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Nernst and seebeck coefficients in bismuth at high magnetic fields
Advanced Energy Conversion,
Vol. 3, pp. 525-528. Pergamon Press, 1963. Printed in Great Britain
NERNST A N D SEEBECK COEFFICIENTS IN BISMUTH AT HIGH MAGNETIC FIELDS T. C. HARMAN and J. M. HONIG*
(Received 20 March 1963) Abstract--Measurements of the Nernst coefficient of bismuth have been carried out for various magnetic fields and several different temperatures. Values up to + 3.4 mV/deg have been observed at 97°K and 50 kG. The data have been interpreted in terms of a two-band overlap model. The magnetic field variation of a diagonal component of the Seebeck tensor and the apparent onset of an oscillatory behavior at high magnetic field are also briefly reported. THERE has been a tendency [1] in the past to discount the possibility of utilizing the Nernst effect in energy conversion devices. It is generally felt that the Nernst effect is too small to allow such devices to be operated at reasonable efficiencies. We therefore wish to report briefly on some experiments which show that a large Nernst coefficient can be observed. The relationship between the efficiency of devices and the Nernst coefficient has been discussed in several recent publications [2-8]. At the same time, we wish to call attention to an apparent oscillatory effect, which sets in at high magnetic fields, whose nature is not well understood at this time. In any anisotropic material, one must carefully distinguish between odd and even (with respect to magnetic field) entries of the generalized Seebeck tensor. Accordingly, we have defined [5] the Nernst and Seebeck coefficients as n y N ~ , ~- (V~(~/e)/V~T)o, Pzz =-- (Vz(~le)/VxT)e, n ~ N x ~ -- (V~(~/e)/V~T)o
and e , , =- (V~(~/e)/V~T),, where ~ is the electron electrochemical potential, o or e designate the odd and even portions of the matrix entries with respect to magnetic field direction, H is the magnetic field, e the magnitude of the electronic charge, T the temperature, and V the N a b l a operator. A single crystal of undoped, high purity Bi was mounted between two massive copper blocks and oriented such that the bisectrix (y), binary (x), and trigonal (z) axes were respectively parallel to Hu, VxT, and Vz(Ue). A special cell with provisions for two heaters and four thermocouples was utilized (a) to minimize possible errors in temperature measurements due to radiation, convection, or extraneous conduction of heat, and (b) to reverse the direction of the temperature gradient. With the four-thermocouple arrangement, the Righi-Leduc coefficient HuMzx = (VzT/VxT) was found to be zero. It follows [5] that the adiabatic and isothermal O D C N T t are approximately equal. The variation of HuNzz with magnetic field is shown in Fig. 1. It is seen that the Nernst coefficient rises roughly * Lincoln Laboratory (operated with support from the U.S. Army, Navy and Air Force) Massachusetts Institute of Technology, Lexington 73, Massachusetts, U.S.A. t Off Diagonal Components of the Nernst Tensor. 525
526
T.C. HARM~ and J. M. HONIO
linearly with magnetic field up to 50 kG, at which point the observed Nernst coefficient has the large positive value of + 3 . 4 mV/deg at 97°K. The corresponding values at 200 and 283 °K are + 0 . 7 9 mV/deg and + 0 . 2 8 mV/deg. This shows that HyN~ is quite sensitive to temperature. Beyond 30 kG, one observes the onset of oscillations whose amplitude is not much greater than the experimental error. Within this rather limited accuracy, one obtains linear plots of integers designating successive maxima or minima vs. 1/Hu, which indicates -36 - 3,4 0
-2z
!, I
-3.0 -2B
•e
0
0
m
o
•
-2.6 E
-2~
z N -2.2
i,-- -20 Z
O0 -174 LO -tO -7 -06 -04 I -02 0 0
I 5
I I0
I 15
__1
20
I 25
MAGNETIC
I 30
FIELD
l 35
I 40
[ 45
50
Hy,kG
FIG. 1. A typical plot of (ODCNT) HyN~zvs. Hy; arrows indicate approximate position of successive maxima. AH N~x should be positive.
that these oscillations are analogous to the Shubnikov-de Haas effect. The period of the oscillations was found to be 2-5 × 10 -5 G -1. These oscillations were also observed in bismuth at 102°K and in Bi-Sb (6 per cent) alloys. F r o m crystal symmetry considerations [9], it can be shown that P~x and HuN~x should vanish. Actually, P~x < 1/~V/deg, and HuNxz = 30/zV/°K, at 50 kG. These quantities are less than 1 per cent of Pzz. and HuNz:~, respectively, and can readily be ascribed to crystal misorientation. A plot of Pzz vs. Hu is shown in Fig. 2. Pxx has a value of --35/~V/deg in zero field, then passes through zero at 6 k G and finally becomes slowly varying beyond 30 kG. Due to the relatively small values of Pzz, the authors feel that no particular significance can be attached to the deviations of the points from the smooth curve entered in Fig. 2. The large observed HuNzx value~ make it unlikely that these transport phenomena can be interpreted in terms of a one-band model. Since the band structure and scattering mechanism in Bi are extremely complicated, we have resorted to use of an isotropic twoband model [10] to obtain a qualitative understanding of the dependence of HuNz:r on
Nernst and Seebeek Coeffacientsin Bismuth at High Magnetic Fields
527
Hu and on T. Defining ~b~ ~ I R~I o~H (i = 1 for electrons, i = 2 for holes) where R~ and oi are the contribution of the i t~ band to the Hall coefficient and magneto-conductivities o f the sample, a n d g ~ al/a2, one can show [11, 12] that H N i s optimized f o r g = ~bl/~b2= 1. This is equivalent to requiring I Rt [ = I R2 ] and ~1 = 02, so that the electron and hole
/
"'"'°
",,"°
-20 ~25
FIG. 2. A plot of the diagonal component of the Seebeck tensor Pxx vs. Hr. carrier mobilities in the magnetic field and the carrier concentrations must be equal. Under these conditions, the optimal value is HNopt = HN1/2 + H N 2 / 2 - - (~1/2)( I e l I + I P2 I ). It is known that [13] H N , --~ 0 in high fields, so that HNov t ~ - - (~bl/2)( [ P1 I + I Pz I ). Indeed the experimental Nernst coefficient is large in magnitude and varies roughly linearly with H. Chandrasekhar [14] has found that I P I I ~ I P z I ~ 100 t~V/deg. Assuming further that P, and o, vary slowly with H, the mobilities are of the order of 105 cm2/V-sec. The variation of the Nernst coefficient with the temperature conforms roughly to the T -2 power law. This dependence is due essentially to a variation of carrier mobility, or ~bl, with temperature; Zitter [15] has reported that/~, depends on temperature as T -2. The large observed Nernst coefficients can only be interpreted on the basis that both holes and electrons are in the high magnetic field region. The above results demonstrate that high Nernst coefficients are indeed observable in bismuth. Further consideration is being given to the utilization of this material in energy conversion devices. Acknowledgements--The authors gratefully acknowledge the use of the facilities of the National Magnet
Laboratory and wish to thank Mr. A. E. Paladino for his assistance in the measurements. REFERENCES [1] F. E. JAUMOT,Jr., Proc. Inst. Radio Engrs. 46, 538 (1958). [2] A. W. RrDDIrORD,B.Se. Thesis, Cornell University (1961). [3] B. J. O'B~r~ and C. S. WALLACe,J. AppL Phys. 29, 1010 (1958). [4] M. R. EL-SADEN,J. AppL Phys. 33, 1800 (1962). [51 T. C. I-IAm~t~Nand J. M. Hot-o, J. AppL Phys. 33, 3178, 3188 (1962); 34, 189, 239 (1963); also to be published.
528
T . C . HARMANand J. M. HONIG
[6] D. A. W~GIJT, Brit. J. AppL Phys. 13, 582 (1962). [7] R. W. UP,E, Proc. IEEE. 51, 699 (1963). [8] S. ANGRIST,d1". Heat T r a ~ . 85c, 41 (1963). [9] W. H. KLEINER,personal communication. [10] E. H. PUTLEY, The Hall Effect and Related Phenomena. Butterworth, London (1960). [11] T. C. HARMAN,Appl. Phys. Letters, 2, 13 (1963). [12] R. T. DELVES,Brit. J. Appl. Phys. 13, 440 (1962). [13] C. HERRING,T. H. GEBALLEand J. E. KUNZLER,Phys. Rev. 111, 36 (1958). [14] B. S. CHANDRASEKHAR,J. Phys. Chem. Solids 11, 268 (1959). [15] R. N. ZITrER, Phys. Rev. 127, 1471 (1962).
R~sum6----Des mesures du coefficient de Nernst de bismuth ont 6t6 effectu6es pour des champs magn6tiques divers et A plusieurs temp6ratures. Des valeurs jusqu' A +3,4 mV/deg, ont 6t6 observ6es A97 ° K et 50 kG. Ces faits ont 6t6 compris relatifs ~tun mod61e de recouvrement ~ deux bandes. La variation du champ magn6tique dans une composante diagonale du tenseur de Seebeck ainsi que le d6part apparent de facteurs oscillatoires ~t champs magn6tique 61ev6 y sont r~sum6s. Zusammenfassung--Messungen des Nernst Koeffizienten von Wismut sind fiir mehrere magnetische Felder und einige verschiedene Temperaturen ausgeftihrt worden. Werte bis +3,4 mV/Grad wurden bei 97°K and 50 kG beobaehtet. Die Angaben sind in Bezug auf ein zweib~ndig iiberlappendes Modell interpretiert worden. Die Ver~aderung einer Diagonalkomponenten des Seebeck Tensors im Magnetfeld und der offenbare Beginn eines Schwingungsverhalten in einem starken magnetisehen Feld werden ebenfalls kurz ertirtert.
Vol. 3, pp. 525-528. Pergamon Press, 1963. Printed in Great Britain
NERNST A N D SEEBECK COEFFICIENTS IN BISMUTH AT HIGH MAGNETIC FIELDS T. C. HARMAN and J. M. HONIG*
(Received 20 March 1963) Abstract--Measurements of the Nernst coefficient of bismuth have been carried out for various magnetic fields and several different temperatures. Values up to + 3.4 mV/deg have been observed at 97°K and 50 kG. The data have been interpreted in terms of a two-band overlap model. The magnetic field variation of a diagonal component of the Seebeck tensor and the apparent onset of an oscillatory behavior at high magnetic field are also briefly reported. THERE has been a tendency [1] in the past to discount the possibility of utilizing the Nernst effect in energy conversion devices. It is generally felt that the Nernst effect is too small to allow such devices to be operated at reasonable efficiencies. We therefore wish to report briefly on some experiments which show that a large Nernst coefficient can be observed. The relationship between the efficiency of devices and the Nernst coefficient has been discussed in several recent publications [2-8]. At the same time, we wish to call attention to an apparent oscillatory effect, which sets in at high magnetic fields, whose nature is not well understood at this time. In any anisotropic material, one must carefully distinguish between odd and even (with respect to magnetic field) entries of the generalized Seebeck tensor. Accordingly, we have defined [5] the Nernst and Seebeck coefficients as n y N ~ , ~- (V~(~/e)/V~T)o, Pzz =-- (Vz(~le)/VxT)e, n ~ N x ~ -- (V~(~/e)/V~T)o
and e , , =- (V~(~/e)/V~T),, where ~ is the electron electrochemical potential, o or e designate the odd and even portions of the matrix entries with respect to magnetic field direction, H is the magnetic field, e the magnitude of the electronic charge, T the temperature, and V the N a b l a operator. A single crystal of undoped, high purity Bi was mounted between two massive copper blocks and oriented such that the bisectrix (y), binary (x), and trigonal (z) axes were respectively parallel to Hu, VxT, and Vz(Ue). A special cell with provisions for two heaters and four thermocouples was utilized (a) to minimize possible errors in temperature measurements due to radiation, convection, or extraneous conduction of heat, and (b) to reverse the direction of the temperature gradient. With the four-thermocouple arrangement, the Righi-Leduc coefficient HuMzx = (VzT/VxT) was found to be zero. It follows [5] that the adiabatic and isothermal O D C N T t are approximately equal. The variation of HuNzz with magnetic field is shown in Fig. 1. It is seen that the Nernst coefficient rises roughly * Lincoln Laboratory (operated with support from the U.S. Army, Navy and Air Force) Massachusetts Institute of Technology, Lexington 73, Massachusetts, U.S.A. t Off Diagonal Components of the Nernst Tensor. 525
526
T.C. HARM~ and J. M. HONIO
linearly with magnetic field up to 50 kG, at which point the observed Nernst coefficient has the large positive value of + 3 . 4 mV/deg at 97°K. The corresponding values at 200 and 283 °K are + 0 . 7 9 mV/deg and + 0 . 2 8 mV/deg. This shows that HyN~ is quite sensitive to temperature. Beyond 30 kG, one observes the onset of oscillations whose amplitude is not much greater than the experimental error. Within this rather limited accuracy, one obtains linear plots of integers designating successive maxima or minima vs. 1/Hu, which indicates -36 - 3,4 0
-2z
!, I
-3.0 -2B
•e
0
0
m
o
•
-2.6 E
-2~
z N -2.2
i,-- -20 Z
O0 -174 LO -tO -7 -06 -04 I -02 0 0
I 5
I I0
I 15
__1
20
I 25
MAGNETIC
I 30
FIELD
l 35
I 40
[ 45
50
Hy,kG
FIG. 1. A typical plot of (ODCNT) HyN~zvs. Hy; arrows indicate approximate position of successive maxima. AH N~x should be positive.
that these oscillations are analogous to the Shubnikov-de Haas effect. The period of the oscillations was found to be 2-5 × 10 -5 G -1. These oscillations were also observed in bismuth at 102°K and in Bi-Sb (6 per cent) alloys. F r o m crystal symmetry considerations [9], it can be shown that P~x and HuN~x should vanish. Actually, P~x < 1/~V/deg, and HuNxz = 30/zV/°K, at 50 kG. These quantities are less than 1 per cent of Pzz. and HuNz:~, respectively, and can readily be ascribed to crystal misorientation. A plot of Pzz vs. Hu is shown in Fig. 2. Pxx has a value of --35/~V/deg in zero field, then passes through zero at 6 k G and finally becomes slowly varying beyond 30 kG. Due to the relatively small values of Pzz, the authors feel that no particular significance can be attached to the deviations of the points from the smooth curve entered in Fig. 2. The large observed HuNzx value~ make it unlikely that these transport phenomena can be interpreted in terms of a one-band model. Since the band structure and scattering mechanism in Bi are extremely complicated, we have resorted to use of an isotropic twoband model [10] to obtain a qualitative understanding of the dependence of HuNz:r on
Nernst and Seebeek Coeffacientsin Bismuth at High Magnetic Fields
527
Hu and on T. Defining ~b~ ~ I R~I o~H (i = 1 for electrons, i = 2 for holes) where R~ and oi are the contribution of the i t~ band to the Hall coefficient and magneto-conductivities o f the sample, a n d g ~ al/a2, one can show [11, 12] that H N i s optimized f o r g = ~bl/~b2= 1. This is equivalent to requiring I Rt [ = I R2 ] and ~1 = 02, so that the electron and hole
/
"'"'°
",,"°
-20 ~25
FIG. 2. A plot of the diagonal component of the Seebeck tensor Pxx vs. Hr. carrier mobilities in the magnetic field and the carrier concentrations must be equal. Under these conditions, the optimal value is HNopt = HN1/2 + H N 2 / 2 - - (~1/2)( I e l I + I P2 I ). It is known that [13] H N , --~ 0 in high fields, so that HNov t ~ - - (~bl/2)( [ P1 I + I Pz I ). Indeed the experimental Nernst coefficient is large in magnitude and varies roughly linearly with H. Chandrasekhar [14] has found that I P I I ~ I P z I ~ 100 t~V/deg. Assuming further that P, and o, vary slowly with H, the mobilities are of the order of 105 cm2/V-sec. The variation of the Nernst coefficient with the temperature conforms roughly to the T -2 power law. This dependence is due essentially to a variation of carrier mobility, or ~bl, with temperature; Zitter [15] has reported that/~, depends on temperature as T -2. The large observed Nernst coefficients can only be interpreted on the basis that both holes and electrons are in the high magnetic field region. The above results demonstrate that high Nernst coefficients are indeed observable in bismuth. Further consideration is being given to the utilization of this material in energy conversion devices. Acknowledgements--The authors gratefully acknowledge the use of the facilities of the National Magnet
Laboratory and wish to thank Mr. A. E. Paladino for his assistance in the measurements. REFERENCES [1] F. E. JAUMOT,Jr., Proc. Inst. Radio Engrs. 46, 538 (1958). [2] A. W. RrDDIrORD,B.Se. Thesis, Cornell University (1961). [3] B. J. O'B~r~ and C. S. WALLACe,J. AppL Phys. 29, 1010 (1958). [4] M. R. EL-SADEN,J. AppL Phys. 33, 1800 (1962). [51 T. C. I-IAm~t~Nand J. M. Hot-o, J. AppL Phys. 33, 3178, 3188 (1962); 34, 189, 239 (1963); also to be published.
528
T . C . HARMANand J. M. HONIG
[6] D. A. W~GIJT, Brit. J. AppL Phys. 13, 582 (1962). [7] R. W. UP,E, Proc. IEEE. 51, 699 (1963). [8] S. ANGRIST,d1". Heat T r a ~ . 85c, 41 (1963). [9] W. H. KLEINER,personal communication. [10] E. H. PUTLEY, The Hall Effect and Related Phenomena. Butterworth, London (1960). [11] T. C. HARMAN,Appl. Phys. Letters, 2, 13 (1963). [12] R. T. DELVES,Brit. J. Appl. Phys. 13, 440 (1962). [13] C. HERRING,T. H. GEBALLEand J. E. KUNZLER,Phys. Rev. 111, 36 (1958). [14] B. S. CHANDRASEKHAR,J. Phys. Chem. Solids 11, 268 (1959). [15] R. N. ZITrER, Phys. Rev. 127, 1471 (1962).
R~sum6----Des mesures du coefficient de Nernst de bismuth ont 6t6 effectu6es pour des champs magn6tiques divers et A plusieurs temp6ratures. Des valeurs jusqu' A +3,4 mV/deg, ont 6t6 observ6es A97 ° K et 50 kG. Ces faits ont 6t6 compris relatifs ~tun mod61e de recouvrement ~ deux bandes. La variation du champ magn6tique dans une composante diagonale du tenseur de Seebeck ainsi que le d6part apparent de facteurs oscillatoires ~t champs magn6tique 61ev6 y sont r~sum6s. Zusammenfassung--Messungen des Nernst Koeffizienten von Wismut sind fiir mehrere magnetische Felder und einige verschiedene Temperaturen ausgeftihrt worden. Werte bis +3,4 mV/Grad wurden bei 97°K and 50 kG beobaehtet. Die Angaben sind in Bezug auf ein zweib~ndig iiberlappendes Modell interpretiert worden. Die Ver~aderung einer Diagonalkomponenten des Seebeck Tensors im Magnetfeld und der offenbare Beginn eines Schwingungsverhalten in einem starken magnetisehen Feld werden ebenfalls kurz ertirtert.