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Resistance anomaly due to displacive phase transition in SnTe
Sohd State Commumcatrons, Vol 17, pp. 875-878,
1975
Pergamon Press
Printed m Great Bntam
RESISTANCE ANOMALY DUE TO DISPLACIVE PHASE TRANSITION IN SnTe K L.1 Kobayashr, Y Kate, Y Katayama and K.F Komatsubara Central Research Laboratory, I-htachr Ltd , Kokubunp, Tokyo 185, Japan (Recerved 28 March 1975, m rewed form 9 June 1975 by H Kawamura)
The effect of drsplacrve phase tranntron on electncal transport properties 1smvestrgated m a p-type smgle crystal of SnTe wrth tamer concentration of 1.2 x lose/cm3 at 77 K The resrstmty vs temperature curve shows an anomalous mcrease m the vrcnuty of the transition temperature An attempt 1smade to mterpret the temperature dependence of the reslstrvrty mcrement on the basrs of the carrier-soft TO-phonon mteraction
resrstrvrty vs temperature curve shows a clear kmk at 97 5 K wluch isJust the temperature where the phase transrtron was observed by the neutron scattermg expenments 4 The Hall coeffcrent, on the other hand, gradually mcreases wrth the mcrease of temperature and does not show any anomalous behavrour even around the transition temperature w&in the accuracy of the present measurements The gradual mcrease of the Hall coefficient wrth the mcrease of temperature can be understood by the two earner conductron model.6 The Hall mob&y was found to follow the power law pi a Tom m the temperature region hrgher than 150 K. Thrs enables us to estunate the background resrstrvity pbp which 1snot affected by the phase transrtion with
SINGLE crystals of SnTe have been expected to provrde an example of “dratomrc ferroelectncs” as well as “narrowgap ferroelectnc sennconductors” since Cochran’ pomted out that the cubic structure of the matenal tends to be unstable and a drsplacrve phase tranntron mrght occur at low temperatures Really Pawley et al. 2 observed the softenmg of the zone center TO-phonon, but phase transrtron Itself was not observed m their experiments The displacrve phase transitron from cubic to rhombohedral m SnTe has recently been observed by X-ray diffracttona neutron Bragg scattermg,’ and Raman scattermg 5 A vanety of tranmtion temperature TC’s are reported for crystals wrth vanous tamer concentrations The transrtron temperature Seems to mcrease wrth the decrease of the earner concentration
Pbg
The measurements of the electrical resistmty and the Hall coefficient were made m p-type smgle crystals of SnTe prepared by the solutron growth method The samples were cut from the same crystal that was used m the neutron scattermg measurements reported prevrously 4 The tamer concentratron of the sample m the present expenment 1s almost equal to the prevrous expenment, i.e., 8 8 x 10’*/cm3 at room temperature and 1.2 x 1020/cm3 at 77 K
=
4-i AT-‘-*
-I-B
(a cm)
where A = 1.0518x 10sandB=604x 10 The sohd curve m Fig 1 representmg equation (1) well fits the expenmental values at temperatures hrgher than 150 K By subfractmg pbg from the measured values of the resstrvity, we get the resrstivrty mcrement Ap related to the phase transrtron, as shown m Frg 2 Takmg mto account the facts that the renstrvrty mcrement Ap becomes extremely large around the transrtron temperature Te and that the zone center TO-phonons becomes soft also around the T,, an
The electrical resrstrvrty and the Hall coefficient are plotted agamst the temperature in Fig 1 The 875
876
DISPLACIVE PHASE TRANSITION IN SnTe
T,
=975K
2t
-2
1
1
TEMPERATURE
1
I
1
0
Vol 17, No 7
I
I
I
100
200
300
TEMPERATURE
(K)
FIG 2 Resrshvlty mcrement Ap vs temperature T
( K)
FIG 1. Electrrcal resrstrvrty p (dots) and Hall coeffiuent RH (open nrcles) vs temperature Arrow trnhcates the kmk m p at the T, of 97 5 K.
attempt 1s made to mterpret the temperature dependence of Ap based on a model of the carrier-soft TO-phonon mteracnon, as one of possible mecharusms In the hgh temperature phase, SnTe takes on NaCl structure As each consutuent atom 1s at the center of mversron symmetry m thrs structure, the mteracuon of free earner wrth TO-phonon via deformation potent& varushes for carners at symmetry points m the Bnlloum zone Thrs mteracuon, however, 1s Brute for carriers away from symmetry pomts. We estimate the magmtude of the mteracuon m SnTe m the k - p perturbatron scheme and the two band approximation The matnx element for the scattermg of the hole wrth the wave vector k measured from the L points by the TO-phonon wrth the wave vector q measured from the P pomt 1s obtamed as
constant Dl. Here a, p, e,, z+, at andNt,, are lattice constant, denstty of the crystal, polanzatron vector, dtsplacement, frequency and occupation number of the t-th phonon, respectively. In the derrvatlon of equatton (2), we have used the unperturbed wavefunctrons for Lf conduction band and L: valence band gtven by Mitchell and Walhs.’ The z axrs IS taken r.n the [ 11 l] crystallograpluc duectron As the condttrons E&
103K) % kT (- 10’K) % huTocrV 10 K)
are well saustied III the temperature repon of mterest, we can use the following approxrmanons m the calculation of the reslstrv:ty (1) Hrgh temperature approxrmauon for phonon hstnbuuon. 1 e NT0.a
= NTo,q +
1g
z.
(2) Elastic scattermg approxunatlon, 6 (Ek’ -EL
I e
+ h‘dT0) 1 S(Eg - Ek)
(3) Degenerate statistics for earner drstnbutron
usmg the transverse momentum matnx element Pl and the transverse mterband deformation potentral
We also srmphfy the q-dependence of TO-phonon around the transtuon temperature as follows the frequency of a TO-phonon wrth a wave vector q smaller than q,, 1surdependent of q and approaches zero urnformly as T + T,, where q. 1s a certam wave vector greater than 2kF 8 Then we get the temperature
877
DISPLACIVE PHASE TRANSITION IN SnTe
Vol. 17, No. 7
‘TEMPERATURE ( K)
for T > TE
BP a y(TTTC)
(5)
T
a
TAT-T,)
FIG 3. Reatstrvrty mcrement Ap vs T/(T - T,) Open crrcles represent expenmental data The ratio of the gradrent of the two sohd hnes for T > T, and T < Tf 111the figure 1sexactly 2 1 dependence of the resrstrvrty mcrement due to the phase trannuon as
kT Oc&zl
(3)
Accordmg to the standard theory of the second order tisplacrve phase transrtron, the TO-phonon frequency vanes as 40
=
r(T-
Td
= 2y(T,-T)
for T> T, forT
(4)
where 7 IS a constant mdependent of temperature Usmg relauon (4), we obtam the temperature dependence of the resrstivrty mcrement Ap as.
2tiTa_
Tc)
forT < TE
We replot the rensttvrty mcrement Ap agamst T/(T - T,) III Rg 3. A good agreement between expenments and the theoretical estrrnatron 1s obtained both 111the temperature regron above 110 K and below 90 K. Expenmental data devrate’from the relatron (5) near the transtron temperature T,, wluch may be due to the break of the perturbational treatment or the devratron of the temperature dependence of +o from equatmn (4) m thrs temperature regron The q dependence of the soft TO-phonon may also be responsible for the disagreement Smce the ratro (kFP&#&)’ * EFIEG UI equatron (3) is in the order of umty m the present experunents, equation (3) grves reasonable order of magnrtude of Ap assuming the deformatron potentral of about 10 eV Furthermore from equation (3) one notrces that the resistivity anomaly 1s enhanced when EC becomes small. In such a small gap semrconductor, the strong carrier-TO-phonon mterachon should affect the lattrce mstabrhty as well as electromc states lo* As other possble mechamsms one may come to the idea of tamer scattenng by longrtudmal phonons If the longrtudmal modes become soft near the transihon temperature, therr effect on tamer scattenng process 1s expected to be greater than m the case of TO-phonon ‘* At present, however, we have no expenrnental evrdence of the softenmg of the long tudmal modes m SnTe. Acknowldegements - Authors thank Dr J Umeda, Dr E Yamada and Dr T Uda for then valuable drscussrons
REFERENCES COCHRAN W , Ferroelecmczty (Edited by WELLER E.F.,) p 62 Elsevrer, NY (1967) PAWLEY G.S , COCHRAN W , COWLEY R A & DOLLING G., Phys Rev Lett 17,753 (1966) MUIDAWER L, J Nonmetds 1,177 (1975) IIZUMI M., HAMAGUCHI Y., KOMATSUBARA K.F & KATO Y , J Phys Sot Japru 38,443 (1975) BRILLSON LJ., BURSTEIN E & MULDAWER L , Phys Rev B9,1547 (1974) SAVAGE H.T , HOUSTON B & BURKE J R , Jr , Phys Rev B6,2292 (1972)
878
DISPLACIVE PHASE TRANSITION IN SnTe
Vol. 17, No 7
7
MITCHELL D.L. & WALLIS R.F ,Phys Rev 151,581 (1966)
8
The Fermi wave vector kF III SnTe of the present expenments 1s- 1 x 10’ cm-’ whle the value of q0 1sesfimated as Z 0 1 x (2a/a) w 1 x 10’ cm- ’ from the dlsperslon curves of TO-phonon determmed by neutron diffraction expenments 2
9
We thank Dr A Naton for her comment on the temperature dependence of aTO at low temperature side of the T,
10 11
KRISTOFFEL N. & KONSIN P , Ferroelectncs 6,3 (1973) KAWAMUR4 H., TAKANO S., HOTTA S., NISHI S , KATO Y., KOBAYASHI K L I & KOMATSUBARA K F , 12th Int Conf Phys. Semcond. Stuttgart p 55 1 (1974)
hc
12
KAWAMURA H. ( nvate commumcation). There 1sa posslbtity of the softenmg of the lower branch of the coupled plasmon- !O -phonon mode m the crystal ~rlth h@ tamer concentration because this branch approaches the TO-phonon at zone center
1975
Pergamon Press
Printed m Great Bntam
RESISTANCE ANOMALY DUE TO DISPLACIVE PHASE TRANSITION IN SnTe K L.1 Kobayashr, Y Kate, Y Katayama and K.F Komatsubara Central Research Laboratory, I-htachr Ltd , Kokubunp, Tokyo 185, Japan (Recerved 28 March 1975, m rewed form 9 June 1975 by H Kawamura)
The effect of drsplacrve phase tranntron on electncal transport properties 1smvestrgated m a p-type smgle crystal of SnTe wrth tamer concentration of 1.2 x lose/cm3 at 77 K The resrstmty vs temperature curve shows an anomalous mcrease m the vrcnuty of the transition temperature An attempt 1smade to mterpret the temperature dependence of the reslstrvrty mcrement on the basrs of the carrier-soft TO-phonon mteraction
resrstrvrty vs temperature curve shows a clear kmk at 97 5 K wluch isJust the temperature where the phase transrtron was observed by the neutron scattermg expenments 4 The Hall coeffcrent, on the other hand, gradually mcreases wrth the mcrease of temperature and does not show any anomalous behavrour even around the transition temperature w&in the accuracy of the present measurements The gradual mcrease of the Hall coefficient wrth the mcrease of temperature can be understood by the two earner conductron model.6 The Hall mob&y was found to follow the power law pi a Tom m the temperature region hrgher than 150 K. Thrs enables us to estunate the background resrstrvity pbp which 1snot affected by the phase transrtion with
SINGLE crystals of SnTe have been expected to provrde an example of “dratomrc ferroelectncs” as well as “narrowgap ferroelectnc sennconductors” since Cochran’ pomted out that the cubic structure of the matenal tends to be unstable and a drsplacrve phase tranntron mrght occur at low temperatures Really Pawley et al. 2 observed the softenmg of the zone center TO-phonon, but phase transrtron Itself was not observed m their experiments The displacrve phase transitron from cubic to rhombohedral m SnTe has recently been observed by X-ray diffracttona neutron Bragg scattermg,’ and Raman scattermg 5 A vanety of tranmtion temperature TC’s are reported for crystals wrth vanous tamer concentrations The transrtron temperature Seems to mcrease wrth the decrease of the earner concentration
Pbg
The measurements of the electrical resistmty and the Hall coefficient were made m p-type smgle crystals of SnTe prepared by the solutron growth method The samples were cut from the same crystal that was used m the neutron scattermg measurements reported prevrously 4 The tamer concentratron of the sample m the present expenment 1s almost equal to the prevrous expenment, i.e., 8 8 x 10’*/cm3 at room temperature and 1.2 x 1020/cm3 at 77 K
=
4-i AT-‘-*
-I-B
(a cm)
where A = 1.0518x 10sandB=604x 10 The sohd curve m Fig 1 representmg equation (1) well fits the expenmental values at temperatures hrgher than 150 K By subfractmg pbg from the measured values of the resstrvity, we get the resrstivrty mcrement Ap related to the phase transrtron, as shown m Frg 2 Takmg mto account the facts that the renstrvrty mcrement Ap becomes extremely large around the transrtron temperature Te and that the zone center TO-phonons becomes soft also around the T,, an
The electrical resrstrvrty and the Hall coefficient are plotted agamst the temperature in Fig 1 The 875
876
DISPLACIVE PHASE TRANSITION IN SnTe
T,
=975K
2t
-2
1
1
TEMPERATURE
1
I
1
0
Vol 17, No 7
I
I
I
100
200
300
TEMPERATURE
(K)
FIG 2 Resrshvlty mcrement Ap vs temperature T
( K)
FIG 1. Electrrcal resrstrvrty p (dots) and Hall coeffiuent RH (open nrcles) vs temperature Arrow trnhcates the kmk m p at the T, of 97 5 K.
attempt 1s made to mterpret the temperature dependence of Ap based on a model of the carrier-soft TO-phonon mteracnon, as one of possible mecharusms In the hgh temperature phase, SnTe takes on NaCl structure As each consutuent atom 1s at the center of mversron symmetry m thrs structure, the mteracuon of free earner wrth TO-phonon via deformation potent& varushes for carners at symmetry points m the Bnlloum zone Thrs mteracuon, however, 1s Brute for carriers away from symmetry pomts. We estimate the magmtude of the mteracuon m SnTe m the k - p perturbatron scheme and the two band approximation The matnx element for the scattermg of the hole wrth the wave vector k measured from the L points by the TO-phonon wrth the wave vector q measured from the P pomt 1s obtamed as
constant Dl. Here a, p, e,, z+, at andNt,, are lattice constant, denstty of the crystal, polanzatron vector, dtsplacement, frequency and occupation number of the t-th phonon, respectively. In the derrvatlon of equatton (2), we have used the unperturbed wavefunctrons for Lf conduction band and L: valence band gtven by Mitchell and Walhs.’ The z axrs IS taken r.n the [ 11 l] crystallograpluc duectron As the condttrons E&
103K) % kT (- 10’K) % huTocrV 10 K)
are well saustied III the temperature repon of mterest, we can use the following approxrmanons m the calculation of the reslstrv:ty (1) Hrgh temperature approxrmauon for phonon hstnbuuon. 1 e NT0.a
= NTo,q +
1g
z.
(2) Elastic scattermg approxunatlon, 6 (Ek’ -EL
I e
+ h‘dT0) 1 S(Eg - Ek)
(3) Degenerate statistics for earner drstnbutron
usmg the transverse momentum matnx element Pl and the transverse mterband deformation potentral
We also srmphfy the q-dependence of TO-phonon around the transtuon temperature as follows the frequency of a TO-phonon wrth a wave vector q smaller than q,, 1surdependent of q and approaches zero urnformly as T + T,, where q. 1s a certam wave vector greater than 2kF 8 Then we get the temperature
877
DISPLACIVE PHASE TRANSITION IN SnTe
Vol. 17, No. 7
‘TEMPERATURE ( K)
for T > TE
BP a y(TTTC)
(5)
T
a
TAT-T,)
FIG 3. Reatstrvrty mcrement Ap vs T/(T - T,) Open crrcles represent expenmental data The ratio of the gradrent of the two sohd hnes for T > T, and T < Tf 111the figure 1sexactly 2 1 dependence of the resrstrvrty mcrement due to the phase trannuon as
kT Oc&zl
(3)
Accordmg to the standard theory of the second order tisplacrve phase transrtron, the TO-phonon frequency vanes as 40
=
r(T-
Td
= 2y(T,-T)
for T> T, forT
(4)
where 7 IS a constant mdependent of temperature Usmg relauon (4), we obtam the temperature dependence of the resrstivrty mcrement Ap as.
2tiTa_
Tc)
forT < TE
We replot the rensttvrty mcrement Ap agamst T/(T - T,) III Rg 3. A good agreement between expenments and the theoretical estrrnatron 1s obtained both 111the temperature regron above 110 K and below 90 K. Expenmental data devrate’from the relatron (5) near the transtron temperature T,, wluch may be due to the break of the perturbational treatment or the devratron of the temperature dependence of +o from equatmn (4) m thrs temperature regron The q dependence of the soft TO-phonon may also be responsible for the disagreement Smce the ratro (kFP&#&)’ * EFIEG UI equatron (3) is in the order of umty m the present experunents, equation (3) grves reasonable order of magnrtude of Ap assuming the deformatron potentral of about 10 eV Furthermore from equation (3) one notrces that the resistivity anomaly 1s enhanced when EC becomes small. In such a small gap semrconductor, the strong carrier-TO-phonon mterachon should affect the lattrce mstabrhty as well as electromc states lo* As other possble mechamsms one may come to the idea of tamer scattenng by longrtudmal phonons If the longrtudmal modes become soft near the transihon temperature, therr effect on tamer scattenng process 1s expected to be greater than m the case of TO-phonon ‘* At present, however, we have no expenrnental evrdence of the softenmg of the long tudmal modes m SnTe. Acknowldegements - Authors thank Dr J Umeda, Dr E Yamada and Dr T Uda for then valuable drscussrons
REFERENCES COCHRAN W , Ferroelecmczty (Edited by WELLER E.F.,) p 62 Elsevrer, NY (1967) PAWLEY G.S , COCHRAN W , COWLEY R A & DOLLING G., Phys Rev Lett 17,753 (1966) MUIDAWER L, J Nonmetds 1,177 (1975) IIZUMI M., HAMAGUCHI Y., KOMATSUBARA K.F & KATO Y , J Phys Sot Japru 38,443 (1975) BRILLSON LJ., BURSTEIN E & MULDAWER L , Phys Rev B9,1547 (1974) SAVAGE H.T , HOUSTON B & BURKE J R , Jr , Phys Rev B6,2292 (1972)
878
DISPLACIVE PHASE TRANSITION IN SnTe
Vol. 17, No 7
7
MITCHELL D.L. & WALLIS R.F ,Phys Rev 151,581 (1966)
8
The Fermi wave vector kF III SnTe of the present expenments 1s- 1 x 10’ cm-’ whle the value of q0 1sesfimated as Z 0 1 x (2a/a) w 1 x 10’ cm- ’ from the dlsperslon curves of TO-phonon determmed by neutron diffraction expenments 2
9
We thank Dr A Naton for her comment on the temperature dependence of aTO at low temperature side of the T,
10 11
KRISTOFFEL N. & KONSIN P , Ferroelectncs 6,3 (1973) KAWAMUR4 H., TAKANO S., HOTTA S., NISHI S , KATO Y., KOBAYASHI K L I & KOMATSUBARA K F , 12th Int Conf Phys. Semcond. Stuttgart p 55 1 (1974)
hc
12
KAWAMURA H. ( nvate commumcation). There 1sa posslbtity of the softenmg of the lower branch of the coupled plasmon- !O -phonon mode m the crystal ~rlth h@ tamer concentration because this branch approaches the TO-phonon at zone center