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Absolute thermoelectric power of fluid cesium in the metal-nonmetal transition range
Volume 43A number 2
PHYSICS LETTERS
26 February 1973
ABSOLUTE THERMOELECTRIC POWER OF FLUID CESIUM IN THE METAL-NONMETAL TRANSITION RANGE H.P. PFEIFER, W.F. FREYLAND and F. HENSEL Institut fur Physikalische Chernie und Elektrochemie, UniversitätKarlsruhe, Germany Received 3 January 1973 The thermoelectric power and the electrical conductivity of cesium are measured at subcritical and supercritical temperatures to 1800°Cand at different pressures. The results are interpreted in terms ofMott’s description of metalnonmetal transition in n~ncrysta1linematerials.
In a crystalline monovalent metal a splitting of the conduction band into two subbands, an occupied and and empty band,has been predicted as a consequence of the electron-electron interaction if with decreasing density of the crystal the width of the conduction band decreases below a critical value. This MottHubbard-transition has been discussed in numerous papers [1—3].However, for fluid metals the situation is more complicated. Mott, in a discussion of the metalnonmetal transition of expanded fluid cesium [4] suggested that in such disordered systems of monovalent atoms the gap between the two Hubbard bands should be replaced by a minimum in the density of states N(E) in the vicinity of the Fermi energy EF, a so-called “pseudogap”. Measuring the depth of N(EF) by the quantity g = N(EF)IN(EF) free (N(EF)free = free electron density of states), he also predicted that when expansion of the fluid has reduced g to values smaller than 0.3, localized states should occur giving a gradual transition to a nonmetallic state, i.e. a mobility gap is formed. This transition should be accompanied by a drastic decrease in the electrical conductivity which was indeed observed for potassium and cesium near the liquid—vapour critical point [1,61. However, from these data no information could be obtained concerning the occurence of a mobility gap. Therefore the absolute thermoelectric power S and the conductivity a of cesium were measured siniultaneously at sub- and super-critical temperatures and pressures. The critical data are = 1750°Cand p~= 100 bar [6]. The applied experimental technique was dictated by the chemical incompatibility of cesium at high temperatures with nearly all electrically insulating materials. Therefore metallic tungsten 26% rhenium -
tubes with thin walls were used as containers for the fluid cesium. At the ends of the tubes two thermocoupled (97% W/3% Re 74% W/26% Re) were fixed. The thermopower S~,and the electrical resistance of this specimen, the tube parallel to the cesium sample, were measured simultaneously. The absolute thermo power ~ and the resistance ~ of cesium were obtamed using the equations valid for conductors in parallel [7], especially, ~ç ~ ~ + SW..Re, (1) S~s Ssp ~1 tRw R -
)
-
e
where SW..Re and RWRe are the thermopower and the resistance of the tungsten 26% rhenium tube. A description of the experimental details is given in refs. [8,9]. The results of the measur~edo and S are plotted in fig. I as a function of temperature at different constant pressures. An extremely steep decrease of both quantities, especially of 5, within a small temperature range is observed at slightly supercritical pressures. In this range every point is an average of many different independent measurements. Special care was taken in determining the temperature dependence of the conductivity since this value is important for the calculation of 5Cs with eq. (1). During one experiment the reproducibility of a was between 0.5% and 1% and the measured S-values could be reproduced between 10% to 20% at constant T and p and with different temperature gradients. The pronounced temperature dependence of S and a near the critical point, which is surely related to the large expansion of the fluid, can be explained by the Mott description of the metal-nonmetal transition for -
111
Volume 43A, number 2
1CflT1
PHYSICS LETTERS
s~0
sif
Who,
______________________
6IQ
26 February 1973
i~oorq
___
1O~ ~Ir,i~~500bar
200bor r
300 bar
-200
200bar 175 bar 150 bar
102
11 -300
130 bar 115 bar
i01
Fig. 1. Electrical conductivity and ferent pressures.
absolute thermoelectric
power of cesium at subcritical and supercritical temperature and dif-
disordered systems. For a larger than 2500 ohm—1 cm’ and S larger than —50 pV/K the electron mean free path L is larger than the interatomic distance a and a and S should be-described by Ziman’s N.F.E.theory [9].For a between 2500 and 300 ohm1cm~ and the corresponding S between —50 and —100 jiV/K possibly a “pseudogap” opens and according to Mott [10] a and S should be described by: a_S~e2ag2/l21r3h
(2)
s
(3)
=
2 ~2
4T (a in g e
For a smaller than 300 ohm—1 cm—1, i.e. for gcalculated from eq. (2) smaller than 0.3, the states near EF should become localized. An analysis of the experimental S with eq. (3) showed that at the onset of localization (0 lng/OE)E~ 1.1 (eV)1 is positive, i.e. EF lies at higher energies than the minimum value of N(E). Thus the main current should be due to thermally activated electrons which is consistent with the negative sign of S. For electron conduction in a conventional liquid semiconductor the following equation should be valid [3, 11], 0 = 0~exp (÷~— . S
+ 1).
(4)
A plot of in a versus S for cesium at conductivities
112
130 bar
smaller than 300 ohm—1 cm1 gives a slope of In a/oS (1.7 ±0.5) X l0~(K/volt) instead of
a
e/kB = 1.2 X 10~(K/volt). This deviation may mdicate that hole conduction plays a certain role in the semiconducting range of fluid cesium. Financial support of this work by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References [l~ N.F. Mott, Phil. Mag. 6 (1961) 287. [2] J. Hubbard, Proc. Roy. Soc. A281 (1964) 401. [3] N.F. Mott and E.A. Davies, Electronic processes in noncrystalline materials (Clarendon Press, Oxford, 1971) Chapter 5 and the cited papers. [4] N.F. Mott, Phil. Mag. 13 (1966) 989. 15] W.F. Freyland and F. Hensel, Ber. Bunsenges. Physik. Chem. 76 (1972) 347. [6] H. Renkert, F. Hensel and E.U. Franck, Ber. Bunsenges.
Chem. 75 (1971) 507. [7] Physik. D.K.C. MacDonald, Thermoelectricity (J. Wiley
and
Sons, N.Y., 1962). [81 W.F. Freyland, F. Hensel, Ber. Bunsenges. Physik. Chem. 76 (1972) 16. [9] W.F. Freyland, Thesis, Karisruhe University, 1972. [10] N.F. Mott, Phil. Mag. 26 (1972) 505. [11] R.W. Schmutzler, F. Hensel, Ber. Bunsenges. Physik. Chem. 76 (1972) 531.
PHYSICS LETTERS
26 February 1973
ABSOLUTE THERMOELECTRIC POWER OF FLUID CESIUM IN THE METAL-NONMETAL TRANSITION RANGE H.P. PFEIFER, W.F. FREYLAND and F. HENSEL Institut fur Physikalische Chernie und Elektrochemie, UniversitätKarlsruhe, Germany Received 3 January 1973 The thermoelectric power and the electrical conductivity of cesium are measured at subcritical and supercritical temperatures to 1800°Cand at different pressures. The results are interpreted in terms ofMott’s description of metalnonmetal transition in n~ncrysta1linematerials.
In a crystalline monovalent metal a splitting of the conduction band into two subbands, an occupied and and empty band,has been predicted as a consequence of the electron-electron interaction if with decreasing density of the crystal the width of the conduction band decreases below a critical value. This MottHubbard-transition has been discussed in numerous papers [1—3].However, for fluid metals the situation is more complicated. Mott, in a discussion of the metalnonmetal transition of expanded fluid cesium [4] suggested that in such disordered systems of monovalent atoms the gap between the two Hubbard bands should be replaced by a minimum in the density of states N(E) in the vicinity of the Fermi energy EF, a so-called “pseudogap”. Measuring the depth of N(EF) by the quantity g = N(EF)IN(EF) free (N(EF)free = free electron density of states), he also predicted that when expansion of the fluid has reduced g to values smaller than 0.3, localized states should occur giving a gradual transition to a nonmetallic state, i.e. a mobility gap is formed. This transition should be accompanied by a drastic decrease in the electrical conductivity which was indeed observed for potassium and cesium near the liquid—vapour critical point [1,61. However, from these data no information could be obtained concerning the occurence of a mobility gap. Therefore the absolute thermoelectric power S and the conductivity a of cesium were measured siniultaneously at sub- and super-critical temperatures and pressures. The critical data are = 1750°Cand p~= 100 bar [6]. The applied experimental technique was dictated by the chemical incompatibility of cesium at high temperatures with nearly all electrically insulating materials. Therefore metallic tungsten 26% rhenium -
tubes with thin walls were used as containers for the fluid cesium. At the ends of the tubes two thermocoupled (97% W/3% Re 74% W/26% Re) were fixed. The thermopower S~,and the electrical resistance of this specimen, the tube parallel to the cesium sample, were measured simultaneously. The absolute thermo power ~ and the resistance ~ of cesium were obtamed using the equations valid for conductors in parallel [7], especially, ~ç ~ ~ + SW..Re, (1) S~s Ssp ~1 tRw R -
)
-
e
where SW..Re and RWRe are the thermopower and the resistance of the tungsten 26% rhenium tube. A description of the experimental details is given in refs. [8,9]. The results of the measur~edo and S are plotted in fig. I as a function of temperature at different constant pressures. An extremely steep decrease of both quantities, especially of 5, within a small temperature range is observed at slightly supercritical pressures. In this range every point is an average of many different independent measurements. Special care was taken in determining the temperature dependence of the conductivity since this value is important for the calculation of 5Cs with eq. (1). During one experiment the reproducibility of a was between 0.5% and 1% and the measured S-values could be reproduced between 10% to 20% at constant T and p and with different temperature gradients. The pronounced temperature dependence of S and a near the critical point, which is surely related to the large expansion of the fluid, can be explained by the Mott description of the metal-nonmetal transition for -
111
Volume 43A, number 2
1CflT1
PHYSICS LETTERS
s~0
sif
Who,
______________________
6IQ
26 February 1973
i~oorq
___
1O~ ~Ir,i~~500bar
200bor r
300 bar
-200
200bar 175 bar 150 bar
102
11 -300
130 bar 115 bar
i01
Fig. 1. Electrical conductivity and ferent pressures.
absolute thermoelectric
power of cesium at subcritical and supercritical temperature and dif-
disordered systems. For a larger than 2500 ohm—1 cm’ and S larger than —50 pV/K the electron mean free path L is larger than the interatomic distance a and a and S should be-described by Ziman’s N.F.E.theory [9].For a between 2500 and 300 ohm1cm~ and the corresponding S between —50 and —100 jiV/K possibly a “pseudogap” opens and according to Mott [10] a and S should be described by: a_S~e2ag2/l21r3h
(2)
s
(3)
=
2 ~2
4T (a in g e
For a smaller than 300 ohm—1 cm—1, i.e. for gcalculated from eq. (2) smaller than 0.3, the states near EF should become localized. An analysis of the experimental S with eq. (3) showed that at the onset of localization (0 lng/OE)E~ 1.1 (eV)1 is positive, i.e. EF lies at higher energies than the minimum value of N(E). Thus the main current should be due to thermally activated electrons which is consistent with the negative sign of S. For electron conduction in a conventional liquid semiconductor the following equation should be valid [3, 11], 0 = 0~exp (÷~— . S
+ 1).
(4)
A plot of in a versus S for cesium at conductivities
112
130 bar
smaller than 300 ohm—1 cm1 gives a slope of In a/oS (1.7 ±0.5) X l0~(K/volt) instead of
a
e/kB = 1.2 X 10~(K/volt). This deviation may mdicate that hole conduction plays a certain role in the semiconducting range of fluid cesium. Financial support of this work by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References [l~ N.F. Mott, Phil. Mag. 6 (1961) 287. [2] J. Hubbard, Proc. Roy. Soc. A281 (1964) 401. [3] N.F. Mott and E.A. Davies, Electronic processes in noncrystalline materials (Clarendon Press, Oxford, 1971) Chapter 5 and the cited papers. [4] N.F. Mott, Phil. Mag. 13 (1966) 989. 15] W.F. Freyland and F. Hensel, Ber. Bunsenges. Physik. Chem. 76 (1972) 347. [6] H. Renkert, F. Hensel and E.U. Franck, Ber. Bunsenges.
Chem. 75 (1971) 507. [7] Physik. D.K.C. MacDonald, Thermoelectricity (J. Wiley
and
Sons, N.Y., 1962). [81 W.F. Freyland, F. Hensel, Ber. Bunsenges. Physik. Chem. 76 (1972) 16. [9] W.F. Freyland, Thesis, Karisruhe University, 1972. [10] N.F. Mott, Phil. Mag. 26 (1972) 505. [11] R.W. Schmutzler, F. Hensel, Ber. Bunsenges. Physik. Chem. 76 (1972) 531.