What does an atom look like? http://www.biologie.uni-hamburg.de/b-online/e16/hydrogen.htm

Recap hydrogen atom (r)/r x Ylm(q,f) En Fnlm ~ Rnl a 2p orbital (r)/r x Ylm(q,f) En Fnlm ~ Rnl We studied the hydrogen atom spectrum in detail. Let’s look at other elements of the periodic table

What changes for other atoms? Multiple electrons, single nucleus Consider Helium. 2 electrons, 2 protons. Simple expectation: En = –Z2/n2 x 13.6 eV Lowest energy -54.4 eV for Z=2, n=1 n=1 n=2 Vacuum Ionization Energy He+ + hn  He++ + e- 54.4 eV He+ n=1 n=2 Vacuum Ionization Energy He + hn  He+ + e- 25 eV He We’re off by 30 eV ! But look at He+ instead

What changes for other atoms? So el-el interaction raises energy levels by 30 eV This is the HARD term to include in our Hamiltonian n=1 n=2 Vacuum Ionization Energy He+ + hn  He++ + e- 54.4 eV He+ n=1 n=2 Vacuum Ionization Energy He + hn  He+ + e- 25 eV He We’re off by 30 eV ! But look at He+ instead

Electron-Electron interaction … … r3 Blue el feels an interaction from (N-1) other electrons with a potential that’s not spherically symmetric U(r2,r3,…,rN) This means we can’t separate variables any more !!! r2 rN r1

Simplify … … But we must simplify things, shouldn’t we? r3 So we smear out all other electrons into a spherically symmetric U(r) that must be calculated self-consistently, but scaled for (Z-1) electrons r2 rN r1

Simplify Self-consistent Field theory or Mean-Field theory r1

Calculating the charge Just like the hydrogen atom with the extra potential Uee(r) F = fnlm(r)Ylm(q,j)/r -ħ2/2m.d2f/dr2 + [U(r) + l(l+1)ħ2/2mr2 + Uee(r)]f = Ef So given Uee, we can find f(r) and thus charge distribution n(r) = |fnlm(r)|2 (over occupied states only) Σ nlm For every atom, need to know electronic configuration to see which states are occupied. eg. Si 1s22s22p63s23p2

Energy Level diagram ∑ l = 0, 1, 2,..., n-1 m = -l, -(l-1), -(l-2),... (l-1), l (2l+1) multiplets s = -½, ½ (2 of them) So one orbit can hold 2 (2l+1) = 2n2 electrons = 2, 8, 18, 32, 50,... l = 0 l = n-1 ∑

Energy Level diagram ∑ Si: 14 electrons 2 in first orbit (n=1, l=0)  1s 8 in second orbit (n=2, l=0,1)  2s, 2p 4 in third orbit (n=3, l=0,1,2)  3s, 3p, 3d 2 (2l+1) = 2n2 electrons = 2, 8, 18, 32, 50,... l = 0 l = n-1 ∑

Energy Level diagram Si: 14 electrons 2 in first orbit (n=1, l=0)  1s 8 in second orbit (n=2, l=0,1)  2s, 2p 4 in third orbit (n=3, l=0,1,2)  3s, 3p, 3d Si 1s22s22p63s23p2

Calculating the potential Getting Uee(r) from n(r)  Pure Coulomb electrostatics “Hartree” approximation (ECE309) s(r) = qn(r) Uee(r) = (q2/4pe0)[∫s(r’)dr’/r’ + ∫s(r’)dr’/r](Z-1)/Z Self-consistency: Uee  f(r)  n(r)  s(r)  Uee(r) etc This works quite well for the periodic table!

Need additional terms  let’s look at them now % -------------------------------------------------------------------------------- % MATLAB code used to generate the figures in the book: % % "Quantum Transport: Atom to Transistor," by Supriyo Datta % published by Cambridge University Press, May 2005 % (ISBN-10: 0521631459 | ISBN-13: 9780521631457) % http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521631459 % THIS FILE FOR: Chapter 3, Figure 3.1.4 % Copyright (c) 2005 Supriyo Datta clear all %Constants (all MKS, except energy which is in eV) hbar=1.055e-34;m=9.110e-31;epsil=8.854e-12;q=1.602e-19; %Lattice Np=200;a=(10e-10/Np);R=a*[1:1:Np];t0=(hbar^2)/(2*m*(a^2))/q; %Hamiltonian,H = Kinetic,T + Potential,U + Ul + Uscf T=(2*t0*diag(ones(1,Np)))-(t0*diag(ones(1,Np-1),1))-(t0*diag(ones(1,Np-1),-1)); UN=(-q*14/(4*pi*epsil))./R;% Z=14 for silicon l=1;Ul=(l*(l+1)*hbar*hbar/(2*m*q))./(R.*R); Uscf=zeros(1,Np);change=1; while change>0.1 [V,D]=eig(T+diag(UN+Uscf));D=diag(D);[DD,ind]=sort(D); E1s=D(ind(1));psi=V(:,ind(1));P1s=psi.*conj(psi);P1s=P1s'; E2s=D(ind(2));psi=V(:,ind(2));P2s=psi.*conj(psi);P2s=P2s'; E3s=D(ind(3));psi=V(:,ind(3));P3s=psi.*conj(psi);P3s=P3s'; [V,D]=eig(T+diag(UN+Ul+Uscf));D=diag(D);[DD,ind]=sort(D); E2p=D(ind(1));psi=V(:,ind(1));P2p=psi.*conj(psi);P2p=P2p'; E3p=D(ind(2));psi=V(:,ind(2));P3p=psi.*conj(psi);P3p=P3p'; n0=(2*(P1s+P2s+P3s))+(6*P2p)+(2*P3p); n=n0*(13/14); Unew=(q/(4*pi*epsil))*((sum(n./R)-cumsum(n./R))+(cumsum(n)./R)); %Uex=(-q/(4*pi*epsil))*((n./(4*pi*a*R.*R)).^(1/3));%Unew=Unew+Uex; change=sum(abs(Unew-Uscf))/Np,Uscf=Unew; end [E1s E2s E2p E3s E3p] %analytical solution for 1s hydrogen a0=4*pi*epsil*hbar*hbar/(m*q*q); P0=(4*a/(a0^3))*R.*R.*exp(-2*R./a0); hold on h=plot(R,P1s,'b'); h=plot(R,P0,'bx'); h=plot(R,P3p,'bo'); set(h,'linewidth',[2.0]) set(gca,'Fontsize',[25]) xlabel(' R ( m ) --->'); ylabel(' Probability ---> '); axis([0 5e-10 0 0.08]); grid on Si1s Si3p H1s Need additional terms  let’s look at them now

Exchange-Correlation Beyond Coulomb terms within SCF “Hartree-Fock”, “Density Functional Theory” Uee(r) = (q2/4pe0)[∫s(r’)dr’/r’ + ∫s(r’)dr’/r](Z-1)/Z Uxc(r)  -(q2/4pe0r0), 4pr03/3 = 1/n Self-consistency: Uee  f(r)  n(r)  s(r)  Uee(r) etc Correlation “Hole” (radius r0) El Nearby els Coulomb Blockade