C. Buluty Topics on Semiconductor Physics In This Lecture: Electronic Bndstructure: Generl Info
C. Buluty Topics on Semiconductor Physics Electronic Bndstructure Acronyms FPLAPW: Full-potentil linerized ugmented plne wve ll-electron (FPLAPW) PAW vlence electron PAW: Projector ugmented wve LMTO: Linerized muffin tin orbitl EPM: Empiricl pseudopotentil method ETB: Empiricl tigh-binding b-initio semi-empiricl plnewve pseudopotentil (FP) LMTO EPM ETB k p
C. Buluty Topics on Semiconductor Physics Perfect Crystl Hmiltonin (cgs units) H ' ' = p j 1 e Pi 1 e ZiZi' e Z i m + r r + M + R R r R j j j, j ' j j ' i i i, i' i i' i, j j i over ll e s over ll nuclei e-nuclei 1 st Approximtion: core vs. vlence e s semicore Still Eq. Above Applies with: core e s+nucleus e s ion core vlence e s
C. Buluty Topics on Semiconductor Physics nd Approximtion: Born-Oppenheimer or dibtic pprox. ions re much hevier (> 1000 times) thn e s So, for e s: ions re essentilly sttionry (t eql. lttice sites {R j0 }) for ions: only time-verged dibtic electronic potentil is seen In other words, using the dibtic pproximtion, we seprte the (in principle non-seprble) perfect crystl Hmiltonin
C. Buluty Topics on Semiconductor Physics Under dibtic pproximtion ( ) (, ) (, ) H = H R + H r R + H r δ R ion i e j i0 e ion j i phonon spectrum electronic bnd structure e-phonon interction (resistnce, superconductivity ) Ref: Yu-Crdon
C. Buluty Topics on Semiconductor Physics Electronic Hmiltonin H e ' p j 1 e e Zi = + m r r r R j j j, j ' j j ' i, j j i0 over ll vlence e s >10 3 cm -3 3 rd Approximtion: Men-field Approximtion p H1 e = + V ( r ); V ( r + R) = V ( r ) m Density Functionl Theory V H +V x +V c A direct lttice vector
C. Buluty Topics on Semiconductor Physics Trnsltionl Symmetry & Brillouin Zones T f ( x) f ( x + R); T : Trnsltionl Opertor R R ikx Introduce function, ψ ( x) e u ( x), where u ( x + nr) = u ( x) with n Z k k k k ikr T ψ ( x) = ψ ( x + R) = e ψ ( x) R k k k [ H T ] Since, = 0, 1e R eigenvlues of T R Eigenfunctions of H cn be expressed lso s 1e eigenfunctions of T R π π n NB : k is only defined modulo, i.e., k nd k + re equivlent R R
C. Buluty Topics on Semiconductor Physics One-dimensionl Lttice x Rel spce Direct Lttice k Momentum spce Reciprocl lttice k=-π/ k=π/ 1 st BZ [Rel Spce] Primitive lttice vector: [Mom. Spce] Primitive lttice vector: π/
C. Buluty Topics on Semiconductor Physics Three-dimensionl Lttice [Rel spce] Primitive lttice vectors:,, 1 3 A generl direct lttice vector (DLV): R = n + n + n ; n Z [Mom. spce] Primitive lttice vectors: b, b, b 1 3 1 1 3 3 i j k where, bi π ( ) 1 3 Volume of the rel spce primitive cell A generl reciprocl lttice vector (RLV): G = n b + n b + n b ; n Z 1 1 3 3 i NB : b = πδ i j ij 1 st Brillouin zone is the Wigner-Seitz cell of the reciprocl lttice
C. Buluty Topics on Semiconductor Physics Direct vs. Reciprocl Lttices SC Primitive trnsltion vectors Direct Lttice ˆ 1 = x = y ˆ Reciprocl Lttice b 1 b 3 = zˆ b 3 π = x ˆ π = yˆ π = z ˆ lso SC lttice! 1 st BZ of SC lttice is gin cube
C. Buluty Topics on Semiconductor Physics BCC Direct Lttice ˆ ˆ 1 = ( y + zˆ x) ˆ ˆ = ( zˆ + x y) ˆ ˆ 3 = ( x + y zˆ ) is the side of the conventionl cube Reciprocl Lttice π b ˆ ˆ 1 = ( y + z) π b ˆ ˆ = ( z + x) π b ˆ ˆ 3 = ( x + y) forms n fcc lttice! 1 st BZ of bcc lttice Ref: Kittel
C. Buluty Topics on Semiconductor Physics FCC Direct Lttice ˆ ˆ 1 = ( y + z) ˆ ˆ = ( z + x) ˆ ˆ 3 = ( x + y) is the side of the conventionl cube Reciprocl Lttice π b ˆ ˆ 1 = ( y + zˆ x) π b ˆ ˆ = ( zˆ + x y) π b ˆ ˆ 3 = ( x + y zˆ ) forms n bcc lttice! 1 st BZ of fcc lttice Ref: Kittel
C. Buluty Topics on Semiconductor Physics fcc 1 st BZ Crdbord Model Truncted Octhedron Ref: Wikipedi Ref: Yu-Crdon
C. Buluty Topics on Semiconductor Physics More on Symmetry Symmetry of the direct lttice Symmetry of the reciprocl lttice The str of k-point All hve the sme energy eigenvlues Wvefunctions cn be expressed in form such tht they hve definite trnsformtion properties under symmetry opertions of the crystl Selection rules: certin mtrix elements of certin opertors vnish identiclly Forml nlysis is remedied by the use of Group theory
C. Buluty Topics on Semiconductor Physics Symmetry Points & Plotting the Bnd Structure EPM Bndstructure of Si Dimond BZ 5 0 Energy (ev) -5-10 -15 Γ X W L Γ K
C. Buluty Topics on Semiconductor Physics Bloch Functions vs Wnnier Functions ψ Felix Bloch provided the importnt theorem tht the solution of the Schroedinger eqution for periodic potentil must be of the specil form: ħk Cell-periodic functions ( ) ik r = e r u ( r ), with u ( r ) = u ( r + R), nk nk nk nk is the crystl momentum (more on this lter) Wnnier functions 1 ik R i n( r; Ri ) = e ψ ( r ), nk N ψ nk 1 ik R i ( r ) = e n( r; Ri ) N R i k Orthonormlity: ψ * ( r ) ψ ( r ) dr = δ δ nk n k nn kk ll spce Trnsformtion reltions
C. Buluty Topics on Semiconductor Physics Bloch vs. Wnnier Functions x Bloch functions re extended x Wnnier fn s re loclized round lttice sites R i Wnnier form is useful in describing impurities, excitons But note tht the Wnnier functions re not unique!
C. Buluty Topics on Semiconductor Physics Crystl Momentum ψ nk + = ( ) ik r r T e ψ ( r ) nk k determines the phse fctor by which BF is multiplied under trnsltion in rel spce k lbels different eigensttes together with the bnd index n k is determined up to reciprocl lttice vector; this rbitrriness cn be removed by restricting it to 1 st BZ A typicl conservtion lw in xtl: k + q = k + G Any rbitrriness in lbelling the BFs cn be bsorbed in these dditive RLVs w/o chnging the physics of the process Physiclly, the lttice supplies necessry recoil momentum so tht liner momentum is exctly conserved
C. Buluty Topics on Semiconductor Physics Be wre of the Complex Bndstructure nk e ik r u ( r ) Rel bndstructure of Si Wht if we llow k to become complex? Ref: Chng-Schulmn PRB 198
C. Buluty Topics on Semiconductor Physics Complex Bndstructure (cont d) Ref: Brnd et l SST 1987 Evnescent modes ply n importnt role in low-dimensionl structures They re required in mode mtching t the boundries etc