Vibrational Properties - Physics of...

Vibrational PropertiesVibrational Properties

From: P.Y. Yu and M. Cardona, Fundamental of Semiconductors, Springer Verlag,

N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt-Saunders Int. Editors.

The physical properties of a crystal can be roughly devided into those that aredetermined by the electrons and those that relate to the mouvement of the atoms abouttheir equilibrium positions.

This division is qualitatively easy to justify: the motions of atomic nuclei are, due to theirhigh mass, much slower than the motions of the electrons.

If the atoms are displaced from their equilibrium positions, then the electrons adopt anew distribution (with higher total energy). If the initial positions of the nuclei arerestored, then the energy is recovered.

Vibrational PropertiesVibrational Properties

∑∑∑∑∑≠≠

−+

−−

−++=

jiji jiij ji

j

jjjj jj

jj

j j

j

i i

i

rr

e

Rr

eZ

RR

eZZ

M

P

m

pH

,

2

,

2

, '

222

2

1

2

1

22'

'

'

The Hamiltonian describing a perfect crystal can be written as

ri denotes the position of the ith electronRj is the position of the jth nucleusZj is the atomic number of the nucleuspi and Pj are the momentum operators of the electrons and nuclei.

Obviously the many-particle Hamiltonian can not be solved without a large number ofsimplifications:

1)separate the electrons into two groups: valence and core electrons. The coreelectrons are assumed to move rigidly with the nuclei.

2) adiabatic approximation (Born-Oppenheimer). Since the frequencies of the ionicvibrations are typically < 1013 s-1 while the frequencies of the electronic motion are ≈ 1015

s-1 the electrons can respond to ionic motion almost instantaneously or to the electronsthe ions are stationary. On the other hand, ions can not follow the motion of theelectrons and they see a time-averaged adiabatic electronic potential.

BornBorn--Oppenheimer approximationOppenheimer approximation

),(),()( 0 jiionejiejions RrHRrHRHH δ−++=

Hion is the Hamiltonian describing the ionic motion under the influence of ionic potentialsplus the time-averaged adiabatic electronic potentials.

He is the Hamiltonian for the electrons with the ions frozen in their equilibrium positions(Rj0).

He-ion describes the change in the electronic energy as a result of the displacements(δRj) of the ions from their equilibrium positions. He-ion is known as the electron-phononinteraction.

By using the Born-Oppenheimer approximation the Hamiltonian can be expressed inthree terms:

Hamiltonian forHamiltonian for the the ions ions

We are now interested in the vibrational properties of semiconductors then we are goingto find the solutions of the Hion (equation motion for the ions).

),...,(2

),...,( 1

2

1 nej j

jnion RRE

M

PRRH +=∑

Ee (R1,... ,Rn) represents the total energy of the valence electrons with ions heldstationary at the positions R1,... ,Rn.

With help of supercomputer is possible to calculate Ee and then solve ab initio forthe motion of the ions.

Since much of the work on lattice dynamics was carried out before supercomputerwere available it relied on a phenomenological approach.

Harmonic approximationHarmonic approximation

In this approach an equation motion for the ions is obtained by expanding Hion as afunction of their displacement (δRj) from their equilibrium positions Rj0:

),....,(),......,( 010'

010 nnoion RRHRRHH δδ+=

H0 is the Hamiltonian of the crystal with all nuclei at their equilibrium positionsH’ is the change in Hion due to displacements of the nuclei by small amount (δR) fromthe equilibrium positions.

We expand Hion around R10, ..,Rn0. Since Rj0 are the equilibrium positions of the ions,the first-order terms in δRj vanish. In addition, when δRj are identical, the crystal isuniformly displaced and not distorted. Thus, the lowest order terms in the expansionrelevant to the vibration of the crystal are the second order terms in δ(Rj-Rk).

If we keep only the terms quadratic in H’, the motion of the nuclei is described by acollection of simple harmonic oscillators (harmonic approximation).

ForceForce constants constants

ukl is the displacement of the ion k in the unit cell l from equilibrium and the matrixΦ(kl,k’l’) contains the force constants describing the interaction between the ionsdenoted (kl) and (k’l’).

The force constants contain two parts:the first part is the direct ion-ion interaction due to their Coulomb repulsion, the second isan indirect interaction mediated by the valence electrons. Infact, the motion of one ioncauses the electron distribution to change and this rearrangement of the electronsproduces a force on the neighboring ions.

The determination of the lattice dynamics described by the above Hamiltonian can becarried out in two steps.

First we treat the Hamiltonian classically and solve the equation motion. In this classicalapproximation the H describes the energy of a collection of particles executing small-amplitude oscillations (normal modes).

''''

2

)'',(2

1

2

1)(' lk

lkkl

klkkl ulkkludt

duMuH ∑ Φ+

=

Within this approximation H’ can be expressed as:

Normal modesNormal modes and and phonons phonons

In the second step we quantized the energies of the normal modes. Each quantum oflattice vibration is called phonon.The procedures for determining the quantized energy levels of one-dimensional simpleharmonic oscillators can be found in most of quantum mechanics texbooks .

Instead now we want to determine the vibrations of the normal modes described by theH’.Since the matrix Φ(kl,k’l’) possesses translational symmetry, we expect that the atomicdisplacements (ukl) which diagonalize the H’(ukl) can be expressed in terms of planewaves similar to the Bloch functions for electrons in a crystal:

)exp(),( 0 tRqiuqu lkkl ωω −=rrrr

ukl is the displacement of the kth ion in the lth unit cell and it can be related to uk0 of acorresponding ion in the unit cell located at the origin by a Bloch wave and where q andω are, respectively the wave vector and the frequency of the wave.

There is however an important difference between electrons and phonon Bloch waves.Electrons can be anywhere in the crystal, ion positions are descrete and follow theperiodicy of the crystal.The phonon frequencies in the first BZ are identical in the others(phonon dispersion curves).

Dynamical matrixDynamical matrix

[ ]

[ ] 0)(D

as written becan eq. the

exp))('0,()(D

as of ansformFourier tr modified-mass a gIntroducin

)exp()0',(

)exp()0',(

0'k'

'2

kk'

2/1'kk'

0'''

02

0'''

=−

−Φ=

Φ

−Φ=

−Φ=

∑

∑

∑

∑

−

kkk

mm

kk

kmlk

kk

kmlk

kok

uq

RqiMMkmkq

uRqikkmuM

uRqikkmuM

δω

ω

r

rrr

rrr

r&&

By substituting ukl in to the Hamiltonian H’ we obtain the equation for uk0(Newton law):

Dkk’(q) is know as the dynamical matrix. The vibrational frequencies ω(q) are found bysolving the secular equation:

The vibrational amplitudes uk0 are obtained by substituting the solutions of the secularequation into the above equation.The degrees of freedom or the number of indipendent waves is equal to three timesthe number of atoms in the crystal.

0)(D '2

kk' =− kkq δωr

1D 1D monoatomicmonoatomic linear chain linear chain

)2( 11 nananana uuufuM −+= −+&&

qM

faq

=)(ω

. . . . .(n-2)a (n-1)a na (n+1)a (n+2)a

We consider a set of ions of mass M distributed along a line a points separated by adistance a

We see solutions u(na,t) proportional to eiq(na-wt) where q=n 2π/a

For k<<π/a

( )( )

2sin2

cos12)(

cos12

)2(2

2

qa

M

f

M

qafq

qafM

eefM iqaiqa

=−

=

−−=−

−−−=− −

ω

ω

ω

Diatomic Linear ChainDiatomic Linear Chain

We are going to discuss the case of a diatomic linear chain (1D). This case haslittle in common with a real crystal but it is frequently discussed because ot itssimple mathemathics.We consider a linear chain in which all nearest neighbors are connnected by idealsprings with force constant f. The unit cell contains two atoms of masses M1 andM2.

k and k’ are 1,2 and we are in 1D and m can take only the values n+1,n,n-1. One thusobtains the following equations:

)2(

)2(

11222

12111

+

−

++−=

++−=

nnn

nnnn

uuufuM

uuufuM

&&

&&


By inserting this plane wave solution in the system of equations we have:

( )

( ) 02

11

0112

22

11

1

21

21

21

12

1

=

−++−

=+−

− −

uM

fue

MMf

ueMM

fuM

f

qa

qa

ω

ω

2,1 )(exp)(),( =−= atqaniququn ωω αα

We use the plane wave solution:


The dynamical matrix Dkk’(q) is therefore

2 )1(

)1( 2

221

211

+−

+−

−

−

M

fe

MM

f

eMM

f

M

f

iqa

iqa

Setting the determinant of the system equal to zero leads to the dispersion relation (ωversus q):

2/1

2

21

2

2121

2

2sin

411

11

−

+±

+=

qa

MMMMf

MMfω

This dispersion relation is clearly periodic in q with a period given by:

aq

qa ππ

2 ,

2==

The solution must satisfy the condition:

m2rG and )()(

have weefurthermor and )()(

πωω

ωω

=⋅=−

+=rrrr

rrr

qq

Gqq


Taking these facts together we can represent ω(q) in the region

20

Gq ≤≤

For example we show the twodispersion relations (branches) for amass ratio ofM1/M2=5The branch that goes to zero at smallq is know as a the acoustic branch(for small q the ω ∝q).The upper branch is called opticalbranch.1/µ= 1/M1+1/M2

For the diatomic linear chain we onlyallowed displacements ot the atomsalong the direction of the chain(longitudinal waves). For a 3D crystalthere are also transverse waves

acoustic branch

opticalbranch

Wave solutionsWave solutions at q=0 at q=0

)(exp),(

)(exp),(

22

11

tqaniAqu

tqaniAqu

ωω

ωω

−=

−=For q<<π/aω∝qa A1=A2 (acoustic mode)ω=(2f/µ)1/2 A1=-A2 (optical mode)

The first belongs to the acoustic mode and describes a motion in which the two ions inthe cell move in phase with one another.The minus belongs to the high-frequency optical mode, and describes a motion in whichthe two atoms in the cell are 1800 out of phase.

Wave solutionsWave solutions at q= at q=ππ/a/a

ω=(2f/M1)1/2 A1=A2

ω=(2f/M2)1/2 A1=-A2

When q=π/a, the motion changes by 1800 from cell to cell. The ions in each cell are inphase for the acoustic and 1800 out of phase for the optical.

3D3D phonon dispersion curves for phonon dispersion curves for Si Si

For Si ( 2 atoms per primitive cell) in 3D there are six phonon branches (threeacoustic branches and three optical branches). Along high-symmetry directionsthe branches can be classified as transverse and longitudinal branches.A clear separation of the vibrational modes into longitudinal and transverse is onlypossible in certain symmetry directions of the crystal.

3D3D phonon dispersion curves for GaAs phonon dispersion curves for GaAs

Fig. 3.2

For GaAs and other zincblende-type semiconductors, the LO phonon has higherenergy than TO phonons. At exactly at the zone center, the TO and LO phonons inthe zinc-blend crystal are not degenerate because of the cubic symmetry of the zinc-blend structure. The difference from Si comes into the partially ionic nature of thebonding in zinc-blend crystals.

TA

LA

L0

TO

The The DebyeDebye Model Model

Debye derived a simple model of the heat capacity due to phonons. This involvesa parameter called the Debye temperature, θD.

In this model the solid is approximated by an isotropic, continuous medium. Thevibrations possess a relation

ω=vΦk where vΦ is the speed of sound.

Since g(k) is proportional to k2

G(ω)=Hω2

The Debye model constrains that the total number of modes is 3Nl where N isthe number of primitive unit cells and l is the number of atoms in the unit cell.

ωωω

ωωωωω

ωω

ω

ω

ωωω

ω

ω

ω

ω

ω

de

Nld

egU

Nlg

NlH

HdNl

d

b

d

b

d

Tkd

Tk

d

d

d

∫∫

∫

⋅⋅=⋅⋅=

=

=

==

0/

22

0/

22

2

2

0

2

191)()(

energy total the9

)(

9

33

hhhh


The heat capacity is:

Making the substitutions:

For low T θD/T becomes large the upper of the integral becomes infinity and integral is anumber than

The Debye T3 lawFor high T θD/T becomes small , x also is small; ex is about (1+x)

ωω

ωω

ω

ω

ω

de

e

TkV

Nl

dT

dE

VC

d

b

b

Tk

Tk

bd∫ −

==0

2/

/42

3 )1(

91)(

h

hh

( )dx

e

exTk

kxandkT

x

x

Db

bbdD

D

∫−

=

==/

02

43

1V

Nl9C

T/ /θ

θ

ωωθ hh

3

4

V

Nl

5

12C

=

Db

Tk

θπ

b

T

Db k

V

Nldxx

Tk

D

3V

Nl9C

/

0

2

3

=

= ∫

θ

θ

The The DebyeDebye model model

The Debye theory does not describe the experimental data because: The approximation ω=vΦk is not true

As consequence g(ω) prop ω2 is not correct

K must integrated in the Brillouin zone The theory works quite well at T much lower than θd where only phonon

with low energy phonons will be present.

The number of phonons excited at a T is given by:

dxe

xTkTk

NlU

dxe

xTkNlN

Tkx

de

NldfgU

de

NldfgN

T

xb

bd

T

xb

dph

b

Tkd

Tkd

ph

d

d

d

b

d

b

∫

∫

∫∫

∫∫

−

=

−

=

=

−=⋅=

−==

/

0

33

3

/

0

3

3

0/3

0/3

)1(

9

)1(

9

/

)1(

19)()(

energy U totalthe

)1(

19)()(

ϑ

ϑ

ω

ω

ω

ω

ω

ω

ω

ωωω

ωωωω

ωω

ωωω

h

h

h

hhh

h


– At low T T<< θd upper limit of the integral is infinity

– At high T θd is small ex is about (1+x)

TkN

U

Tdxe

xTkNlN

bph

xb

dph

∝

∝−

= ∫∞

3

0

3

3 )1(

9

hω

sbph

b

T

b

d

s

T

b

d

T

b

dph

kN

U

TNlkdxxTkNl

U

TT

NldxxTkNl

dxxTkNl

N

s

ss

θ

ω

θωωθ

θθ

3

2

39

2

999

/

0

23

3

/

0

3

3

/

0

3

3

=

=

=

∝=

=

∫

∫∫

h

hh

The The DebyeDebye model model

The Debye model provides a useful approximation which enables thephonon distribution to the heat capacity to be derived at low T.

The relevant energy scale in the Debye model is given by the Debyetemperature θd.

Furthermore the Debye phonon frequency provides a reasonable estimateof the most energetic phonon in a material therefore is a guide to thehardness (i.e. an hard material will have stiffer bonds and therefore moreenergetic phonons)

Li θd = 400KC 1860 KGe 360 KSi 625 K

Phonon SpectroscopyPhonon Spectroscopy

The momentum and energy conservation laws that govern the interaction of light withthe crystal can be used to advantage for the experimental determination of phonondispersion curves.

The two most commonly employed electromagnetic techniques, each with its limitations,are the inelastic scattering of X rays and of visible light. Inelastic scattering of thermalneutrons and atoms can also be used for phonon determination.

X-Ray photons can be inelastically scattered with emission and/or absorption of one ormore phonons. However the change in energy is extremely difficult to measure:⇒X ray energy is several 103eV while phonon energy is several 10-3eV.

Usually one can only measure the total scattered radiation of all frequencies, as afunction of scattering angle, and determining the angles away from those satisfying theBragg condition.

On the contrary, if photons of visible light are scattered with the emission or absorptionof photons, the energy shifts are still very small but they can be measured.

Phonon SpectroscopyPhonon Spectroscopy

The inelastic scattering of light in and around the visible region is known either asRaman scattering, or, when the interaction is with the acoustic waves, as Brillouinscattering.

The scattering is produced by the polarization of the atoms in the radiation field andthe subsequent emission of the dipole radiation.In the frequency range of visible light the maximum wave vector transfer is:

i.e., ∝1/1000 of a reciprocal lattice vector (G≈Å-1). Thus the Raman scattering canonly be used to study lattice vibrations near to the center of the Brillouin zone(q=0).

This is not the case for inelastic scattering of X-rays (E=104 eV, λ=1 Å) were themomentum transfert is ∝1/λ≈G= Å-1 (reciprocal vector).

130 102

42 −− Α≈= &xk

λπ

ScatteringScattering Amplitudes Amplitudes

The solution of the equations of motion of the lattice have the the form of plane waves.In analogy to the wave-particle dualism of quantum mechanics, one might ask whetherlattice waves can also be interpreted as particles.Any such particle aspect would manifest itself, for example, in the interactions withother particles, i.e. electrons, atoms, neutrons, and photons (light).

In the diffraction experiment from a crystal one can make use of X-ray, electrons,neutrons and atoms. The scattering amplitude scattered from a rigid lattice ρ(r)integrating over the entire scattering region is:

To simplify the matemathics we consider a primitive lattice and assume the atoms tobe point-like scattering centers situated at the time-dependent positions rn(t). Thus wewrite:

∫ −−∝ dreretA rKKitiB

o )( 0)()( ρω

∑∝

−∝

−

∑

n

trKKitiB

nn

neeA

trrtr

)()( 00

))(())((

ω

δρ

InelasticInelastic scattering scattering

We separate the rn(t) into the lattice vector rn and a displacement from the latticesite sn(t)

with this one obtains

for small displacement s(t) we can make the expansion

we obtain the elastic scattering espression:

Thus are scattered waves for which ω differs from that of the primary waves (ω0) byexactly the frequency of the lattice waves.

)()( tsrtr nnn +=

ti

n

tsKKirKKi eeeA nn 000 )()()( ω∑ −−∝

[ ][ ]tqrqi

n

ti

nn

rKKi

n

n

uets

etsKKeA

)(

0)(

)(

)()(1 00

ω

ω

−⋅±

−

∝

⋅−+∝∑planewaves

[ ] tqi

n

rqKKi euKKeA n ))((0

)( 00 )( ωω ±±− ⋅−∝ ∑

MomentumMomentum and and energy conservation energy conservation

The scattered waves must obey a further conditions relating to the wave vector:

by multipying both equation by

one sees that the second of these classical equations can be interpreted quantummechanically as the conservation of energy.

The plus sign corresponds to the excitation of a lattice wave by the primary wave; theminus sign apply to the process in which a lattice wave loses energy to the primarywave (this occurs only when the lattice wave has sufficient energy).

The first condition can be interpreted as the conservation of quasimomentum if it isassumed that the q is the quasimomentum of the lattice wave.In this sense we can regard the lattice waves as a particle (phonon) with aquasimomentum (defined within an arbitrary reciprocal lattice vector).

)(0

0

q

GqKK

ωωω ±=

=±−

0)(

0

0

0

=±−

=−±−

q

GqKK

ωωω hhh

hhhh

h

Raman SpectroscopyRaman Spectroscopy

The interaction with visible light with solid occurs via the polarizability of the valenceelectrons. The electric field E0 of the incident light wave induces, via thesusceptibility tensor χ a polarization P. P=ε0χE0 and E0=E0cosωot

The susceptibility tensor depends from the collective excitations that they can beregarded as a perturbation. By just considering the first two terms:χ=χ0+(δ χ /δX)X If the susceptibility itself oscillates with the frequency ω(q) of an elementary excitation(e.g. phonon) then the oscillation of polarization induced by the incident radiation (ω0) ismodulated by ω(q) .

The periodic modulation of P leads tothe emission of a wave (scattered)


Elastic process

Stoke and anti-stoke lines.

The modulation in the polarization leads to contributions in the scattered light at ω0

±ω(q). The plus and minus corresponds to scattered light that have, respectively,absorbed the energy of, and lost energy to, the relevant elementary excitation [ω(q),q].

Raman spectrumRaman spectrum of of GaAs GaAs at 5K at 5K

υ=λ-1

Elastic contribution


• Excitation source: High power laser (Neodymium, Krypton, Argon) hv=2-4eV (v=1015

Hz)

• Spectrometer with very high resolution (meV).

• Detectors: highly sensitive photomultipliers.

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