasing in the +y

direction). Determine the following properties of the

semiconductor bar from these data :

(a) The sign of the dominant charge carriers.

(b) The concentration of the dominant charge carriers.

(c) The electrical conductivity σ.

(d) The mobility μ of the dominant charge carriers.

11.6Using Eq.(11.59), estimate the increase △n in the electron

concentration in an n-type semiconductor due to the uniform

absorption of light with α=105m-1, I0=1W/m2, and hω=1e V, a

quantum efficiency η=1, and a minority-carrier lifetime ηp=10-3s.

Eq.(11.59): 11.7 Using the definition of the Hall mobility μH =︱ζR H ︱and the

expression for R H for an intrinsic semiconductor given in Eq.(11.49), show that μH =︱μh -μe ︱.

Eq.(11.49): 11.8 Consider the structural transformation of a binary crystal AB

from the hexagonal wurtzite crystal structure to the cubic zincblende crystal structure in which the density of the atoms remains constant. Find the lattice constant of the resulting cubic crystal if the lattice constants of the initial wurtzite crystal are a =0.3400nm and c =0.5552nm.

11.11 List all of the local tetrahedral bonding units, A-B 4, which are

present in the ternary semiconducting compounds Cu 2SiTe 3, Cu 3PS 4, and CuSi 2P 3. Note that each tetrahedron must contain an average of four bonding electrons per atom.

11.13 Derive the expression for the shift △E of the electron energy

bands from one side of a p-n junction to the other under zero bias as given in Eq.(11.93). Calculate the magnitdde of the built-in electric potential V B =△E/e for Si at T=300K for N d =N a =2×1024m -3. Using these same parameters, calculate the depletion width d and the maximum electric field Q/∈A for a Si p-n junction at T=300 K.

ω

τηατω p

p L I G p n 0)(==?=?)

(e h e

h H ne R μμμμ+-=

Eq.(11.93): .

Chap 12、Metals and Alloys 12.1 Referring to Section 12.5, show that the condition for the tangency

of the Fermi sphere to the Brillouin zone boundary for the FCC lattice is N =1.36.

12.2 Derive Eq.:

12.3 Derive Eq.:

Chap 13、Ceramics

13.1 For the silicon oxynitride compound Si 2N 2O, assume that Si, N,

and O atoms have their usual valences (4, 3, and 2) and that the N and O atoms do not form covalent bonds with each other. (a) Given a local bonding unit Si-N x O y for Si with x+y =4,

determine x (and y ) for this crystal structure.

(b) What are the local bonding units for N and O?

13.2 For the Si x N y O z ternary phase diagram, locate the following

compounds :SiO 2, Si 3N 4, Si 2N 2O, and Si 3N 2O 3.

13.3 Find the average number of bridging oxygens, b, and nonbridging

oxygens, n, for the following glasses:

(a)CaO?SiO 2, and

(b) soda-lime(i.e., 2CaO?3Na 2O?15SiO 2) n i

a d B v c a d B g N N T k N N N N T k E E 2ln ln =+=?)163253(02222∈-=πF F k e m k N U )]1(356[822

02ws c ws Coul r r r Nze U --∈=π

Chap 14、Polymers

14.1 A polymer whose viscoelastic properties are described by

Eq.(14.40) (i.e., the Maxwell model) is subjected to a time-dependent stress σ=σ0exp(-i ωt). Find the steady-state strain. Compare this result to that of a polymer that obeys the Voigt model, given by Eq. (14.37).

Eq.(14.40): Eq. (14.37): 14.2 Consider an elastomer consisting of monomers that are optically

anisotropic [i.e., they have a polarizability α11(ω) for light parallel to the chain axis and α┴(ω) for light polarized perpendicular to the chain axis]. Assume that there are N chains per unit volume. Let )(ωn be the mean index of refraction of the material. The elastomer is stretched with a steching parameter s , as defined in Secti on 14.5. Show that the elastomer will have a birefringence given by

)]()()[1()(]2)([452)()()(1122

11ωαωαωωπωωωδ⊥--⊥--+=-=s s n n N n n n

Obtain an expression for the stress optical coefficient. C≡δn(ω)/ζ, where ζ is the applied stress.

Chap 15、Dielectric and Ferroelectric Materials

??=+ε

ησσG η

στεε=+?

15.1 Given the Landau free-energy density for a ferroelectric of the form Where b>c. Let a=a 0(T-T C ) and assume that b and c are constant. Find P z and χ as a function of T for the state of thermal equilibrium. 15.2 Design a piezoelectric actuator that can be used to sweep an

STM head over the surface of a solid. What is the area that can practically be covered?

15.3 Adapt Weiss molecular field theory (see Chapter 9) to describe a

ferroelectric. Assume that there are just two orientations for the electric-dipole moment of a unit cell and that NN cells interact via an exchange interaction. Obtain the hysteresis curve and values for the coercive field E c , saturation polarization P sat , and remanent polarization P rem .

15.4 BaTiO 3 is a paraelectric for T>T C =130℃ and has a Curie constant

C=76,000K.

(a)If the lattice constant for the cubic unit cell of BaTiO 3 is a=0.401nm, calculate the electric-dipole moment μ of this unit cell. (b)What would the corresponding polarization P=μn be at T=0 K?

Chap 16、Superconductors

16.1 (a) Derive expression for the difference in entropy

△S(T)=S n (T)-S s (T) and the difference in specific heat

Z y x x z z y z y x o EP P P P P P P c P P P b P a g g -+++++++=)(2

)(422222224442

△C(T)=C n (T)-C s (T) between the normal and superconducting states in terms of the critical magnetic field H c (T) and its first derivative dH c /dT and second derivative d 2H c /dT 2. [Hint: Use Eq.(16.3) and standard thermodynamic relationships.]

(b) Evaluate these expressions for △S(T) and △C(T) for the case where H c (T) can be approximated by H c0[1-(T/T c )2] and show that : (ⅰ) △S(T c )=△S(0)=0

(ⅱ) △S(T)>0 for 0

(ⅲ)△C(T c )=-4μ0H c02/T c . Calculate △C(T c ) from this expression

using T c = 1.175 K and H c0 = 105 Oe = 8360 A/m for Al and compare with the measured result -225 Jm -3K -1 for Al.

(ⅳ) △C=0 for T=T c /3 and T=0K.

Eq.(16.3): 16.2 (a) Using Eq.(16.5), calculate the condensation energy in J/m 3 and in

eV per electron at T=0K for the superconductor Pb for which H c0=6.39×104A/m.

(b) Compare your result from part (a) with the expression ε(0)(ε(0)/E F ) where the superconducting energy gap 2ε(0) = 2.6 meV for Pb. Here ε(0)/E F is the fraction of conduction elections whose energies are actually affected by the condensation .

Eq.(16.5): 16.3 Consider the London penetration depth λL defined in Eq.(16.10).

?+=-=H s s s s H T G dH H M T G T H G 02002

),0()(),0(),(μμ2

)(),0(),0(2

0T H T G T G c s n μ=-

(a) Calculate λL (0K) for the superconducting Al, Pb, and Nb. (b) If a superconductor has a London penetration depth λL (0K)=200nm, what is the concentration n s of superconducting electrons at T=0.5T c .

Eq.(16.10): 16.4 When transport current i flows through a superconducting wire

of radius R, its path is confined to a region of thickness λ, the penetration depth, just inside the surface of the wire.

(a) In this case show that the critical current density J c =i c /A eff is independent of R and can be expressed in terms of the critical field H c by J c =H c /λ. Here A eff is the effective area through which the current flows, with A eff <<πR 2.

(b) Calculate J c for superconducting Pb at T=0K. [Note: H c0 = 803 Oe = 63919 A/m and λ(0) = 39 nm]

(c) Sketch J c (T)/J c (0) from T = 0 K to T c using the temperature dependencies of H c and λ given in Eqs.(16.6) and (16.11), respectively .

Eq.(16.6):

Eq.(16.11): 16.5 A type Ⅱ superconductor has T c =125k, ΘD =250K, and κ(T c )=50. On

the basis of standard theories [free-electron model, Debye model, BCS theory, G-L theory, Pauli limit for H c2 given in Eq.(16.33)], 2

0)()(e T n m

T s L μλ=)1()(220c c c T T H T H -=4

)/(1)

0()(c L L T T T -=λλ

estimate the following:

(a)The superconducting energy gap 2ε(0).

(b) The upper critical field H c2(0)=H p .

(c)The co herence length ξ(0) and the penetration depth λ(0).

(d)The thermodynamic critical field H c0=H c (0).

(e)The coefficients γ and A of the electronic and phonon contributions to the specific heat , γT and AT 3, respectively.

Eq.(16.33): 16.7 Use Eq.(16.20) to find the limiting values of λ(l ) and ξ(l ) (a) in the

clean limit where the electron mean free path l >>ξ0, and (b) in the dirty limit where l <<ξ0.

Eq.(16.20): and 16.8 (a) Calculate the density of vortices per unit area B/Φ0 for the

following values of B, the average flux density present in the mixed state of a superconductor. Take H c2=1.6MA/m.

(i) B = μ0H c2/2. (ii) B ≈ B c2 = μ0H c2.

(b)Calculate the average separation d between the vortices from your answers in part (a) and compare your answers with the conherence length ξ. [Hint: You can obtain ξ with the help of Eq.(16.22).]

Eq.(16.22): 16.10 Calculate the number of holes N hole per Cu ion in the CuO 2

copper-oxygen layers in the superconductor YBa 2Cu 3O 7-x for the B

p T T H μμε0)

()(≈l l 11)(10+=ξξl

l ξλλ+∞=1)()()

(2)(200

2T T H c ξπμΦ=

cases of x = 0, 0.25, and 0.5. Assume the following ionic charge states for the ions in this structure: Y 3+, Ba 2+, Cu 2+, and O 2-.

16.11 For the compound with the chemical formula La 1.7Sr 0.3CuO 3.9:

(a)what is the total number of electrons per formula unit outside closed shells?

(b)How many electrons are contributed by each ion to the CuO 2 layers?

(c)what is the average valence of the copper atoms?

(d)Assuming that all copper ions have a charge of +2e, what is the number of holes per formula unit?

16.12 Derive Eq.(16.33) for the Pauli limiting field H p by setting

G n (H)=G s (H) at H=H p and using the Pauli paramagnetic

susceptibility χp =μ0μB 2ρ(E F ) of the conduction electrons in the

normal state. [Hint: Use G n (H)-G n (0)=-μ0χp H 2/2, and the BCS result

G n (0)-G s (0)=ρ(E F )ε(0)2/2.]

Eq.(16.33):

Chap 17、Magnetic Materials

17.1 Consider a single-domain uniaxial ferromagnetic particle

magnetized along its easy axis with M=M s in zero applied magnetic field. The magnetic anisotropy energy density is given by E a =K sin 2θ where K >0 and θ is the angle between the B

p T T H μμε0)()(≈

magnetization M and the easy axis. A magnetic field H is now applied at 90° to the easy axis.

(a)Show that the sum of the anisotropy and magnetostatic energy densities for this particle is ｕ(θ)=K sin 2θ-μ0MHsinθ.

(b)Find the angle θ be tween M and the easy axis as a function of the magnitude of the field H by minimizing ｕwith respect to θ .(Note that it will be important to check for the stability of the solution by requiring that ?2ｕ/?θ2>0.)

(c) Show that the resulting magnetization curve (i.e., the plot of the

component of M in the direction of H versus the applied field H ) is a straight line (with slop χ=μ0M 2s /2K ) up to H =H k =2k/μ0M s , at which point the magnetization is saturated in the direction of H . Here H k is the effective magnetic anisotropy field in Eq.(17.14).

Eq.(17.14): 17.4 Prove that when the shape anisotropy constant K s is <0 (i.e., when

N ⊥

17.5 Calculate the radius at which a spherical Fe particle behaves

superparamagnetically at T =300K by setting K 1V=K B T , where K 1≈4.2×104J/m 3 is the first-order magnetocrystalline anisotropy coefficient for Fe and V is the volume of the sphere.

17.6 Calculate t he increase in temperature ΔT of a magnetic material with

s

K M K

H 02μ=

a square magnetization loop, with M s =1370KA/m and H c =1100kA/m, when the loop is traversed once, Assume that the material is thermally-isolated from its surroundings and that its specific heat is 4×106J/m 3K.

17.7 For a magnetically isotropic m aterial with magnetostriction λ, prove

that B 2(C 11-C 12)=2B 1C 44. Show, in fact, that if the material is also elastically isotropic, then B 1=B 2. (Hint: See Section 10.8)

17.8 Show that B (t) and M (t) both lag the applied magnetic field

H(t )=H 0e -iωt by the same phas e angle δ when μ0H 0<

Eq.(17.33): and

Chap 18、Optical Materials

18.1 Suppose that a quantum dot has the shape of a two-dimensional

circular disk. A model that is often used to describe the potential of an electron confined in such a dot is V(r)=m ﹡ω02r 2/2. Suppose a magnetic induction B is imposed perpendicular to the plane of the dot. Show that the electron energy levels are given by the formula where n=0, 1, 2,…………and l=……, -2, -1, 0, 1, 2,……..

18.2 Consider a Lorentz oscillator model for an electron moving in )(0

)(δω+-=t i e B t B )

(0)(φω+-=t i e M t M *

*-+++=m eB l m eB l n E l n 2)2(

)()12(220, ω

one-dimensional anharmonic potential described by the Toda potential V(x)=Ae -ax +Bx, where A, a and B are constants. The equation of motion is

Derive expressions for the linear polarization P , at frequency ω and the nonlinear polarization at frequency 2ω, P (2ω).

18.3 Consider a particle of mass m moving in the anharmonic

symmetric potential V(x)=Acosh[a(x-x o )] subject to a damping force –γv and a driving force qE cosωt . Find the Fourier coefficients for the dipole moment at frequencies ω and 3ω. 18.4 Using Vegard′s law , derive an expression for the bandgap energy

E g (x,y) of a layer of In 1-x Ga x As y P 1-y which is lattice matched to an InP substrate. Compare your expression with the experimental result given in Eq.(18.15) and comment on any differences.

Eq.(18.15): E g =1.35-0.72y+0.12y 2 eV

t qE e e aA dt

dx dt x d m o ax ax ωγcos )()(22+-=+--