Materials Science and Engineering - Electronic and Mechanical Properties of Materials Part 3 pot

3.225 15 • Compton, Planck, Einstein – light (xrays) can be ‘particle-like’ • DeBroglie – matter can act like it has a ‘wave-nature’ • Schrodinger, Born – Unification of wave-particle duality, Schrodinger Equation Wave-particle Duality: Electrons are not just particles © E. Fitzgerald-1999 3.225 16 Light is always quantized: Photoelectric effect (Einstein) • Photoelectric effect shows that E=hν even outside the box I,E,λ e- metal block Maximum electron energy, E max ν ν c E max =h( ν - ν c ) ! For light with ν<ν c , no matter what the intensity, no e- © E. Fitzgerald-1999 8 9 3.225 17 DeBroglie: Matter is Wave • His PhD thesis! • λ=h/p also for matter • To verify, need very light matter (p small) so λ is large enough • Need small periodic structure on scale of λ to see if wave is there (diffraction) • Solution:electron diffraction from a crystal Nλ=2dsinθ For small θ , θ ~ λ /d, so λ must be on order of d in order to measure easily © E. Fitzgerald-1999 3.225 18 Ψ must be able to represent everything from a particle to a wave (the two extremes) Unification: Wave-particle Duality wave particle ( tkxi Ae ω− =Ψ k and p known exactly ( txki n n nn ea ω− ∑ =Ψ ∞= n to create a delta function in ψ 2 generalized ( txki n n nn ea ω− ∑ =Ψ © E. Fitzgerald-1999 ) ) ) 3.225 19 Quantum Mechanics - Wave Equation Classical Hamiltonian QM Operators EzyxV m p =+ ),,( 2 2 ∆       = i p h ti E ∂ ∂       −= h 1. ψ and ψ∆ must be finite, continuous and single valued. 2. * ψψ real with dV * ψψ = probability of finding particle in volume dV. 3. Average or expectation value of variable ∫ = v op dV ψαψα * ti zyxV m ∂ ∂− −=+∆− ψ ψψ hh ),,( 2 2 2 © H.L. Tuller-2001 3.225 20 Time and Spatial Dependence of ψ Assume ψ (x,y,z,t) separable into ψ(x,y,z) and φ(t) Applying separation of variables: ε φ φψ ψ = ∂ ∂ −=+ ∇− ti V m 1 2 22 hh = constant Time-Dependent Equation: () ( titi AeAet ωε φ −− == h ωε h = Time-Independent Equation: ( 0 2 2 2 =−+∇ εψ V m h Solutions: ψ n -eigenfunctions; ε n -eigenvalues ⇒ © H.L. Tuller-2001 ) ) ψ 10 3.225 21 Free Particle • One dimensional OV = ψψ εψ 2 22 2 2 k m dx d −=−= h ikx Ae = ψ )( ),( tkxi Aetx ω ψ − = • Momentum kdx xi p x h h = ∂ ∂ = ∫ ψ ψ * m p m k 22 222 == h ε kp h= Crystal Momentum © H.L. Tuller-2001 3.225 22 Particle in Box 2 2 2 ; h ε ψ m kBeAe ikxikx =+= − Boundary Conditions: 0)()0( == d ψψ 0)0( =+= BA ψ BA −=⇒ 0)()( =−= − ikdikd eeAd ψ 02 =⇒ ASinkd d n k π = 3,2,1 = n       = d xn d π ψ sin 2 3,2,1= n V 0 = x dx = ∞ • • • © H.L. Tuller-2001 ∞ 11 3.225 23 Particle in Box zkykxkAzyx n 321 sinsinsin),,( ⋅⋅= ψ 2 3 2 2 2 1 2 kkkk ++= ; d n k i i π = 3,2,1 = i n m k nnn md n 2 )( 8 22 2 3 2 2 2 1 2 2 hh =++=ε n = Quantum numbers • Degeneracy First excited state 112, 211, 121 • Ground state E ; 1 321 === nnn not zero! © H.L. Tuller-2001 3.225 24 Consequence of Electrons as Waves on Free Electron Model Standing wave picture Traveling wave picture 0 0 L L n k e ee Lxx ikx Lxikikx π2 1 )()( )( = = = +Ψ=Ψ + Just having a boundary condition means that k and E are quasi-continuous, i.e. for large L, they appear continuous but are discrete © E. Fitzgerald-1999 L 12 13 3.225 25 Representation of E,k for 1-D Material m p m k E 22 222 == h E k ∆k=2π/L Quasi-continuous k m k E m k dk dE ∆=∆ = 2 2 h h states electrons E n E n-1 E n+1 m=+1/2,-1/2 All e- in box accounted for E F k F k F Total number of electrons=N=2*2k F *L/2π © E. Fitzgerald-1999 1 3.225 1 Representation of E,k for 1-D Material 2 1 2 221 )( 2 − === = E m k m LdE dk dk dN Eg Lk N F hh ππ π g(E)=density of states=number of electron states per energy per length • n, the electron density, the number of electrons per unit length is determined by the crystal structure and valence • n determines the energy and velocity of the highest occupied electron state at T=0 2 or 22 2 π ππ n k mE k L N n F F F ==== h m k dk dE mE k m k E 2 22 2 ; 2 h h h = == © E. Fitzgerald-1999 3.225 2 Representation of E,k for 2-D Material E(k x ,k y ) k x k y m kk E yx 2 )( 222 + = h © E. Fitzgerald-1999 3.225 3 Representation of E,k for 3-D Material k x k y k z Ε (k x ,k y ,k z ) 2 π /L m kkk E zyx 2 )( 2222 ++ = h Fermi Surface or Fermi Sphere k F m k v F F h = m k E F F 2 22 h = B F F k E T = 32 2 )( h mEm Eg π = ( 3 1 2 3 nk F π= © E. Fitzgerald-1999 ) 3.225 4 So how have material properties changed? • The Fermi velocity is much higher than kT even at T=0! Pauli Exclusion raises the energy of the electrons since only 2 e- allowed in each level • Only electrons near Fermi surface can interact, i.e. absorb energy and contribute to properties T F ~10 4 K (T room ~10 2 K), E F ~100E class , v F 2 ~100v class 2 © E. Fitzgerald-1999 2 3 3.225 5 Effect of Temperature (T>0): Coupled electronic-thermal properties in conductors • Electrons at the Fermi surface are able to increase energy: responsible for properties • Fermi-Dirac distribution • NOT Bolltzmann distribution, in which any number of particles can occupy each energy state/level Originates from: N possible configurations T=0 T>0 E F 1 1 )( + = − Tk EE b F e f If E-E F /k b T is large (i.e. far from E F ) than Tk EE b F ef )( −− = © E. Fitzgerald-1999 3.225 6 Fermi-Dirac Distribution: the Fermi Surface when T>0 µ~E F f(E) 1 T=0 T>0 0.5 k b T k b T E All these e- not perturbed by T f Boltz Boltzmann-like tail, for the larger E-E F values v v T U c       ∂ ∂ = Heat capacity of metal (which is ~ heat capacity of free e- in a metal): ( [ ( ( ) 2 ~~~ TkEgTkEgTkNEU bFbFb ⋅⋅⋅∆⋅∆ U=total energy of electrons in system TkEg T U c bF v v 2 )(2 ⋅⋅=       ∂ ∂ = Right dependence, very close to exact derivation © E. Fitzgerald-1999 ) ] ) 3.225 7 Electrons in a Periodic Potential • Rigorous path: HΨ=EΨ • We already know effect: DeBroglie and electron diffraction • Unit cells in crystal lattice are 10 -8 cm in size • Electron waves in solid are λ=h/p~10 -8 cm in size • Certain wavelengths of valence electrons will diffract! © E. Fitzgerald-1999 3.225 8 Diffraction Picture of the Origin of Band Gaps • Start with 1-D crystal again λ~a a 1-D θλ sin2 dn = d=a, sin θ =1 a n k k an π λ π λ = = = 2 2 Take lowest order, n=1, and consider an incident valence electron moving to the right x a i oo x a i ii e a k e a k π π ψ π ψ π − =−= == ; ; Reflected wave to left: Total wave for electrons with diffracted wavelengths: x a i x a oia ois oi π ψψψ π ψψψ ψψψ sin2 cos2 =−= =+= ±= a kkk oi π 2 =−=∆ © E. Fitzgerald-1999 4 . 22 2 π ππ n k mE k L N n F F F ==== h m k dk dE mE k m k E 2 22 2 ; 2 h h h = == © E. Fitzgerald-1999 3. 225 2 Representation of E,k for 2-D Material E(k x ,k y ) k x k y m kk E yx 2 )( 222 + = h © E. Fitzgerald-1999 3. 225 3 Representation of E,k for 3- D Material k x. 3, 2,1 = n       = d xn d π ψ sin 2 3, 2,1= n V 0 = x dx = ∞ • • • © H.L. Tuller-2001 ∞ 11 3. 225 23 Particle in Box zkykxkAzyx n 32 1 sinsinsin),,( ⋅⋅= ψ 2 3. Fitzgerald-1999 2 3 3.225 5 Effect of Temperature (T>0): Coupled electronic- thermal properties in conductors • Electrons at the Fermi surface are able to increase energy: responsible for properties •