NiO - hole doping and bandstructure of charge transfer insulator
Solids and Bandstructure
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Transcript of Solids and Bandstructure
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ECE 663-1, Fall ‘08
Solids and Bandstructure
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ECE 663-1, Fall ‘08
QM of solids
QM interference creates bandgaps and separatesmetals from insulators and semiconductors
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ECE 663-1, Fall ‘08
Recall numerical trick
xn-1 xn xn+1
n-1 n n+1
-t Un-1+2t -t
H = -t Un+2t -t
-t Un+1+2t -t
t = ħ2/2ma2
-t
-t
Periodic BCsH(1,N)=H(N,1)=-t
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ECE 663-1, Fall ‘08
Extend now to infinite chain
1-D Solid
-t -t
-t -t
-t -tH =
-t
Onsite energy (2t+U)-t: Coupling (off-diag. comp. of kinetic energy)
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ECE 663-1, Fall ‘08
Extend now to infinite chain
1-D Solid
-t -t
-t -t
-t -tH =
-t
Let’s now find the eigenvaluesof H for different matrix sizes N
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ECE 663-1, Fall ‘08
Eigenspectra
N=2 4 6 8 10 20 50 500
If we simply find eigenvalues of each NxN [H] and plot them in a sortedfashion, a band emerges!Note that it extends over a band-width of 4t (here t=1).The number of eigenvalues equals the size of [H] Note also that the energies bunch up near the edges, creating large DOS
there
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ECE 663-1, Fall ‘08
Eigenspectra
If we simply list the sorted eigenvalues vs their index, we getthe plot below showing a continuous band of energies.
How do we get a gap?
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ECE 663-1, Fall ‘08
Dimerized Chain
H =
-t1 -t2
-t2 -t1
-t1 -t2
-t2 -t1
-t2
-t1
-t1
Once again, let’s do this numerically for various sized H
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ECE 663-1, Fall ‘08
Eigenspectra
t1=1, t2=0.5
N=2 4 6 8 10 20 50 500
If we keep the t’s different, two bands and a bandgap emerges
Bandgap
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ECE 663-1, Fall ‘08
One way to create oscillations
+ + + +
Periodic nuclear potential(Kronig-Penney Model)
Simpler abstraction
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ECE 663-1, Fall ‘08
Solve numerically
Un=Ewell/2[sign(sin(n/(N/(2*pi*periods))))+1];
Like Ptcle in a boxbut does not vanishat ends
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ECE 663-1, Fall ‘08
Matlab code
• hbar=1.054e-34;m=9.1e-31;q=1.6e-19;ang=1e-10;• Ewell=10;• alpha0=sqrt(2*m*Ewell*q/hbar^2)*ang;• period=2*pi/alpha0;• periods=25;span=periods*period;• N=505;a=span/(N+0.3);• t0=hbar^2/(2*m*q*(a*ang)^2);• n=linspace(1,N,N);• Un=Ewell/2*(sign(sin(n/(N/(2*pi*periods))))+1);• H=diag(Un)+2*t0*eye(N)-t0*diag(ones(1,N-1),1)-t0*diag(ones(1,N-1),-1);• H(1,N)=-t0;H(N,1)=-t0;• [v,d]=eig(H);• [d,ind]=sort(real(diag(d)));v=v(:,ind);• % figure(1)• % plot(d/Ewell,'d','linewidth',3)• % grid on• % axis([1 80 0 3])• figure(2)• plot(n,Un);• %axis([0 500 -0.1 2])• • hold on• • for k=1:N• plot(n,real(v(:,k))+d(k)/Ewell,'k','linewidth',3);• hold on• axis([0 500 -0.1 3])• end
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ECE 663-1, Fall ‘08
Bloch’s theorem
(x) = eikxu(x)
u(x+a+b) = u(x)
Plane wave part
eikx
handles overall X-alPeriodicity
‘Atomic’ part u(x)handles local
bumpsand wiggles
(x+a+b) = eik(a+b)(x)
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ECE 663-1, Fall ‘08
Energy bands emerge
~0.35
~1-1.35
~1.7-2.7
E/Ewell
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ECE 663-1, Fall ‘08
Can do this analytically, if we can survive the algebra
N domains2N unknowns (A, B, C, Ds)
Usual procedureMatch , d/dx at each of the N-1 interfaces(x ∞) = 0
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ECE 663-1, Fall ‘08
Can’t we exploit periodicity?
Bloch’s Theorem
This means we can work over 1 period alone!
Need periodic BCs at edgesSolve transcendental equations graphically
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ECE 663-1, Fall ‘08
Allowed energies appear in bands !
Like earlier, but folded into -/(a+b) < k < /(a+b)
The graphical equation:Solutions subtended between black curve and red lines
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ECE 663-1, Fall ‘08
Number of states and Brillouin Zone
Only need points within BZ(outside, states repeatthemselves on the atomic grid)
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ECE 663-1, Fall ‘08
The overall solution looks like
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ECE 663-1, Fall ‘08
More accurately...
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ECE 663-1, Fall ‘08
Why do we get a gap?
E
k/a-/a
At the interface (BZ), we have two counter-propagating waves eikx,
with k = /a, that Bragg reflect and form standing waves
Its periodicallyextended partner
Let us start with a free electron in a periodic crystal,but ignore the atomic potentials for now
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ECE 663-1, Fall ‘08
Why do we get a gap?
E
k/a-/a
-+
Its periodicallyextended partner
+ ~ cos(x/a) peaks at atomic sites
- ~ sin(x/a) peaks in between
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ECE 663-1, Fall ‘08
Let’s now turn on the atomic potential
The + solution sees the atomic potential and increases its energy
The - solution does not see this potential (as it lies between atoms)
Thus their energies separate and a gap appears at the BZ
k/a-/a
+
-
|U0|
This happens only at the BZ where we have standing waves
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ECE 663-1, Fall ‘08
Nearly Free Electrons
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ECE 663-1, Fall ‘08
What is the real-space velocity?
Superposition of nearby Bloch waves
(x) ≈ Aei(kx-Et/ħ) + Aei[(k+k)x-(E+E)t/ħ]
≈ Aei(kx-Et/ħ)[1 + ei(kx-Et/ħ)]Fast varyingcomponents
Slowly varyingenvelope (‘beats’)
k
k+k
time
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ECE 663-1, Fall ‘08
Band velocity
(x) ≈ Aei(kx-Et/ħ)[1 + ei(kx-Et/ħ)]
Envelope (wavepacket) moves at speed v = E/ħk = 1/ħ(∂E/∂k)
i.e., Slope of E-k gives real-space velocity
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ECE 663-1, Fall ‘08
Band velocity
v = 1/ħ(∂E/∂k)
Slope of E-k gives real-space velocity
This explains band-gap too!
Two counterpropagating waves give zero net group velocity at BZ
Since zero velocity means flat-band, the
free electron parabola must distort at BZ
Flat bands
Flat bands
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ECE 663-1, Fall ‘08
Effective mass
v = 1/ħ(∂E/∂k), p = ħk
F = dp/dt = d(ħk)/dt
a = dv/dt = (dv/dk).(dk/dt) = 1/ħ2(∂2E/∂k2).F
1/m* = 1/ħ2(∂2E/∂k2)
Curvature of E-k gives m*
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ECE 663-1, Fall ‘08
Approximations to bandstructure
Properties important near band tops/bottoms
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ECE 663-1, Fall ‘08
What does Effective mass mean?
1/m* = 1/ħ2(∂2E/∂k2)
Recall this is not a free particle butone moving in a periodic potential.
But it looks like a free particle near the band-edges, albeit with an effective massthat parametrizes the difficulty faced bythe electron in running thro’ the potential
m* can be positive, negative, 0 or infinity!
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ECE 663-1, Fall ‘08
Band properties
Electronic wavefunctions overlapand their energies form bands
http://fermi.la.asu.edu/ccli/applets/kp/kp.html
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ECE 663-1, Fall ‘08
Els between bound and free
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ECE 663-1, Fall ‘08
Band properties
Electronic wavefunctions overlapand their energies form bands
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ECE 663-1, Fall ‘08
Band properties
Shallower potentials give bigger overlaps.
Greater overlap creates greater bonding-antibonding splitting of
each multiply degenerate level, creating wider bandwidths
Since shallower potentials allow electrons to escape easier, they correspond to smaller effective mass
Thus, effective mass ~ 1/bandwidth ~ 1/t (t: overlap)
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ECE 663-1, Fall ‘08
• Nearly free-electron model, Au, Ag, Al,...
Parabolic electron bands distort near BZto open bandgaps (slide 32)
• Tight-binding electrons, Fe, Co, Pd, Pt, ...
Localized atomic states spill over so that theirdiscrete energies expand into bands
(slides 9, 38)
Two opposite limits invoked to describe bands
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ECE 663-1, Fall ‘08
(For every positive J2 or J3 component, there is an equal
negative one!)
Electron and Hole fluxes
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ECE 663-1, Fall ‘08
Electron and Hole fluxes
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ECE 663-1, Fall ‘08
How does m* look?
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ECE 663-1, Fall ‘08
Xal structure in 1D
(K: Fourier transform of real-space)
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ECE 663-1, Fall ‘08
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ECE 663-1, Fall ‘08
Bandstructure along -X direction
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ECE 663-1, Fall ‘08
Bandstructure along -L direction
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ECE 663-1, Fall ‘08
3D Bandstructures
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ECE 663-1, Fall ‘08
GaAs Bandstructure
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ECE 663-1, Fall ‘08
Constant Energy Surfaces for conduction band
Tensor effective mass
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ECE 663-1, Fall ‘08
4-Valleys inside BZ of Ge
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ECE 663-1, Fall ‘08
Valence band surfaces
These are warped (derived from ‘p’ orbitals)
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ECE 663-1, Fall ‘08
In summary
• Solution of Schrodinger equation tractable for electrons in 1-D periodic potentials
• Electrons can only sit in specific energy bands. Effective mass and bandgap parametrize these states.
• Only a few bands (conduction and valence) contribute to conduction.
• Higher-d bands harder to visualize. Const energy ellipsoids help visualize where electrons sit