• 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