Polar Scattering from Optical Phonon

Chapter 6 N-type PbS doped with Cl

6.4 Polar Scattering from Optical Phonon

115

Figure 6.7. zT at different temperatures as function of Hall carrier density for n-type PbS. Error bars represent 10% uncertainty.

Figure 6.8. A comparison of a) Seebeck coefficient, b) Hall mobility, and c) zT of n-type PbSe and n-type PbS at 800 K. all lines calculated using single Kane band model.

If the lattice contains more than one species of atoms meaning an electronegativity difference between two neighboring atoms, carriers can also be scattered by the changing dipole moment due to optical vibration. The polar scattering differs from the deformation potential scattering form optical phonons in that the interaction is of an electro-static interaction nature rather than a perturbation in lattice potential. There are two important quantities for polar optical scattering. The first is the dimensionless polar coupling constant αpo, which governs the magnitude of interaction between carriers and polarization of optical phonons^{95, 181}:

!_po= e²

4"!( m^* 2!#_l⁾

1/2($^!1_"!$₀^!1) Equation 6.9

where ε₀, ε_∞ are the static and high frequency dielectric constant (with unit F/m, not relative values).

The second is the optical phonon temperature kBΘ = ħωl (close to the Debye temperature).

Generally materials with large effective mass and low optical phonon Debye temperature Θ are likely to have strong carrier-polarization interaction. Also materials with large ε0 also tend to have strong interaction. In III-V109, 182, 183 and II-VI^{184, 185} semiconductors, ε₀ is in fact not large but close to ε∞ so the last term in Equation 6.9 is also large, which makes the polar scattering by optical phonons important, even dominant in certain cases.

For general cases a universal τ can’t be defined due to the inelastic nature of the polar scattering, and the transport parameters are calculated using variational method. Detailed calculations have been given by researchers such as Howarth and Sondheimer¹⁰⁴, and Ehrenreich¹⁸⁶. For most good thermoelectric materials with high temperature application, when T > Θ, a relaxation time can be defined.

Since the rigorous derivation of transition rate following a textbook has not been carried out, it is better here to only list a few key points during the derivation while leaving all details to the reference in Ziman or Askerov. The calculation of interaction Hamiltonian is essentially to determine the electromagnetic field induced by propagation polarization waves that perturbs the electronic states. So the Hamiltonian has the form:

H_po=e! Equation 6.10

The field strength is determined from its gradient, which is linked to the polarization vector P(r).

4!P(r)=!" Equation 6.11

117 P(r) is further related to the displacement vector u(r):

P(r)=1 2

!!

2"V

!

"

# $

%& 1

#_'(1

#0

!

"

# $

%&

)

*

+ ,

- .

1/2

e_q,n

! "!!

(a_q,ne^iqr(a_q,n^* e^(iqr)

q

/

Equation 6.12

So the final form for the Hamiltonian is:

Hpo=2!e !! 2"V

!

"

# $

%& 1

#'

(1

#0

!

"

# $

%&

)

*+ ,

-.

1/2 1

qeq,n

! "!!

(aq,ne^iqr(aq,n

* e^(iqr)

q

/

=2!e !! 2"V

!

"

# $

%& 1

#'

(1

#0

!

"

# $

%&

)

*+ ,

-.

1/21

q(aq,ne^iqr(aq,n

* e^(iqr)

Equation 6.13 The dot product qŸe implies the interaction is only between electrons and longitudinal optical branch so the summation of different branch is omitted.

The rest of the derivation is largely similar to that for acoustic phonon deformation potential scattering.

Since thermoelectric materials are usually heavily doped, the screening of polarity vibration by free electrons must also be considered. Ravich’s derivation takes into account this together with the band nonparabolicity in lead chalcogenides, which gives Equation 6.1, notice that in this equation all parameters are independent measurable. The static dielectric constant ε0 of a conductor is probably difficult to measure directly, but can be derived using the ratio of the longitudinal optical phonon frequency over that of the transverse optical phonon at Brillion Zone center k = 0, via the Lyddane- Sachs-Teller relation.

Equation 6.1 has been used by other researchers when studying the scattering mechanism in PbTe106, 187, 188 and Bi2Te3.¹⁸⁹ It should also be a reasonable expression for such scattering mechanism in other systems with Kane band behavior, such as CoSb3 at high temperature (Θ for CoSb3 is ~300 K).

Qualitatively from Equation 6.1:

!_po!m^*"1/2T^"1/2"^1/2 Equation 6.14

Compared to Equation 3.47 for acoustic phonon scattering, relaxation time governed by polar scattering from optical phonons has a weaker dependence on temperature and effective mass. It will increase with carrier energy ε, instead of decrease as for the case of acoustic phonon scattering. This

implies it would be less important for most thermoelectric materials above room temperature. In more general case, the exponent r in τpo ~ ε^r is plotted against Θ/T by Ehrenreich, r changes greatly¹⁸⁶ with T and there is a singularity around T = Θ /2.

Table 6.3. The polar coupling constant for a few compound semiconductors

ε0 ε∞ Θ αpo comment

PbTe 414 33 160 0.29

m^* use 300 K value from Seebeck data

PbSe 204 23 190 0.36

PbS 169 17 300 0.45

CoSb₃ 42 cal^b 32 cal 25 exp

306 0.07 For p type, m^* use 0.15 m_e

Bi₂Te₃

290 (//c) 75 ( c)

85 (//c) 50 ( c)

164

0.13 (//c) 0.07 ( c)

m^* from Seebeck data from CRC

handbook

GaAs 13 11 344 0.08

InSb 17 16 203 0.01

ZnO 8 4 660 1.02

CdTe 10 7 158 0.41

Lead chalcogenides are unique compounds in term of their extraordinarily large static dielectric constants. For instance for PbTe, ε0 around 400 has been reported by different groups from different measurement techniques^{190, 191}. In contrast, ε₀ for most III-V and II-V compounds are usually¹²⁴ from 10 to 20. Considering the low Debye temperatures in lead chalcogenides, large polar coupling constants α_po would be expected in these compounds. In the table below α_po is compared for a few semiconductors. Lead chalcogenides are seen to have larger αpo compared to other typical thermoelectric materials as well as III-V compounds, whereas some II-V compounds show the largest αpo, which stems from their small yet different dielectric constants.

From the result shown in Table 6.3 the polar scattering is important around room temperature in lightly doped lead chalcogenides. Its magnitude in other compounds would be less as can be judged from the values of αpo. For most heavily doped thermoelectric materials neglecting the contribution of polar scattering from optical phonons should not lead to drastic error in modeling and the acoustic phonon scattering assumption can be considered valid.

! ! !

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