Transport Property Modeling

Chapter 4 P-type PbSe with Na doping

4.3 Transport Property Modeling

77

Figure 4.5. zT as function of temperature for p-type PbSe.

Figure 4.6. a) Seebeck coefficient and b) Hall mobility as function of Hall carrier density for both p-type and n-type PbSe at 300 K. Data from literature are also included.

On the other side it is seen that the mobility of n-type and p-type samples are very different, with the p-types much lower in mobility than the n-types. The same effective masses for both, the elastic constant Cl are further found independent of the type of doping by measuring the speed of sound, which turned out to be identical (vl ≈ 3200 m/s, vt ≈ 1700 m/s). Thus the difference in mobility indicates a difference in the deformation potential coefficient Ξ, which for the L conduction band was found 25 eV, and for the L valence band it is determined to be 35 eV.

It is necessary to note that the effective mass determination from Seebeck coefficient is subject to a fair amount of uncertainty due to factors such as uncertainty in Seebeck measurement and Hall measurement, as well as the scattering mechanism(s) used. By comparing data from Caltech as well as most of published data on PbSe, as shown in Figure 4.6 a), the majority of results has suggested a same effective mass for the L conduction band and valence band. Bear in mind a possible small difference beyond the resolution of this method, the difference in mobility seen in Figure 4.6 b) could be possibly in part due to such difference as well. However, the difference seen is about a factor of 2 for the acoustic phonon scattering dominated regime, which would require a 30%

difference in effective mass to explain and would be certainly detectable in the Pisarenko relation.

Thus, it is possible that the conclusion about identical effective masses for the L conduction band and valence band is inaccurate, and the difference in Ξ is not as significant, but the general finding of Ξ  being  larger  in  the  L  valence  band  than  that  in  the  L  conduction  band  should  be  solid.

So far, it is also known that the bands at L are non-parabolic Kane bands, and the effective mass for the conduction band increases with temperature with dlnm^*/dlnT = 0.5, this should be the same for

p type

Caltech, Na (Se) Wang, Ag n type

Caltech, Br (Pb) Prokofeva, Cl

Androulakis, Cl Vinogradova, Na

a

p type

Caltech, Na (Se) Allgaier, Se Smirnov, Se Schliting, Se Wang, Ag n type

Caltech, Br (Pb) Chernik, Cl Prokofeva, Cl Androulakis, Cl Smirnov, Pb

b

79 the L valence band. On the temperature dependence of band gap, our most recent study¹⁴² (Figure 4.7) has provided the most accurate estimate of:

E_g(eV)=E_g,0+3!10^"4T Equation 4.2

The zero temperature band gap Eg,0 is 0.17 eV taken from low temperature measurement results.

The  Σ valence band is assumed to be parabolic, with effective mass independent of temperature and is isotropic for each valley.

Figure 4.7. Temperature dependent band gap of PbSe and PbS measured at Caltech and re- interpreted from literature. Image taken from Appl. Phys. Lett 103, 262109, 2013.

The expression of transport parameters in a multi-band system given by Putley¹⁴³ are used here:

n= n_i

i=Lc,Lv,!v

"

Equation 4.3

! = !i i=Lc,Lv,!v

"

Equation 4.4

R_H= R_H,i!_i²

i=Lc,Lv,!v

"

^!i

i=Lc,Lv,!v

"

#

$%% &

'((= eA_in_iµ_i²

i=Lc,Lv,!v

"

^niµi

i=Lc,Lv,!v

"

#

$%% &

'(( Equation 4.5

n_H =1 /eR_H Equation 4.6

µH =! /en_H =!R_H = e²A_in_iµi 2

i=Lc,

"

Lv,!v Equation 4.7

S= S_i!_i

i=Lc,Lv,!v

"

^!i

i=Lc,Lv,!v

"

Equation 4.8

!"#

!$#

!%#

&$ '

( )* ( )+ ( ), ( )- ( ).

!/0#

&$ '/

12 34 " 567 / 8

&$ 9/

( * . ( , ( ( - . ( : ( ( ; . (

T<!=#

( )* . ( )+ . ( ), . ( )- . ( ). .

E

g

86 3/ %> <! / 0 #

&$ '<?" 5 > / %@

&$ '<A6 $ B2 1

&$ '<C2 1 / B

&$ '/ <?" 5 > / %@

&$ '/ <A6 $ B2 1

&$ '/ <D/ E F " > G 2 E "

( * . ( , ( ( - . ( : ( ( ; . (

T<!=#

( )+ ( ( )+ . ( ), ( ( ), . ( )- (

E

g

86 3/ %> <! / 0 #

?" 5 > / %@

HI 9" J $ / 3 <* K : : A6 $ B2 1 <* K . + '" " G F " 1 <* K : :

! = !L+ L_i"iT

i=Lc,Lv,!v

"

^{+T Si2"i}

i=Lc,Lv,!v

"

^{# Si"i}

i=Lc,Lv,!v

"

$

%&& ' ())

2

"i i=Lc,Lv,!v

"

* + ,,

- . //

Equation 4.9

The results are shown in Figure 4.8. With the contribution from Σ valence band taken into account, the transport properties of p-type PbSe can be well modeled throughout the temperature range 300 – 800 K. The deformation potential coefficient of L valence band is set at 35 eV while to model the 800 K result it is allowed to increase by 10 %, just as for the L conduction band. Interestingly it doesn’t require additional adjustment, which one would expect if the inter-valley scattering gets intensified at higher temperatures. The parameters determined for the Σ valence band are: the effective mass for each valley mb* = mI* = 0.8 me; the deformation potential 28 eV; and its maximum is separated from that of L valence band by:

!E=0.32"2.2#10^"4T Equation 4.10

Figure 4.8. a) Seebeck coefficient and b) Hall mobility as function of Hall carrier density for p- type PbSe at different temperatures. Solid curves calculated with multi-band model. Literature

data included (not marked) in 300 K mobility.

To some extent, getting information about the Σ  band  from  modeling  is  not  convincing  because   the  result  is  subject  to  the  formulism  of  the  modeling  and  its  input.  The  bad  news  is  that  there   hasn’t   been   a   way   to   directly   observe   the   Σ   band.   For   example   for   the   energy   gap   measurement,  the  primary  direct  transitions  happen  at  much  larger  probabilities  since  they   do  not  require  phonon  participation.  They  would  dominate  the  absorption  spectra  making  the   L–Σ   indirect   transitions   hard   to   detect.   It   is   even   more   difficult   to   get   information   about   effective   masses   and   deformation   potential   experimentally.   To   some   theorists   such   a   two-­‐

band   picture   is   even   wrong   because   the   two   local   maximum   are   actually   from   the   same   eigenstate   thus   should   be   regarded   as   a   single   band   with   complex   energy   contours¹⁴⁴.  

300 K 450 K 600 K

800 K

Vinogradova, Na:PbSe

a

_{300 K}

450 K 600 K

800 K

b

81 Nonetheless,   such   a   simple   model   has   been   proven   to   explain   the   experiments   at   different   temperatures  very  well  and  successfully  guided  experiments  to  improve  the  performance  of   PbSe.  

The advantage of having the Σ band is directly shown in the Seebeck coefficient. Figure 4.9 compares the modeling result on Seebeck coefficient and mobility with and without contribution from Σ band. At 300 K, the influence from the Σ band is evident from modeling, but is arguably observable due to the uncertainty of Seebeck measurements. As temperature increases to 450 K, the difference becomes noticeable and the experiment results clearly suggested the importance of the Σ band. At higher temperatures the Seebeck coefficient is enhanced even at low carrier densities and at 800 K the enhancement is over 30%. On the other hand, the Hall mobility doesn’t change much by the presence of the Σ band, especially below 600 K where almost no difference is seen. The total Hall mobility is proportional to the sum of conductivity from each band weighed by mobility. Thus at lower temperatures due to large band offset σ_L >> σ_Σ, in addition µ_c,L >> µ_c,Σ because of the big difference in effective mass, as a result the overall Hall mobility is dominated by the L band.

Figure 4.9. a) Seebeck coefficient and b) Hall mobility as function of Hall carrier density modeled with and without the contribution from the Σ band.

Even though the transport properties are well explained by the current model and combination of parameters, it is also important to investigate how much the overall properties would change if the parameters for the Σ  band  were  chosen  differently.  This  result  is  shown  in  Figure 4.10.  Error   bars  represent  5%  uncertainty  in  property  measurements.  The  result  indicates:  changing  mb*,   Ξ,  ΔE0K   for   sigma   band   by   ±25%,   15%,   and   13%   leads   to   no   difference   in   the   mobility   modeling,  whereas  the  difference  seen  in  Pisarenko  relations  indicates  that  all  combinations  

300 K 450 K 600 K

800 K

dashed: L band only

300 K

450 K 600 K

800 K dashed: L band only

a b

of     {mb*,   Ξ,  ΔE0K}   within   the   range   of   {0.7±0.2   me,   30±4   eV,   0.34±0.02   eV}   would   give   reasonably  good  fit  to  the  experiment  result.  This  is  the  suggested  range  for  parameters  of  the   Σ  band.  

Figure 4.10. Influence of different parameters for the Σ  band  on  overall  transport  properties  at   300  K  and  600  K.  

Figure 4.11. zT  calculated  from  3-­‐band  model  for  n-­‐type  and  p-­‐type  PbSe  at  850  K  as   function  of  Hall  carrier  density.

Figure 4.11 plots the zT of both n-type and p-type PbSe calculated at 850 K, based on the 3-band model calculation. It generally predicted the same maximum zT for both types with experimentally suggested ones, although for some unknown reason the Hall carrier densities corresponding to the maximum zT are lower than the optimum carrier density determined experimentally, for both n-type

300 K 600 K

m* = 0.6 m* = 1.0

a b

300 K

600 K

m* = 0.6 m* = 1.0

300 K

600 K

ΞΣ = 24 eV ΞΣ = 32 eV

d

ΞΣ = 24 eV ΞΣ = 32 eV 300 K

600 K

c

300 K 600 K

ΔE = 0.28 eV ΔE = 0.36 eV

e

300 K

600 K

ΔE = 0.28 eV ΔE = 0.36 eV

f

zT

nH n-type PbSe

p-type PbSe

p-type PbSe only L band

83 and p-type. It also shows the zT calculated for the L valence band only, assuming there is no Σ band. Due to its larger deformation potential coefficient, the maximum zT if there is no Σ band would be much smaller around 0.6.