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• Quantum wells and superlattices
y
Quantum wells
4
2. Quantum Well States (QWS) and Quantum SizeEffects
Qualitative explanation…
yikxikn
yx eezzyx )(),,( !="
2
kk
2),,(
2
y
2
x
2
22 ++=
D
nkknE yx
!
zk
xk
yk
Electronic structure
in parabolic subbands
Dx
z
d
•2D conducting system (x-y plane), size -quantization in z,
d ! λF
•Metals:
•Semiconductors: λF ! 10− 100 nm
λF ! 0.5 nm
•Film deposition technology (e.g. molecular-beam epitaxy) ! quantum wells with c-Si, GaAs
26
• Quantum wells and superlattices
27
•2D electron density of states (per unit area):
gs
DOS2D(ε) = gsgv(2m!
xm!y)1/2
2π!2
gv
spin degeneracy, 2, Si, GaAS
valley degeneracy (degeneracy of states at bottom of conduction band); 1 for GaAs (single valley); 2 or 4 for Si (indirect gap; depending on number of valleys involved)
GaAS:
Si{100}:
direct gap semiconductor
indirect gap semiconductor, more complicated
m!t ! 0.19 me transverse
effective masses fo x and y motion
m!x = m!
y = m!l = 0.067me
m!l ! 0.92 me longitudinal
3D
• Quantum wells and superlattices
28
•Calculate λF
Electron density (areal) na
⇒ kF =2π
λF=
(4πna
gsgs
)1/2
GaAS {100}:
Si {100}:
λF = 40 nm
na = DOS2D(εF )εF = gsgv(m!
xm!y)1/2εF
2π!2
λF = 35− 110 nm (depending on na)
• Quantum wells and superlattices
•Square-well potential
•InfiniteV (z) =
∞ , z > d, z < 0
0 , 0 < z < d
1D particle in a box
Ψz(0) = Ψz(d) = 0
Ψz = A sin kzz , kz =nπ
d
•FiniteV0
d
Maximum number of bound states:
nmax = 1 + Int
[(2m!
z|V0|d2
!2π2
)1/2]
29
• Quantum wells and superlattices
30
ε = εn(z) +!2
2m!x
k2x +
!2
2m!y
k2y
quantized energy levels
kinetic energy for 2D free e- motion
m!x,m!
y effective masses for (x,y)
•Electron energy:
εn(z) =!2π2
2m!zd
2n2
effective mass for motion in z direction
n = 1, 2, 3...
• GaAs,
•
ε1(z) ! 50 meV (d=10 nm)
εn ∝ n2
• Quantum wells and superlattices
infinite well
31
4
2. Quantum Well States (QWS) and Quantum Size
Effects
Qualitative explanation…
yikxik
nyx eezzyx )(),,( !="
2
kk
2),,(
2
y
2
x
2
22 ++=
D
nkknE yx
!
zk
xk
yk
Electronic structure
in parabolic subbands
D
• No allowed state until ε = ε1(z)zero point energy
finite well
• DOS constant until subband starts to be populatedε = ε2(z)infinite well
n = 1, 2, 3...εn subbands
• Quantum wells and superlattices
32
•Sandwitch thin layer GaAs between thick layers AlxGa1-xAs
(d ! 5 nm, εg ! 1.5 e V )
•Similar unit-cell parameters! possible epitaxial growth one over the other.
•The band gap of AlGaAs (large gap) straddles that of GaAs (smaller gap).
•Electrons in conduction band of GaAs layer confined in z direction, by forbidden energy gap of AlGaAs in both sides
conduction
valencebands
(εg ! 1.42 + 1.26x eV ! 2 eV for x ! 0.45)
holes
electrons
• Quantum wells and superlattices
33
•Triangular potential wellSi-SiO2 junction in MOS
GaAS-n-AlGaAs
V (z) =
∞ , z < 0
Ez , z > 0
Triangular wells - infinite
Triangular potential wells are quite common in real
devices.
Examples: Si MOSFETs; single-interface
GaAs/Al0.3Ga0.7As heterojunctions.
Idealization:
εn(z) ! (32π!e)2/3
(E2
2m!z
)1/3
(n +34)2/3 ; n = 0, 1, 2, ...
Quantized energy levels
Linear variation of confining electrostatic potential
• Quantum wells and superlattices
•Optical absorption:
bulk
• Bulk ! absorption involves e- states at conduction band minimum and valence band maximum.
• QW! quantized levels
εn ∝n2
d2square well
εn ∝ E2/3triangular well
Lowest allowed level, n=1 in square well and n=0 in triangular well
• The threshold energy for optical absorption increases ("100 meV) !sharp exciton (bound e-h pair) peak
• GaAs, d=21 nm ! max. number bound states nmax=4 ! 4 peaksd=14 nm ! max. number bound states nmax=3 ! 3 peaks
•Absorption profile: exciton peaks occurring at each discontinuity in DOS (peak when a new subband starts to be populated)
34
absorptionthreshold
•GaAs, "F=10 nm
• Quantum wells and superlattices
35
• Thinest QW !GaAs d=5nm
Fine structure: splitting in exciton peak
one bound state n=, in valence band split into two
• Quantum wells and superlattices
36
εconf ∝1
m!
heavy holes
light holes
j=3/2 states split into two
BulkQW
heavy!flat band The light is the heavy now
Metals, example Pb islands
Covered area
Courtesy: Rodolfo Miranda
Number of islandsBuilding island heights
histograms
R. Otero, A. L. Vázquez de Parga and R. Miranda PRB 66, 115401 (2002)
• Quantum wells and superlattices
37
• Quantum wells and superlattices
38
Superlattices• Molecular beam-epitaxy ! ordered arrays of heterostructures or homojunctions• d1 material (1) + d2 material (2) ! new period d= d1 + d2
Periodic array of QW’s
• Quantum wells and superlattices
39
• Periodic array of semiconductor QW’s.
•d1 : thickness of well
•d2 : thickness of barrier
1- Isolated QW’s, d2 >> d1 # "F Multiple quantum wells (no tunnelling of electrons through barriers)
2- Interacting QW’s, d1 # d2
Superlattice (very different properties from bulk material)
• Two possibilities:
• Quantum wells and superlattices
gaps at
k =nπ
d
• Electronic structure of semiconductor superlattices
• Tight-binding model ! Bloch function, array N QW’s
Ψk(z) = N−1/2∑
n
eikndψn(z − nd) Enveloppe function of nth well centerd at z=nd, overlap with the two neighbours. Equivalent to atomic orbitals or Wannier functions
• Analogy with normal crystals ! energy of electron motion in superlattice
ε(k) = εi − αi − 2βi cos kdi-th level of a well, i=1,2,3...
cos-like dependence, minibandself-energyintegral, >0
overlapintegral, >0 40
energy-width
• Quantum wells and superlattices
41
• Quantum wells and superlattices
42
Density of statesg(ε) =
Nm!
!2π2cos−1
[−(ε− εi + α)
2βi
]
, |ε− εi + αi| < 2|βi|
Smearing corresponding to band width
• Quantum wells and superlattices
Gaps at ∓π
d
vg(k)
• Forbidden gaps at k =nπ
d(d >> a)
! smaller than BZ !no periodic extended state along z
• Bloch oscillations: E d.c. electric field ! oscillatory electron velocity, a.c current in superlattice.
ε(k) = ε0 − ε1 cos kd ! group velocity of electron:
vg(k) =1!
∂ε
∂k=
ε1! sin kd
43
• Quantum wells and superlattices
44
k = ±π/d
Bloch oscillatrions observed if T < τ (scattering relaxation time)
, not possible to observe in bulk crystals (d=a) ; but superlattice
• Consider ε(k) = ε0 − ε1 cos kd
! group velocity of electron:
Periodic, , Bragg scatteringvg = 0 at |k| =π
d
• Apply
! Period for motion in reciprocal space between
T =2π
d
!eE
(previous page figure)
vg(k) =1!
∂ε
∂k=
ε1d
! sin kd
d ! 10− 100a
!E ⇒ !k̇ = −eE ⇒ k = ko − eEt/!
• Quantum wells and superlattices
45
• Stark ladder : applied E field prevents minibands !no Bloch oscillations (no wave function overlap)
field inducedpotential displacement
If eEd ! 4|βi|
miniband width
• Quantum wells and superlattices• Effects on phonon propagation !acoustic phonons folded back at k = ±π/d
Brillouin new boundaries (d: new period superlattice)
- Optical phonons ! confined to one or other layer of material: confined modes
• Doping superlattice !periodic array of homojunctions or p-n junctions (nipi structure)
intrinsic layerbetween n and p- Band-edge modulations determined by doping level:
$ dopant concentration ! $ Eo modulation energy (next page figure)
! “indirect” minimum gap in real space
46
(Phonon dispersion in a crystal)
εeffg = εg − E0
• Quantum wells and superlatticesEo ∝ Nd , doping concentration
spatial variation of n- and p- doping
spatial variation (in z) of conduction and valence bands
nipi semimetallic
47
Nipi structure
! C/A ! εε0d/2
semiconductordielectric constant
Charge per unit area, Q/A = Nddn = Nadp widths of p-type and n-type regions
C = Q/V ⇒ E0 "e2Nddnd
2εε0=
e2Ndd2
4εε0(dn = dp = d/2)
electrostatic potential difference=E0/e
• Quantum wells and superlattices
Nipi structure
• To estimate the modulation consider the homojunction as a capacitor:
(assume the same for - and + regions)
48
• Quantum wells and superlattices
49
• Very high doping ! modulation very pronounced, nipi semimetal
• Band modulation and can be controlled varying charge densities in n- and p- layers ! injection of excess e- or holes electrically or optically reduce E0 by neutralization of charge donors or acceptors.
εeffg
e- -hole pairs after absorption of photons
Light intensity $ ! $εeffg
Nipi structure