Chapter 7 Solid Solutions between Lead Chalcogenides
7.3 Alloy Scattering of Charge Carriers
133 The zT value of PbSe1-xSx samples at different temperatures is plotted in Figure 7.16. The carrier densities of samples shown are carefully controlled: nH,300 K = 3 × 1019 cm-3 (±10%) for alloys with x
< 0.5 and 5 × 1019 cm-3 (±10%) for alloys with x ≥ 0.5. The Seebeck coefficient values at 850 K for these samples are about the same at -190 µV/K (±10%). These will lead to zT values close to the optimized ones at 850 K for all compositions. At 850 K, zT is found to increase when substituting S in PbS with Se, while it is found to decrease when substituting Se in PbSe with S. Neither of these changes, however, is significant especially when compared with the averaged zT from the rule of mixing between PbSe and PbS (the dashed lines).
Figure 7.16. Measured zT versus composition at different temperatures for PbSe1-xSx. Error bars represent 10% uncertainty.
If we define an average lattice potential in a disordered system by,
U=xUA+(1!x)UB Equation 7.1
x being the concentration of atom A in the alloy, so for a lattice site occupied by atom A, the local potential fluctuation would be,
He!alloy,A=UA!U=(1!x)"U Equation 7.2
The scattering rate by A atoms amounting NA can be readily expressed as,
Mk,A2=nA k|He!alloy,A|k' 2
k'
"
=nA[
(1!x)#U]
2=x(1!x)2#U2 Equation 7.3Similarly the scattering rate by B atoms,
Mk,B2=nB k|He!alloy,B|k' 2
k'
"
=nB[
x#U]
2=(1!x)x2#U2 Equation 7.4So the total relaxation time given by Fermi’s golden rule is,
!alloy= !
2!
(
Mk,A2+ Mk,B2)
g(!)!1= !
2!
!2!3
"21/2m*3/2(kBT)1/2!1/2 1 x(1!x)#U2
Equation 7.5
The volume per atom Ω is introduced so the potential ΔU has the unit of energy (eV.)
Harrison219 went through a similar derivation of the transition probability and finally gets to a similar expression of the relaxation time that only differs in the pre-factors.
!alloy= 8!4
3 2!!CA(1"CA)U2md*3/2(kBT)1/2!"1/2 Equation 7.6 Harrison’s use of Ω came from the interatomic distance of III-V compounds, which he derived this equation for. As the relation between interatomic distance and lattice parameter differ with crystal structure, the pre-factor in Equation 7.6 is expected to be different when applied to other systems.
Fortunately, Harrison himself also mentioned219 the arbitrariness on the choice of this distance so that it is not necessary to think of one specific pre-factor should be the only rigorous result.
Difference in pre-factors is quite common in scattering related equations derived by different researchers. In specific for τalloy, after a short literature survey we found that the pre-factor used by
135 Makowski and Glicksman216 is roughly 2 times as large. The one used by Chattopadhyay220 is 0.5 times as large. Mahrotra used221 the pre-factor in Equation 7.5, which is very close to Equation 7.6 numerically. The same situation is seen in the expression of thermal conductivity governed by the Umklapp phonon scattering as well. Caution is needed when comparing results from literature.
For non-parabolic Kane bands, Equation 7.6 becomes:
!alloy= 8!4
3 2"!CA(1"CA)U2md*3/2(kBT)1/2(#+#2$)"1/2(1+2#$)"1 Equation 7.7 Although it also involves potential fluctuation just as deformation potential phonon scattering, the alloy scattering is limited to short range interaction only and phonon is not involved in this process.
With the relaxation time defined for alloy scattering, the mobility of any solid solution composition can be calculated. Beyond the dilute limit, the material properties of the alloys, such as the elastic constants Cl, deformation potential coefficient Ξ, and effective mass, will be different from the constituent compounds. How would these quantities change with solid solution composition is usually not well studied. In the study of Pb chalcogenide solid solutions, we assumed a linear average of each of these properties, and we were able to explain the experimental results very well.
The alloy scattering potential U, according to how these equations are derived, should be the offset of atomic potentials between solvent and solute atoms. How to acquire information about this quantity from well-determined physical parameters is not clear. The value of U should be, according to117 Brooks, related to the band gap difference between the constituent compounds. This is later suggested219, 222 to be inaccurate to calculate U whereas the difference of electron affinity might be a possible alternative. Neither of these could however explain the value of U found for n- type PbTe-PbSe or PbSe-PbS: Figure 7.17 compares these two quantities between different Pb chalcogenides. We see for both n-type (PbTe)1-x(PbSe)x and (PbSe)1-x(PbS)x, the band offset is about 0.1 eV at 0 K. while the alloy scattering potential is found much larger at around 1 eV for both cases.
Figure 7.17. Position of conduction and valence band of Pb chalcogenides relative to each other, plotted according to calculated band energy at 0 K reported by Wei and Zunger.
Even though the alloy scattering potential U is not exactly the band offset ΔE, it is still reasonable to expect U being proportional to ΔE. Then immediately from Figure 7.17 arises an interesting question, will the alloy scattering be negligible in p-type PbSe1-xSx, since the valence band offset223 is only 0.03 eV (at 0 K, at room temperature or above this result may be different)? Trying to probe the answer, we have made undoped PbSe1-xSx with 0.1% extra anions. It turns out all the samples on PbS-rich side are n-type, while only samples rich in PbSe are found p-type and these are listed in Table 7.2. Considering samples without bipolar conduction, we compared the measured mobility with the calculated mobility of p-type PbSe as well as (PbSe)1-x(PbS)x omitting alloy scattering (Ξ of p-type PbS was assigned the value 38 eV based on PbSe result), all with the same carrier density.
The result is listed in Table 7.3. Limited by the number of samples and the scatter of mobility result, no conclusion could be made yet at this stage. However, none of the measured mobilities from all three samples is significantly smaller than calculated mobility for PbSe1-xSx without alloy scattering, suggesting a good chance that the alloy scattering is negligibly weak in p-type PbSe1-xSx. We notice that Ioffe has made a very intuitive argument12 that since the valence band is formed primarily by anion atom orbits, substituting on the anion site will lead to more scattering in p-type solid solutions than n-type ones. Ioffe’s intuitive picture seems to be consistent with what had been found in PbTe1- xSex (since the band offset in the valence band happen to be larger than in conduction band), but most probably would fail when applied to the PbSe1-xSx case.
137 Table 7.3. Mobility for undoped p-type PbSe1-xSx compared with calculated mobility for p-type
PbSe, and PbSe1-xSx without alloy scattering contribution, assuming the same carrier density.
Sample nH (1018 cm-3) µH (cm2/Vs) µH PbSe µH PbSe1-xSx
p-50 2.4 487 784 520
p-60 0.8 593 694 467
p-80 1.2 780 739 606
In general, the value of alloy scattering potential U is of key importance to the magnitude of alloy scattering and the mobility reduction in solid solutions. Values of U, however, are reported only for a few systems220, 221, 224-227. These are shown in Table 7.4 along with some other physical property differences of the alloy components228, 229. The typical magnitude of U is found between 0.6 to 2 eV, which is much smaller than a typical effective deformation potential coefficient Ξ (8 to 35 eV).
Table 7.4. A comparison of alloy scattering potential U in a few solid solutions together with the difference between two constituents in electron affinity ΔX, band gap ΔEg, molar mass ΔM, and
lattice parameter Δα.
Alloy system ΔX (eV)
ΔEg
(eV) ΔM Δα (Å) U (eV) note
n-Al1-xGaxN 3.5 2.69 42.7 0.21 1.5 – 2.0 Δa compare c direction n-Al1-xGaxAs 0.43 1.72 42.7 0.01 1.1
n-Cd1-xZnxTe 0.8 0.88 88.2 0.37 0.8
n-InAs1-xPx 0.5 0.99 44 0.19 0.6
n-Si1-xGex 0.05 0.46 44.5 0.23 0.6 - 1.0 n-PbSe1-xTex 0.1 0.02 48.6 0.33 1.1
n-PbSe1-xSx 0.1 0.12 47.0 0.19 1.0