ELECTRICAL PROPERTIES AND BAND STRUCTURE IN Pb1-xSnxTe

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ELECTRICAL PROPERTIES AND BAND STRUCTURE IN Pb1-xSnxTe

H. Albany, M. Ocio

To cite this version:

H. Albany, M. Ocio. ELECTRICAL PROPERTIES AND BAND STRUCTURE IN Pb1-xSnxTe.

Journal de Physique Colloques, 1968, 29 (C4), pp.C4-125-C4-128. �10.1051/jphyscol:1968417�. �jpa- 00213621�

JOURNAL DE PHYSIQUE Colloque C 4, supplkment au no 11-12, Tome 29, Novembre-Dkcembre 1968, page C 4 - ¹²⁵

ELECTRICAL PROPERTIES AND BAND STRUCTURE IN Pb,-,Sn,Te

H. J. ALBANY and M. OCIO

Service dYElectronique Physique, Centre d'Etudes nuclkaires de Saclay, 91 - Gif-sur-Yvette, France

Abstract. - Strong evidence for the existence of two valence bands in Pb0.7Sn0.3Te and Pbo.sSno.sTe is inferred from the observation of a kink in the variation of the Hall ratio R ~ O O ~ K I R ~ ~ O O K , as a function of hole concentration. This existence is also supported by the fact that the variation of the Seebeck coefficient as a function of hole concentration exhibits a minimum.

Moreover, non parabolic bands are expected from our Hall mobility data, which indicate - ^p-1

instead of p~ - ^p-113 for a parabolic band.

According to the existence of two valence bands, the thermal energy gap behavior in Pb-rich alloys (x < 0.3) is discussed. From electrical results in the intrinsic regions, the thermal energy gaps for different compositions ranging from PbTe to x = 0.3 were found to be of the same order.

Taking into account the interpretation for PbTe, and in correlation with the observed behavior of the energy gap E,, the energy separation E, between the two valence bands Vl and VZ is expected to increase with increasing Sn content and to decrease with increasing temperature. Data on Sn-rich alloys would lead to a better appreciation of a possible shift of Vz relatively to VI and to the conduction band with increasing Sn content.

Rbsumb. - I1 existe de fortes presomptions pour l'existence de deux bandes de valence dans Pb0,7Sno,sTe et Pbo,sSno,sTe d'aprks l'observation d'un maximum dans la variation du rapport de Hall R300 o K / R I o o ~ ~ en fonction de la concentration de trous. Cette existence est confirmee par le fait que la variation du coefficient de Seebeck en fonction de la concentration de trous possede un minimum. De plus, des bandes non paraboliques sont presumks d'aprks nos mesures de mobilite de Hall qui indiquent : p~ - ^p-1 au lieu de p~ - ^p-113 pour une bande parabolique.

Nous discutons le comportement de la bande interdite thermique dans des alliages riches en plomb ( x < 0,3) dans l'hypothkse de deux bandes de valence. D'apres les resultats des mesures electriques dans les regions intrinseques, les bandes interdites thermiques pour diffkrentes compo- sitions allant de PbTe a x = 0,3 se sont rkvelkes Stre du mgme ordre. En tenant compte de l'inter- prttation donnee pour PbTe et en correlation avec le comportement observe de la bande inter- dite Eg, la separation energktique Ev entre les deux bandes de valence ^{V I} et ^V2 devrait augmenter avec le pourcentage d'etain et decroitre quand la tempkrature augmente. Des mesures sur des alliages riches en etain conduiraient a une meilleure comprehension d'un deplacement possible de VZ par rap- port & V I et la bande de conduction quand la quantitk d'etain croit,

We report in this-paper some recent results on the electrical properties of Pbl-,Sn,Te. These data concern particularly the variation of Hall and Seebeck coefficients, and electrical resistivity as a function of hole concentration in two alloys Pbo.,Sno.,Te and Pbo.,Sno.,Te. The existence of two valence bands is inferred from the observed electrical behavior. Pre- vious thermal energy gap results in Pb-rich alloys are discussed, and comparison with reported data on SnTe and PbTe yields information about the band structure in these solutions. An increase of the valence band energy separation E,, in Pb-rich alloys is suggested in correlation with the recent results [I] [2] on the beha- vior of the energy gap E, in these alloys.

Na-doped polycrystals of Pbo.,Sno.,Te and

Pbo.,Sn0.,Te, with room-temperature hole concentra- tions between 10'' and loz1 ~ r n - ~ were investigated in the temperature range SO0-600 OK. For all the sam- ples, the Hall coefficient R increases with increasing temperature. This behavior was observed in a great number of extrinsic semiconductors. It was discussed by Aukerman and Willardson [3] in the case of classi- cal statistics, and was extended by Allgaier [4] to higher carrier concentrations. The possibility of complex single band to explain the temperature depen- dence of R could not be ruled out. However, strong evidence for the existence of two valence bands could be obtained from the observation of a sharp kink [4] [5] in the carrier concentration dependence of the Hall ration RH/R,, where RH and RL are the Hall

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1968417

C 4 - 126 H. 3. ALBANY AND M. OCIO

FIG. 1. - Variation of the Hall ratio R ~ O O ~IC/RIOO as a function of apparent hole concentration p?oo 1 in Pbo,-~Sno.~Te and Pb~.sSno.sTe.

ratio R,,, oK/R,oo as a function of apparent hole concentrationp~oo measured at 100 OK

C 100" K O Pb0,7 Te

300%

- _L9

- ^a

- ^V)

- 2 -3

- ^+..5

3%

-

I .... ^{1 ,} . . ^{1 .... 1} . . . ¹ ....

coefficients measured respectively at two temperatures TH and TL (TH > TL). The Hall ratio first increases then decreases with increasing carrier concentration, the break in the variation occurring when the low- temperature Fermi-level enters the second band.

Such kink is observed in the two alloys investigated.

We have plotted in figure 1, the variation of the Hall

The kink occurs at PToo ⁼ 7 x loi9 and 1.7 x loZ0 cm-3 for Pb,.,Sn,.,Te and Pbo.,Sno+,Te respectively.

No convincing determination of the energy separation E., between the two valence bands could be carried out, for little is known about the band parameters.

Furthermore, the bands are believed to be non para- bolic, as suggested by the hole concentration depen- dence of the Hall mobility ^pH, which varies as p - I

(Fig. 2), unstead of p-lI3 expected for a parabolic band. This problem requires further arialysis, likely on the basis of complex band structure models (Cohen, Kane).

However, the existence of two valence bands is supported by another result. The variation of the Seebeck coefficient a as a function of hole concentra- tion exhibits a minimum for both alloys (fig. 3). The value of cc at the minimum increases with increasing temperature, while the shape of the minimum beco- mes less pronounced, with a tendency to flattening [6].

Observation of such minima a t room-temperature

1oi9

C I

lozo loa 101" loa 1P

p;

o p * ( ~ r n - ~ )

2 .

112e9.

4 0 0

13

1

FIG. 2. - Variation of the Hall mobility PH (PH = Ra) at 100° and 300 OKas a function of apparent hole concentrationp*

measured respectively at 100° and 300 OK.

FIG. 3. -Variation of the Seebeck coefficient u at 200°

and 300 OK as a function of apparent hole concentration p*

measured respectively at 200° and 300 OK.

o . b s a

/'

-

P ⁹

,o

0 Pbm Sn,,Te A Pb, Snb5Te

" " " I ^{1 1 1} 1 1 1 1 1 1

was reported by Efimova and Kolomoets [7]. A two band model have been considered to account for such behavior [6-81. Similar results to the above data

lo2

ELECTRICAL PROPERTIES AND BAND STRUCTURE IN Pb~-~sn,Te C 4 - 127

have been obtained on SnTe [8-111: existence of a Hall ratio kink at aboutp&, = 2.3 x lo2' cm-3 ; a p - l hole concentration dependence of the Hall mobility;

presence of a minimum on the x(p*) curve about the p* value of the Hall ratio kink. Moreover, recent optical study indicates that the variation of the electric- susceptibility effective-mass ratio as a function of hole concentration shows [12] a rather abrupt change at about the above p* value.

On the basis of the existence of two valence bands, one may suggest a variation of E, in the Pb-rich alloys.

From high-temperature electrical results in the intrinsic regions, different alloys ranging from PbTe to Pb,., Sn,.,Te were found to exhibit [13] 1141 thermal energy gaps approximately of the same order (0.34 -f- 0.02 eV). For PbTe [15], the thermal energy gap at high temperature (above 4000K) was correlated to the almost constant region of the optical energy gap, extrapolated to 0.36 eV at 0 OK, while a linear region at low temperature, with dE,/dT ⁼ 4.1 x eV/OK, indicates an extrapolated value of 0.19 eV at OOK.

One may consider a similar two-region behavior in Pb-rich alloys (x 5 0.3). The recent investigations [I]

[2] on the latter (below 300 OK) show a linear variation of E, with temperature, similar to that of PbTe. These low-temperature regions could be extrapolated to higher temperatures up to constant regions of the same order than that of PbTe. A similar interpretation to that considered for PbTe [13] can be assumed : the relative energy separation (E, + E,) between the conduction band BC and the second valence band BV, is supposed to remain approximately constant, while the first valence band BV, shifts away from BC approaching BV, with increasing temperature. At high temperatures, the thermal energy gap represents nearly E,, E, being small or negligible. Therefore, in Pb-rich alloys for whichEg is shown to decrease [I] 121, E, is expected to increase with increasing Sn content.

At liquid helium temperatures, for instance, E, decreases from about 0.19 eV at PbTe to zero at x

-

-

On the basis of relativistic effects in Pb and Sn, Dimmock et al. [I] have proposed a band inversion model to explain the energy gap behavior in Pb, -,Sn, Te : addition of Sn to PbTe gives rise to an upward

shift of the BC edge more pronounced than that of the BY, edge. After their crossover at some interme- diate composition (zero energy gap), the interchanged bands shift away from each other, and an increase of E, up to SnTe takes place. If one considers the values of p * at the Hall ratio kinks for the three alloys investigated (x = 0.3 and 0.5, and SnTe), one may expect a break in the variation of p" as a function of composition. Such break would provide a further support to the band inversion model. Investigations on Sn-rich alloys is of great interest, and would give meaningful information about the relative possible, shift of BV, comparatively to BC and BV, with incre- asing Sn content.

We wish to thank R. S. Allgaier for private commu- nication, and Mme M. Roudier for sample preparation.

References

[I] DIMMOCK (J. C.), MELNGAILIS (I.) and S r ~ ~ u s s (A. J.), Phys. Rev. Letters, 1966, 16, 1193.

[2] MELNGAILIS (I.) and CALAWA (A. R.), Appl. Phys.

Letters, 1966, 9, 304.

[3] AUKERMAN (L. W.) and WILLARDSON (R. K.), J. Appl.

Phys., 1960, 31, 939.

141 ALLGAIER (R. S.), J. Appl. phys., 1965, 36, 2429.

[5] ALLGAIER (R. S.) and HOUSTON (B. B.), J. Appl.

Phys., 1966, 36, 302.

161 BORDE (D.) and ALBANY (H. J.), Compt. Rend. Acad.

Sc., 1967, 264, 466.

[7] EFIMOVA (B. A.) and KOLOMOETS (L. A.), Fiz. Tverd.

Tela, 1965, 7, 424.

[8] BREBRICK (R. F.) and STRAUSS (A. J.), Phys. Rev., 1963, 131, 104.

[9] KAFALAS (J. A.), BREBRICK (R. F.) and STRAUSS (A. J.), Appl. Phys. Letters, 1964, 4, 93.

[lo] EFIMOVA (B. A.), KAIDANOV (V. I.), MOIZHES (B. Y.) and CHERNIK (I. A.), Fiz. Tve~d. Tela, 1966, 7, 2032.

[I 11 ALLGAIER (R. S.), private communication.

[12] RIEDL (H. R.), DIXON (J. R.) and SCHOOLAR (R. B.), Phys. Rev., 1967, 162, 692.

[13] BORDE (D.), Thesis, Paris 1967 (unpublished).

[14] MACHONIS (A. A.) and CADOFF (I. B.), Trans. Metal.

Soc. A. I. M. E., 1964, 230, 333.

[15] TAUBER (R. N.), MACHONIS (A. A.) and CADOFF (I. B.), J. Appl. Phys., 1966, 37, 4855.

DISCUSSION

WOOLLEY, J. C. - For Pb,,,Sn,.,Te will not the optical energy gap at low temperatures decrease with increasing temperature, rather than increase with increasing temperature as postulated here ?

We have attempted to fit R, versus T curves for several Pbl-,Sn,Te samples using a simple 3-band model (2 valence bands, 1 conduction band). Although

C 4 - 128 H. J. ALBANY AND M. OCIO the energy gap value required depends upon the values

assumed for energy gap temperature coefficient, effec- tive mass value mobility ratio, etc, the results tend to indicate that the zero-temperature separation of the valence bands decreases with increasing x.

ALBANY. - The increase of the optical energy gap with temperature considered here is for Pb-rich alloys ( x 5 0.3).

From the above discussion, one may expect a decre- ase of the zero-temperature separation of the valence bands with increasing x in Sn-rich alloys. However, if such a decrease is to be considered in the Pb-rich alloys, on the basis of the band inversion model, BV, would have a more pronounced shift than BY, with increasing Sn content, giving rise to a cc crossing ^M of BV, with BV, and BC.