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

The strange world of Quantum Mechanics “It is impossible for both The Ultimate Answer and the Ultimate Question to be known about in the same universe, as they will cancel each other out and take the Universe with them, to be replaced by something even more bizarre..”

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

3D analogue of drum modes 2p 3p 4p 3d 4d 5d

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 (n+l) rule (6Z) 1/3 = n+l E ~ Z2 Coulomb repulsion ‘screens’ electrons

Energy Level diagram ∑ Si: 14 electrons 2 (2l+1) = 2n2 electrons = 2, 8, 18, 32, 50,... 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 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 E.4pr2 = Qenc/e0 = q∫s(r’)dr’/e0 r Gauss’ Law V(r) = -∫Edr’ = -q∫dr’/4pe0r’2∫s(r’’)dr’’ = q∫s(r’)dr’/4pe0r +∫dr’s(r’)q/4pe0r’ ∞ r r r' ∞ r r ∞ U = qV

Calculating the potential Getting Uee(r) from n(r)  Pure Coulomb electrostatics “Hartree” approximation 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!

% Code to calculate Radial functions and energies of silicon clear all close 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=(5e-10Np);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); % Only count upto 3p2 n=n0*(13/14); % Count 13 out of 14 electrons Unew=(q/(4*pi*epsil))*((sum(n./R)-cumsum(n./R))+(cumsum(n)./R)); change=sum(abs(Unew-Uscf))/Np,Uscf=Unew; End [E1s E2s E3s E2p E3p] Quantum transport: Atom to Transistor (Supriyo Datta)

% -------------------------------------------------------------------------------- % 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

Screening gives Aufbau principle -Zq VCoul ≈ + l(l+1)ħ2/2mr2 4pe0r[1 + r/l]2 Screening length long for weak Z (Thomas Fermi) l = 1.65a0/Z1/3 Madelung, Klechkowski, n+l rule To get larger (n+l) need higher Z Large n+l has smaller l, more screening, higher energy

Screening gives Aufbau principle http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/orbdep.html#c1

Screening in solids (Thomas Fermi) Molecule Solids -Zq VCoul ≈ -Zq e-r/l 4pe0r[1 + r/l]2 4pe0r l = 1.65a0/Z1/3 l = √(e/q2D) 2V = -qn/e 2dV = -qdn/e n = n(qV) dn = (∂n/∂E)qdV = qDdV 2dV = -q2DdV/e = -dV/l2

Variational treatment of He <H> = <YHe|-ħ2/2m(12+22 ) –2q2/4pe0r1 –2q2/4pe0r2 +q2/4pe0r12|YHe>/<YHe|YHe> YH1s(r1) = 1/√(pa03) e-Zr1/a0 YHe(r1,r2) = Z3/(pa03) e-Z|r1+r2|a0 Treat Z as a variational parameter

Estimate for screening <H> = <YHe|-ħ2/2m(12+22 ) –2q2/4pe0r1 –2q2/4pe0r2 +q2/4pe0r12|YHe>/<YHe|YHe> = q2/4pe0a0[Z2–4Z+5Z/8] Zmin = 27/16 ≈ 1.7 (15% screening/electron) <H> = 77.5eV Exact = 25 + 54.4 = 79.4eV

Total Energy For He, added energy of 1st electron in presence of 2nd, and 2nd in isolation Can’t just some SCF energies of 1s, 2s, 3s, 2p etc. Must subtract ½ the interaction energy (ie, only include it for 1 member of each pair)

A nonlocal correction (‘Exchange’) F = fnlm(r)Ylm(q,j)/r -ħ2/2m.d2f(r)/dr2 + [U(r) + l(l+1)ħ2/2mr2 + UH(r)]f(r) ∫UF(r,r’)f(r’)dr’ = Ef(r)

Exchange-Correlation Beyond Exchange terms “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