Electrons in a Weak Periodic Potential EMPTY LATTICE APPROXIMATION. First Brillouin Zone. H H + v. (x) 2M x. (x) E (k) E (k) v. vdx.

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1 Electrons in We Periodic Potentil EMPTY ATTICE APPROXIMATION V = Ψ (x) = Ψ M x Ψ (x) = e ix (x) = m First Brillouin Zone NEARY FREE EECTRON (NFE) APPROXIMATION Now we consider the we periodic potentil H H + v v is we potentil We v mens tht the inetic enery is much lrer thn the potentil enery () () Ψn, v Ψ, () () = + Ψ, Ψ, +,n E () E n ( ) E () E () v () () n, v, Ψn, v Ψ, = Ψ n, vψ ()* (), () () ix ix, v, Ψ, v Ψ, = e v(x)e dx = Which is independent of dx vdx

2 () () Ψn, v Ψ, () () = + Ψ, Ψ, +,n E () E n ( ) E () E () v () () Ψn, v Ψ, = E () +,n E () E n ( ) E () Thus this term does not depend on The potentil is periodic v(x + R) = v(x) v(x) = ix v e () () Ψn, v Ψ, () () = + Ψ, Ψ, +,n E () E n ( ) E () E () v Now let us o throuh the third term () () ()* () n, v, Ψn, v Ψ, = Ψ n, vψ, dx = = = n, = n i x ix ix v e e e dx = v(x) = ix v e i x i(+ )x v e e dx i x i(+ )x e e dx = δ n, v, = v if = +,+ () () Ψn, v Ψ, () () = + Ψ, Ψ, +,n E () E n ( ) E () E () v v = E () +,n n E () E () E ( + ) v = E () +,n n E () E () E ( + ) v = E () + E () E () E () v = E () + n E () n E () E () We cn usin ood pproximtion inore the interctions of C nd D nd bnds with the A bnd. If we only consider the nerest B bnd, then: v = E () + E () E () E () This mens tht the non deenerte perturbtion theory cnnot be loner vlid, nd one hs to use deenerte perturbtion theory insted. v E () E () () () E () E () Ner this point we hve

3 v = E () + E () E () E () Electrons in the we periodic potentil NEARY FREE EECTRON (NFE) APPROXIMATION () () if < E ( + ) > E () () E () < E () () () if > E () > E ( + ) () E () > E () non deenerte perturbtion theory cnnot be loner vlid! Hψ = ψ + U( r) m U( r + R) = U( r) Trnsltion ttice Vector Now suppose we re delin with one dimensionl problem d ψ + U(x) ψ = ψ m dx n( r + R) = n( r) U(x) i x = U e d i x + U e ψ = ψ m dx d i x + U e ψ = ψ m dx ψ (x) = C()e ix ix i( C()e + )x i( C( + )e = )x C( )e ψ (x) = = + + d ψ d ix C()e dx dx = = m m m C() e ix ix ix i(+ )x, U(x) ψ (x) = U e C()e = U C()e m U(x) ψ (x) = U C( )e, ix ix ix C() e + UC( )e = C()e, ix m ix ix ix C() e + UC()e = C()e, i x Now we multiply both sides of the bove eqution by e nd then te interls of both sides i x ix dxe e = δ i( )x ( ) e i x ix dxe e i( ) = = = n, = n i( n n ) i( )x i (n n ) e = e = e = = δ 3

4 C( ) U C( ) C( ) m + = Electrons in the we periodic potentil NEARY FREE EECTRON (NFE) APPROXIMATION C() + UC( ) = C() m C() + UC( ) = m ( λ ) = λ m C() + U C( ) = ( λ ) C() + U C( ) = ( λ ) ( λ ) C() + UC( ) = = C( ) + U C() = =- ( λ ) ( λ ) U = U = U C() + UC( ) = = C( ) + UC() = =- / λ U U = λ λ U U = λ λ + λ λ + λ U = ± + At the ede of the BZ: λ = λ λ = m = m ( / ) λ ( / ) = = = m m m ( / ) ( / ) 4

5 λ + λ λ λ U = ± + λ + λ = ± U Ep = U 5

6 ψ (x) = C()e IF we in eep only two terms due to the NFE pproximtion, then: ix ix ψ (x) = C()e + C( )e i( )x We now prove tht C() = + C(-) or - C(-) t the ede of the BZ: We hd shown tht ( λ ) C() + UC( ) = C( ) λ λ = ± U λ = = ± C() U U λ + λ = ± U λ = λ = λ ± U ix ψ (x) = C()e + C( )e i( )x ix ix i/x i/x ( ) ( ) = C() e ± e = C() e ± e cos( / x) ψ (x) = sin( / x) + cos( / x) ψ ψ (x) sin( / x) ψ Oy. In the NFE model we find tht the bnd structure behves similr to the free electron pproximtion fr from the ede of the Brillouin zone. Indeed the bnd structure of the NFE pproximtion devites from free electron model only in the djcent of the ede of the Brillouin zone. Therefore, it is importnt to lern how one cn plot the bnd structure of n empty thee dimensionl lttice. Bnd Structure plottin of empty 3 dimensionl lttices Before lernin how to plot bnd structure in n empty spce, it is useful to remind you tht how one cn drw vrious zones of Brillouin. 6

7 Oren Cohen, uy Brtl, Hrvoje Buljn, Tl Crmon, Json W. Fleischer, Mordechi Seev nd Demetrios N. Christodoulides Nture 433, 5-53(3 Februry 5), First (reen squre) nd second (four yellow trinles) Brillouin zones of twodimensionl squre lttice with the hih-symmetry points (, X, nd M) mred with white dots. b, Trnsmission spectrum of the first two bnds of two-dimensionl squre lttice with lttice period d. c, Dispersion curves between the symmetry points of the first two bnds. Netive curvture in these curves corresponds to norml diffrction reions. d, Dirm of the opticl induction technique used to obtin the twodimensionl, squre photonic lttice. The blue plnes with hevy blc rrows indicte the plne wves used to opticlly induce the lttice. The red rrow indictes the direction of probe bem enterin the lttice. The orne circle indictes the width of the probe bem. e, Dirm of our set-up for obtinin sptilly incoherent (qusitherml), qusi-monochromtic bem. Drw line connectin this oriin point to one of its nerest neihbors. This line is reciprocl lttice vector s it connects two points in the reciprocl lttice. The locus of points in reciprocl spce tht hve no Br Plnes between them nd the oriin defines the first Brillouin Zone. It is equivlent to the Winer-Seitz unit cell of the reciprocl lttice. In the picture below the first Zone is shded red. Then drw on perpendiculr bisector to the first line. This perpendiculr bisector is Br Plne. Now drw on the Br Plnes correspondin to the next nerest neihbours. Add the Br Plnes correspondin to the other nerest neihbours. The second Brillouin Zone is the reion of reciprocl spce in which point hs one Br Plne between it nd the oriin. This re is shded yellow in the picture below. Note tht the res of the first nd second Brillouin Zones re the sme. Two dimensionl Brillouin zone The first twenty Brillouin zones of D hexonl lttice. The (outside of) 5th Brillouin zone for the simple cubic. The (outside of) 8th Brillouin zone for the fce-centered cubic The (outside of) th Brillouin zone for the bodycentered cubic lttice. 7

8 Three dimensionl Brillouin zone First Brillouin Zone BCC First Brillouin Zone FCC z Bnd structure of simple cubic lttice lon Γ to X direction. Alon this direction we hve: = ( x,,) x X(,,) Γ(,,) Γ(,,) X(,,) y K = + Inside of the BZ Outside of the BZ / = / = / ( x x y y z z ) = m + = m ( ) ( ) ( ) ( ) Thus one would te some llowed shortest vectors nd try to plot their corresponded bnds 8

9 Γ(,,) X(,,) = ( x,,) x = m = = y ( x x y z ) = ( ) ( ) m + = m = x = y = z = z x = m = / = / (,,) / = / x =, y = z = = x + m = x + m 3 = x m = x + m = / = / (,,) = x m / = / x =, y = z = = x m / = / / = / = (,,) = (,,) = (,,) = (,, ) 3 = x m 4 = x + = 5 = 6 = 7 m 3 = x = x + m m = (;;; ) / = / = x m / = / = + + = = = m 8 x 9 4 = 5 = 6 = 7 4 = x + = 5 = 6 = 7 m This bnd is four fold deenerted / = / 3 / = / 4 = 5 = 6 = 7 9

10 = (; ; ; ) = + = = = m x = 3 = 4 = 5 8 = 9 = = = 3 = 4 = 5 8 = 9 = = 4 = 5 = 6 = = 5 = 6 = 7 / = / / = / / = / / = / Find the bnd structure of n empty body centered cubic (bcc) lon Γ() to H() for the first four llowed vectors tin bcc structure fctor into ccount Find the bnd structure of n empty simple cubic (SC) lon () direction ( n x n y n3z ) = ˆ + ˆ + ˆ Γ(,,) Miller indexes h,, l () H () K = + Inside of the BZ Outside of the BZ Γ(,,) H(,,) = (, y,) / = / = / ( x x y y z z ) = m + = m ( ) ( ) ( ) ( ) Thus one would te some llowed shortest vectors nd try to plot their corresponded bnds = = x ( x y y z ) = ( ) ( ) m + = m = x = y = z = z y = m

11 5 ( / ) m 5 ( / ) m 4 ( / ) m 4 ( / ) m 3 ( / ) m ( / ) m ( / ) m = y m 3 ( / ) m ( / ) m ( / ) m,...,5 = ± + y m = y m = / = / = ( ) = (,,, ) y = m = ± + m,...,5 y 5 ( / ) m 4 ( / ) m 3 ( / ) m ( / ) m ( / ) m 6,...,9 = + y m,...,5 = ± + y m = y m 5 ( / ) m 4 ( / ) m 3 ( / ) m ( / ) m ( / ) m,...,3 = ± + y + m 6,...,9 = + y m,...,5 = ± + y m = y m = / = / = (,,, ) = (,,, ) m 6,...,9 = + y + = ± + + m,...,3 y 4 = y m = ± + +,...,3 y m 6,...,9 = + y m,...,5 = ± + y m 5,...,8 = ± + y m 4 = y m = ± + +,...3 y m 6,...,9 = + y m,...,5 = ± + y m = y m = y m = / = / = ( ) = (,,, ) 4 = y m m 5,...,8 = ± + y

12 ,...,3 = ± + y m 9 = y + m 5,...,8 = ± + y m 4 = y m = ± + +,...,3 y m 6,...,9 = + y m,...,5 = ± + y m 9 = y + m 5,...,8 = ± + y m 4 = y m = ± + +,...,3 y m 6,...,9 = + y m,...,5 = ± + y m = ( ) = / = ( / ) m 9 = y + m = (,,, ) = / = ( / ) m,...,3 = ± + y m,...,3 = ± + y m 9 = y + m = ± + 4,...,7 y m,...,3 = ± + y m 9 = y + m 8,...,3 = ± + + y m = ± + 4,...,7 y m 5,...,8 = ± + y m 4 = y m = ± + +,...,3 y m 6,...,9 = + y m,...,5 = ± + y m 5,...,8 = ± + y m 4 = y m = ± + +,...,3 y m 6,...,9 = + y m,...,5 = ± + y m = / = ( / ) m = / = ( / ) m = (,,, ) 4,...,7 = ± + y m = (,,, ) 8,...,3 = ± + y + m Find the followin empty fcc lttice bnd structure. Miller indexes re even or odd. X =,, w =,, 3 3 =,, K =,,