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Shubnikov-de Haas effects in Bi2Se3 with high carrier concentrations
Solid State Communications,
Vol. 13, PP. 257—263, 1973.
Pergamon Press.
Printed in Great Britain
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3 WITH HIGH CARRIER CONCENTRATIONS* G.R. Hyde,t R.O. Dillon, H.A. Beale and I.L. Spain Laboratory for High Pressure Science, Department of Chemical Engineering, University of Maryland, College Park, Maryland 20742, U.S.A. J.A. Woollam NASA, Lewis Research Center, Cleveland, Ohio 44135, U.S .A. and 4 David J. Sellmye Materials Science Center, M.I.T., Cambridge, Massachusetts 02139, U.S.A. (Received 1 March 1973 by E. Burstein)
Shubnikov— de Haas frequencies were measured in highly degenerate n-type Bi 2Se3 having a higher carrier density (-‘ 9 X 10~m~)than previously reported. The Fermi surface was found to be elongated along the trigonal axis, fitting a spheroidal model with an axial ratio of 5.0 for angles up to 0 = 45°.Comparison of the number of carriers obtained from Hall measurements with that obtained from the Shubmkov—de Haas measurement supports the contention that the lowest conduction band minimum is a single surface located in the center of the Brillouin zone. The higher effective mass (0.25 mo) found for these carrier concentrations indicates that the band is non-parabolic. 3 yielding relatively good specimens, and and availability others’ of magnetic fields in the 10—20 tesla the range, galvanomagnetic and Shubnikov—de Haas measurements are beginning to produce reliable data
POTENTIALenergy applications to infrared optical. to-electrical conversion devicesdetectors, and thermoelectric generators have stimulated interest in the bismuth telluride and selenide compounds for several years. However, energy band studies on Bi 2Se3 have been hampered by the lack of good, homogeneous singlecrystal specimens. With recent efforts in this laboratory *
t .1
+
on the energy band structure of Bi2Se3. The purpose of the work reported here was to determine the properties of the conduction band of Bi2Se3 specimens having very high carrier concentrations (~ 9 X 1025 m~);paying particular attention
From a dissertation to be submitted to the Graduate School, Umversity of Maryland by G.R. Hyde, in partial fulfillment of the requirements for the Ph.D. degree in Chemical Engineering. Full time address, Bureau of Mines, College Park, Maryland 20740. .
.
.
.
to amsotropy m the Fermi surface. Earlier4work, has been in. cluding low explained infield termsgalvanomagnetic of a model requiring studies, six equivalent energy minima in the lowest conduction band. More recently, Shubnikov—de Haas data3 has indicated that the lowest conduction band of Bi 2Se3 consists of only
.
Now at Department of Physics, Umversity of Nebraska Lincoln Nebraska. NSF Contract No. GP—21 312.
.
.
a single valley or single ellipsoid. Our present work agrees with the latter result. The Fermi surface for the 257
258
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
Vol. 13, No. 3
Table 1. Conduction band parameters ofBi2Se3 Quantity nH n~fH
cio(4.2 K) PH
fHIIc
m*/mo Kf TD TH(4.2K)
Present work 25m3 (9.1 ±0.5) 0.3) X X 1025m3 l0 (8.8 ± (1.07 ±0.05) X l06(f2—m~’ (7.35 ±0.4) X 102m2/Vsec (2.13 ±0.05)X lO2tesla 0.25 ±0.05 5.0±0.5 (15.4±1)K (7.9±0.5)Xl0’4sec (10.4±l)X 10’4sec
Kohier and Landwehr (reference 3) 1.7 X 10~m3 1.8 X l0~m3 —
— —
0.13 1.54 9.7K 12.5X 10~4sec —
tComputed from measurements with angle 0 less than 45°. conduction band in our degenerate samples of Bi 2Se3 appears to be a prolate effipsoidal shape located at the center of the Brillouin zone with major axis parallel to the trigonal axis. Experimental measurements included galvanomag. netic measurements and exploratory work of Shubnikov —de Haas oscillations at 4.2K in fields up to 22 tesla and detailed Shubnikov—de Haas measurements from 1.1 to 4.2K in magnetic fields up to 11 tesla. Galvano. magnetic data included Hall voltage, resistance and magnetoresistance. From these were computed the number of carriers, ~H, using a single band model and the normal relationship RH = r/nHe where RH is the Hall coefficient, e the electronic charge and r a par. ameter depending on the scatteringmechanism, and its amsotropy, the shape of the 5constant energy surThe value for njq faces and degree computed with r of = degeneracy. 1 is given in Table 1. Mobility values (PH = RH a 0) are given also in the table, where the subscript 0 refer to H = 0. Shubnikov—de Haas frequencies and amplitudes were measured with the field parallel to the trigonal axis and at angles 0 up to 45°from the trigonal axis. From frequency data the cross-sectional area of the Fermi surface was computed as a function of e using the approximate relationship between the oscillatory resistivity ~ and magnetic field H: =
AH’~Texp
[
wch2k (TD + 1’)] 2ir
(~_ eH + (1)
,,
where the relationship w~= eH/m* allows the effective mass m* to be determined. In equation (l),A is a constant, T the absolute temperature and S the extremal cross.sectional area of the Fermi surface perpendicular to H. The number of carriers computed from the volume of the Fermi-surface (flSdH) as fit to a spheroidal model is compared in Table 1 with ~ Also the value for the relaxation time computed from the relationship ,
_____
TD =
2irkTD
(2)
is compared with that obtained from the Hall mobility ‘TH = m*PH/e using the value for m* computed from the Shubnikov—de Haas data. The n-type specimens used for technique these experiments wereBi2Se3 grown by the Bridgman and evaluated by chemical and X-ray analysis (details to be given elsewhere). Galvanomagnetic measurements were made using standard d.c. instrumentation and techniques. Shubnikov—de Haas measurements with H parallel to the trigonal axis were made using standard d.c. techniques. For measurements where H made some angle 0 with the trigonal axis two techniques were used: (1) the angle 0 was set at fixed values between 0 and 45°and the field swept up or down, and (2) the magnetic field was set at fixed values between 8 and 11 tesla and the sample rotated with respect to the field at a fixed rate, from 0 = —90°to 0constant = 90°.In both cases the temperature was kept at a value. The second technique uses the fact that ,?~,,in equation (1) goes through a maximum
Vol. 13, No.3
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
259
when cos [(hS/eR) + ‘y] = 1. That is, each time the (cos) term goes through a maximum the Shubnikov —de Haas frequency has changed by H (S is propor. tional to the frequency). Therefore, if the 0°frequency at one fixed angle is measured from swept field data, then at each angle corresponding to a maximum on the angular swept field data the frequency will have increased by H over the previous maximum. Typical Shubnikov—de Haas data for our specimens of Bi2Se3 ~
at fixed are shown magnetic in Fig. 1. field Data areobtained shown inbyFig. angular 2. sweeps
80
~
90 ~,
The effective mass was computed to be 0.25 m0 Experimental results are summarized in Table 1. (Fig. 3). Only data at 4.2 and 3.2 K were used to compute m*, since equation (1) is an approximation. to the expression. 2 ‘hrS = AH~’2T v (—lyr” 4~ [2ir~rk cos + sinh ~ (TD + T)] ~eH (3)
0.0
FIG. 1. Shubnikov—de Haas oscillations in Bi 2Se3. (A) 1.1 K field parallel to trigonal axis. (B) 1.1 K field 35°to trigonal axis. (C) 4.2 K field parallel to trigonal axis. The figures between 4.2show and 1.1 the Ksmall (large differences TD) and the in amplitudes considerable deterioration of the signal as the field was moved away from the trigonal axis.
I
-50
-40
I
I
-30
-20
I
-10
\
in which the exponential form of equation (1) may be used neglecting harmonics if 2rk(T + TD) > 1. (4) 2,r
I
0 10 8 DEGREES
I
I
I
I
20
30
40
50
FIG. 2. Shubnikov—de Haas oscillations in Bi 2Se3 in a magnetic field of 11.2 tesla varying in angle 0 with the trigonal axis.
260
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
The error in using the exponential form with only the first harmonic [equation (1)] increases rapidly below
+
0
\
\
\
4 K. For example, the error in using the exponential un place of sinh Xis 4.4, 10, 24 and 80 per cent respectively at 4.2,3.2, 2.2 and K. The plot of vs 2) 1/H allowing the 1.1 Dingle temperature in (~~~H” to be calculated from the slope, is shown in Fig. 4. The experimental points satisfactorily lie on a straight. line giving 7’D = 15.4 ±1K. The angular dependence of the inverse square of the Shubnikov—de Haas
\ \
\
frequency is given by: = 1/f2 = ~[I —(1—1/K2) sin2 0]
-
\ -
Vol. 13, No.3
~O
~
5 T(K)
FIG. 3. Determination of m* from Shubiukov—de Haas amplitudes. Only the two lower points were used to determine the slope because equation (1) is an approximation which becomes less reliable at lower temperatures (see text). The magnetic field is parallel to the trigonal axis.
where K is the anisotropy ratio. This equation for our data is plotted as the straight line in Fig. 5. Data obtained at fixed field by rotating the sample allow a much more precise estimate of the angular dependence to be made. The straight line in Fig. 5 fitting the data for. values less than 45°corresponds too an—anisotropy — ~ ratio (K) of 5.0 ±0.5 (K = S(90 )/S(0 ) = S II c/S I c). With such a large amsotropy, K is difficult to determine precisely from this plot. This ratio is much higher than that obtained by Kohler and Landwehr (1.54) in their sample (Table 1).
40C
3Z0~ x
3~0
2.50 095
(5)
+ I
I
I
.1
.100
.105
.110
.115
.120
8.~(teslo~)
FIG. 4. Determination of Dingle temperature TD from amplitudes of Shubnikov—de Haas oscillations in Bi 2Se3 at 1.1 K with magnetic field parallel to the trigonal axis.
Vol. 13, No.3
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
261
22
20
18 16 ‘4.
2iox
8 4,
2
0
I
I
I
01
02
03
I
Q~4
I
05 tl Sin
I
I
I
I
06
0.7
01
01
ID
FIG. 5. Plot of the inverse square of the Shubnikov—de Haas frequency vs sin20 at 1.1 K. The straight line represents the prediction for a spheroid with axial ratio K = 5.0. The dotted line represents an extrapolation for a truncated spheroid with major to minor axis ratio of 3.8 which would correspond with approximately a 20 per cent reduction in the number of enclosed carriers. The relaxation time obtained from the Dingle temperature TD is somewhat less than that obtained from the Hall mobility PH~This is consistent with the fact that small angle scattering is more effective in producing phase randomization in the oscillatory effect than in destroying electric current. The higher Dingle temperatures obtained here compared with Kohler and Landwehr (consistent with our higher electron density), did not enable the frequency measurements to be made for all values of 0 (Fig. 2). Thus, measurements of the deviation from circular symmetry of the planar surface (S II c) could not be made. Since the surface must have at least six-fold symmetry in the plane, significant deviations are not expected. The reduction in signal strength was made more acute in the present case since the effective mass was also higher (Table 1); consistent with a non-parabolic band.
The lack of measurements for our sample with 0 > 45°also prevented an accurate determination of n~H.Using the ratio K = 5.0, and assuming a spheroidal shape, then ~ = 8.8 X 1 025m3. This is within experimental error of that obtained from the single band interpretation of the Hall constant with r = 1 (nH = 9.1 X 1025m~).Thus the result is consistent with a single minimum for the conduction band located at the center of the Brillouin zone with approximately spheroidal shape if the above interpretation of the Hall constant is correct. Kohler and Landwehr3 reported that the Fermi surface corresponded to a truncated spheroid and that this decreased by about 20 per cent the number of carriers calculated from the data based on measurements for angles less than about 45°.A similar
262
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
truncation the number of carriers ns~ to about 7 would X 1025 decrease m3. Figure 5 indicates extrapolated data consistent with this truncation. It is possible that effects from carriers in other higher lying minima in the conduction band could affect the interpretation of the Hall measurement. However, although Kohler and Landwehr3 reported beats in the amplitude of some of their samples, which were attributed to crystal inhomogeneities, none were seen in the present measurements. It may tentatively be concluded that the next highest conduction minima in Bi 2Se3 lie at least 0.1 eV above the bottom of the 2’6 conduction band in contrast to. the caseband for for which higher lying conduction Bi2Te3 minima may be observed in samples with carrier concentrations greater than about 0.5 X 1025m3. The value of CF 0.1 eV is obtained by using a parabolic energy-wave vector relationship with values of kF and m* deduced from the Shubmkov—de Haas measurements. (8 X l08m~and 0.25 m 0 respectively). This value represents a lower limit to the Fermi energy and the non-parabolicity corrections may enhance it by a factor of more than two. Energy band calculations indicate thecenter lowest minima in the conduction band will lie that at the of the Brillouin zone (f’) (single minimum) or along the symmetry directions ~—L or r—x (six equivalent minima).2 Although two equivalent minima along the symmetry direction I~’—Tare not ruled out topographically this possibility may be ruled out from energy considerations. Similarly, minima at T (single equivalent minimum) and X or L (three equivalent minima) may be ruled out. In the limit of extreme degeneracy for a single spheroid with parabolic dispersion relationship the coefficient r is equal to unity if the relaxation time
Vol. 13, No.3
2 >1(r)2 may be expressed as a function of energy. ((r = 1 ).6 This holds even for anisotropic scattering if r is described by a tensor with the same principal axis system as the effective mass tensor.8 If the Fermi surface is non.effipsoidal, then the coefficient r may differ from unity Since the Shubnikov—de Haas results closely fit a spheroidal model for 0 < 0 ~ 45° it is unlikely that r will differ appreciable from unity.5 .~
Since it is unlikely that p-type conduction in the impurity levels could give two-band effects in the Hall coefficient, the Hall and Shubnikov—de Haas results together indicate that the conduction electrons are located in a single at the center ~of the Brillouin zone.minimum The goodlocated agreement between and ~H for the spheroid also indicates that the Fermi surface cannot differ appreciable from this shape. In summary, the present results on the Shubnikov— de Haas effect for a crystal with a much higher density of carriers than has previously been reported, support the contention that the lowest conduction band minima is a single surface located at the center of the Brillouin zone. The properties of the surface described here differ from a crystal with a smaller 3 inthat thatofthe effective mass of the carrier carriers concentration and the surface area ratio in the two principal directions are higher. Acknowledgements Thanks are due to the Center of Materials Research, University of Maryland (administering a grant from the National Science Foundation), for supporting this work and for use of their X-ray and crystal growth facilities. High magnetic field measurements were made at both the National Magnet Laboratory and at the Lewis Research Center. One of us (I.L.S.) also wishes to thank Dr. R. Allgaier for discussions on the interpretation of the Hall constant. —
REFERENCES 1.
GOBRECHT H. and SEECK S., Z. Phys. 222,93 (1969).
2. 3.
CAYWOOD L.P. and MILLER G.R., Phys. Rev. B2, 3209 (1970). KOHLER H. and LANDWEHRG.,Phys. Status Solidi 45, Kl09 (1971).
4.
HASHIMOTO K.,J. Phys. Soc. Japan 16, 1970 (1961).
5. 6.
ALLGAIER R.S.,Phys. Rev. 165, 775 (1968). MALLINSON R.B., RAYNE J.R. and URE R.W.,Phys. Rev. 175, 1049 (1968).
Vol. 13, No.3
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
7.
SMITH R.A., Semiconductors, chapter 5, Cambridge University Press (1959).
8.
HERRING C. and VOGT E.,Phys. Rev. 101,944(1956).
Die in einem höchst degenerierten n-Typ Bi2Se3 gemessenen Shubnikov— als Frequenzen vorher berichtet wurde. Man fand, dass man die Fermi(‘-9 Oberfiache de 3) Haas ergaben em höheres spezifIsches Gewicht X 1025 m elne Ausdehnung entlang det Trigonalachse einem sphäroidischen durch Modell mit einem Achsenverhãltnis von 4.8 für Winkel bis zu 0 = 450 anpassen kann. Em Vergleich, der durch Halls Messungen erhaltene Trãger. anzahl mit den de Haasischen Trägeranzahlmessungen unterstUtzt die Behauptung, dass das niedrigste Leistungsbandminimum eine einzige Oberfiache ist, die im Mittelpunkt der Brillouin zone liegt. Die höhere effektive Masse (0.25 m 0), die für these Trägerkonzentrierung gefunden wurde, lässt erkennen, dass das Band nicht parabolisch ist.
263
Vol. 13, PP. 257—263, 1973.
Pergamon Press.
Printed in Great Britain
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3 WITH HIGH CARRIER CONCENTRATIONS* G.R. Hyde,t R.O. Dillon, H.A. Beale and I.L. Spain Laboratory for High Pressure Science, Department of Chemical Engineering, University of Maryland, College Park, Maryland 20742, U.S.A. J.A. Woollam NASA, Lewis Research Center, Cleveland, Ohio 44135, U.S .A. and 4 David J. Sellmye Materials Science Center, M.I.T., Cambridge, Massachusetts 02139, U.S.A. (Received 1 March 1973 by E. Burstein)
Shubnikov— de Haas frequencies were measured in highly degenerate n-type Bi 2Se3 having a higher carrier density (-‘ 9 X 10~m~)than previously reported. The Fermi surface was found to be elongated along the trigonal axis, fitting a spheroidal model with an axial ratio of 5.0 for angles up to 0 = 45°.Comparison of the number of carriers obtained from Hall measurements with that obtained from the Shubmkov—de Haas measurement supports the contention that the lowest conduction band minimum is a single surface located in the center of the Brillouin zone. The higher effective mass (0.25 mo) found for these carrier concentrations indicates that the band is non-parabolic. 3 yielding relatively good specimens, and and availability others’ of magnetic fields in the 10—20 tesla the range, galvanomagnetic and Shubnikov—de Haas measurements are beginning to produce reliable data
POTENTIALenergy applications to infrared optical. to-electrical conversion devicesdetectors, and thermoelectric generators have stimulated interest in the bismuth telluride and selenide compounds for several years. However, energy band studies on Bi 2Se3 have been hampered by the lack of good, homogeneous singlecrystal specimens. With recent efforts in this laboratory *
t .1
+
on the energy band structure of Bi2Se3. The purpose of the work reported here was to determine the properties of the conduction band of Bi2Se3 specimens having very high carrier concentrations (~ 9 X 1025 m~);paying particular attention
From a dissertation to be submitted to the Graduate School, Umversity of Maryland by G.R. Hyde, in partial fulfillment of the requirements for the Ph.D. degree in Chemical Engineering. Full time address, Bureau of Mines, College Park, Maryland 20740. .
.
.
.
to amsotropy m the Fermi surface. Earlier4work, has been in. cluding low explained infield termsgalvanomagnetic of a model requiring studies, six equivalent energy minima in the lowest conduction band. More recently, Shubnikov—de Haas data3 has indicated that the lowest conduction band of Bi 2Se3 consists of only
.
Now at Department of Physics, Umversity of Nebraska Lincoln Nebraska. NSF Contract No. GP—21 312.
.
.
a single valley or single ellipsoid. Our present work agrees with the latter result. The Fermi surface for the 257
258
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
Vol. 13, No. 3
Table 1. Conduction band parameters ofBi2Se3 Quantity nH n~fH
cio(4.2 K) PH
fHIIc
m*/mo Kf TD TH(4.2K)
Present work 25m3 (9.1 ±0.5) 0.3) X X 1025m3 l0 (8.8 ± (1.07 ±0.05) X l06(f2—m~’ (7.35 ±0.4) X 102m2/Vsec (2.13 ±0.05)X lO2tesla 0.25 ±0.05 5.0±0.5 (15.4±1)K (7.9±0.5)Xl0’4sec (10.4±l)X 10’4sec
Kohier and Landwehr (reference 3) 1.7 X 10~m3 1.8 X l0~m3 —
— —
0.13 1.54 9.7K 12.5X 10~4sec —
tComputed from measurements with angle 0 less than 45°. conduction band in our degenerate samples of Bi 2Se3 appears to be a prolate effipsoidal shape located at the center of the Brillouin zone with major axis parallel to the trigonal axis. Experimental measurements included galvanomag. netic measurements and exploratory work of Shubnikov —de Haas oscillations at 4.2K in fields up to 22 tesla and detailed Shubnikov—de Haas measurements from 1.1 to 4.2K in magnetic fields up to 11 tesla. Galvano. magnetic data included Hall voltage, resistance and magnetoresistance. From these were computed the number of carriers, ~H, using a single band model and the normal relationship RH = r/nHe where RH is the Hall coefficient, e the electronic charge and r a par. ameter depending on the scatteringmechanism, and its amsotropy, the shape of the 5constant energy surThe value for njq faces and degree computed with r of = degeneracy. 1 is given in Table 1. Mobility values (PH = RH a 0) are given also in the table, where the subscript 0 refer to H = 0. Shubnikov—de Haas frequencies and amplitudes were measured with the field parallel to the trigonal axis and at angles 0 up to 45°from the trigonal axis. From frequency data the cross-sectional area of the Fermi surface was computed as a function of e using the approximate relationship between the oscillatory resistivity ~ and magnetic field H: =
AH’~Texp
[
wch2k (TD + 1’)] 2ir
(~_ eH + (1)
,,
where the relationship w~= eH/m* allows the effective mass m* to be determined. In equation (l),A is a constant, T the absolute temperature and S the extremal cross.sectional area of the Fermi surface perpendicular to H. The number of carriers computed from the volume of the Fermi-surface (flSdH) as fit to a spheroidal model is compared in Table 1 with ~ Also the value for the relaxation time computed from the relationship ,
_____
TD =
2irkTD
(2)
is compared with that obtained from the Hall mobility ‘TH = m*PH/e using the value for m* computed from the Shubnikov—de Haas data. The n-type specimens used for technique these experiments wereBi2Se3 grown by the Bridgman and evaluated by chemical and X-ray analysis (details to be given elsewhere). Galvanomagnetic measurements were made using standard d.c. instrumentation and techniques. Shubnikov—de Haas measurements with H parallel to the trigonal axis were made using standard d.c. techniques. For measurements where H made some angle 0 with the trigonal axis two techniques were used: (1) the angle 0 was set at fixed values between 0 and 45°and the field swept up or down, and (2) the magnetic field was set at fixed values between 8 and 11 tesla and the sample rotated with respect to the field at a fixed rate, from 0 = —90°to 0constant = 90°.In both cases the temperature was kept at a value. The second technique uses the fact that ,?~,,in equation (1) goes through a maximum
Vol. 13, No.3
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
259
when cos [(hS/eR) + ‘y] = 1. That is, each time the (cos) term goes through a maximum the Shubnikov —de Haas frequency has changed by H (S is propor. tional to the frequency). Therefore, if the 0°frequency at one fixed angle is measured from swept field data, then at each angle corresponding to a maximum on the angular swept field data the frequency will have increased by H over the previous maximum. Typical Shubnikov—de Haas data for our specimens of Bi2Se3 ~
at fixed are shown magnetic in Fig. 1. field Data areobtained shown inbyFig. angular 2. sweeps
80
~
90 ~,
The effective mass was computed to be 0.25 m0 Experimental results are summarized in Table 1. (Fig. 3). Only data at 4.2 and 3.2 K were used to compute m*, since equation (1) is an approximation. to the expression. 2 ‘hrS = AH~’2T v (—lyr” 4~ [2ir~rk cos + sinh ~ (TD + T)] ~eH (3)
0.0
FIG. 1. Shubnikov—de Haas oscillations in Bi 2Se3. (A) 1.1 K field parallel to trigonal axis. (B) 1.1 K field 35°to trigonal axis. (C) 4.2 K field parallel to trigonal axis. The figures between 4.2show and 1.1 the Ksmall (large differences TD) and the in amplitudes considerable deterioration of the signal as the field was moved away from the trigonal axis.
I
-50
-40
I
I
-30
-20
I
-10
\
in which the exponential form of equation (1) may be used neglecting harmonics if 2rk(T + TD) > 1. (4) 2,r
I
0 10 8 DEGREES
I
I
I
I
20
30
40
50
FIG. 2. Shubnikov—de Haas oscillations in Bi 2Se3 in a magnetic field of 11.2 tesla varying in angle 0 with the trigonal axis.
260
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
The error in using the exponential form with only the first harmonic [equation (1)] increases rapidly below
+
0
\
\
\
4 K. For example, the error in using the exponential un place of sinh Xis 4.4, 10, 24 and 80 per cent respectively at 4.2,3.2, 2.2 and K. The plot of vs 2) 1/H allowing the 1.1 Dingle temperature in (~~~H” to be calculated from the slope, is shown in Fig. 4. The experimental points satisfactorily lie on a straight. line giving 7’D = 15.4 ±1K. The angular dependence of the inverse square of the Shubnikov—de Haas
\ \
\
frequency is given by: = 1/f2 = ~[I —(1—1/K2) sin2 0]
-
\ -
Vol. 13, No.3
~O
~
5 T(K)
FIG. 3. Determination of m* from Shubiukov—de Haas amplitudes. Only the two lower points were used to determine the slope because equation (1) is an approximation which becomes less reliable at lower temperatures (see text). The magnetic field is parallel to the trigonal axis.
where K is the anisotropy ratio. This equation for our data is plotted as the straight line in Fig. 5. Data obtained at fixed field by rotating the sample allow a much more precise estimate of the angular dependence to be made. The straight line in Fig. 5 fitting the data for. values less than 45°corresponds too an—anisotropy — ~ ratio (K) of 5.0 ±0.5 (K = S(90 )/S(0 ) = S II c/S I c). With such a large amsotropy, K is difficult to determine precisely from this plot. This ratio is much higher than that obtained by Kohler and Landwehr (1.54) in their sample (Table 1).
40C
3Z0~ x
3~0
2.50 095
(5)
+ I
I
I
.1
.100
.105
.110
.115
.120
8.~(teslo~)
FIG. 4. Determination of Dingle temperature TD from amplitudes of Shubnikov—de Haas oscillations in Bi 2Se3 at 1.1 K with magnetic field parallel to the trigonal axis.
Vol. 13, No.3
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
261
22
20
18 16 ‘4.
2iox
8 4,
2
0
I
I
I
01
02
03
I
Q~4
I
05 tl Sin
I
I
I
I
06
0.7
01
01
ID
FIG. 5. Plot of the inverse square of the Shubnikov—de Haas frequency vs sin20 at 1.1 K. The straight line represents the prediction for a spheroid with axial ratio K = 5.0. The dotted line represents an extrapolation for a truncated spheroid with major to minor axis ratio of 3.8 which would correspond with approximately a 20 per cent reduction in the number of enclosed carriers. The relaxation time obtained from the Dingle temperature TD is somewhat less than that obtained from the Hall mobility PH~This is consistent with the fact that small angle scattering is more effective in producing phase randomization in the oscillatory effect than in destroying electric current. The higher Dingle temperatures obtained here compared with Kohler and Landwehr (consistent with our higher electron density), did not enable the frequency measurements to be made for all values of 0 (Fig. 2). Thus, measurements of the deviation from circular symmetry of the planar surface (S II c) could not be made. Since the surface must have at least six-fold symmetry in the plane, significant deviations are not expected. The reduction in signal strength was made more acute in the present case since the effective mass was also higher (Table 1); consistent with a non-parabolic band.
The lack of measurements for our sample with 0 > 45°also prevented an accurate determination of n~H.Using the ratio K = 5.0, and assuming a spheroidal shape, then ~ = 8.8 X 1 025m3. This is within experimental error of that obtained from the single band interpretation of the Hall constant with r = 1 (nH = 9.1 X 1025m~).Thus the result is consistent with a single minimum for the conduction band located at the center of the Brillouin zone with approximately spheroidal shape if the above interpretation of the Hall constant is correct. Kohler and Landwehr3 reported that the Fermi surface corresponded to a truncated spheroid and that this decreased by about 20 per cent the number of carriers calculated from the data based on measurements for angles less than about 45°.A similar
262
SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
truncation the number of carriers ns~ to about 7 would X 1025 decrease m3. Figure 5 indicates extrapolated data consistent with this truncation. It is possible that effects from carriers in other higher lying minima in the conduction band could affect the interpretation of the Hall measurement. However, although Kohler and Landwehr3 reported beats in the amplitude of some of their samples, which were attributed to crystal inhomogeneities, none were seen in the present measurements. It may tentatively be concluded that the next highest conduction minima in Bi 2Se3 lie at least 0.1 eV above the bottom of the 2’6 conduction band in contrast to. the caseband for for which higher lying conduction Bi2Te3 minima may be observed in samples with carrier concentrations greater than about 0.5 X 1025m3. The value of CF 0.1 eV is obtained by using a parabolic energy-wave vector relationship with values of kF and m* deduced from the Shubmkov—de Haas measurements. (8 X l08m~and 0.25 m 0 respectively). This value represents a lower limit to the Fermi energy and the non-parabolicity corrections may enhance it by a factor of more than two. Energy band calculations indicate thecenter lowest minima in the conduction band will lie that at the of the Brillouin zone (f’) (single minimum) or along the symmetry directions ~—L or r—x (six equivalent minima).2 Although two equivalent minima along the symmetry direction I~’—Tare not ruled out topographically this possibility may be ruled out from energy considerations. Similarly, minima at T (single equivalent minimum) and X or L (three equivalent minima) may be ruled out. In the limit of extreme degeneracy for a single spheroid with parabolic dispersion relationship the coefficient r is equal to unity if the relaxation time
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2 >1(r)2 may be expressed as a function of energy. ((r = 1 ).6 This holds even for anisotropic scattering if r is described by a tensor with the same principal axis system as the effective mass tensor.8 If the Fermi surface is non.effipsoidal, then the coefficient r may differ from unity Since the Shubnikov—de Haas results closely fit a spheroidal model for 0 < 0 ~ 45° it is unlikely that r will differ appreciable from unity.5 .~
Since it is unlikely that p-type conduction in the impurity levels could give two-band effects in the Hall coefficient, the Hall and Shubnikov—de Haas results together indicate that the conduction electrons are located in a single at the center ~of the Brillouin zone.minimum The goodlocated agreement between and ~H for the spheroid also indicates that the Fermi surface cannot differ appreciable from this shape. In summary, the present results on the Shubnikov— de Haas effect for a crystal with a much higher density of carriers than has previously been reported, support the contention that the lowest conduction band minima is a single surface located at the center of the Brillouin zone. The properties of the surface described here differ from a crystal with a smaller 3 inthat thatofthe effective mass of the carrier carriers concentration and the surface area ratio in the two principal directions are higher. Acknowledgements Thanks are due to the Center of Materials Research, University of Maryland (administering a grant from the National Science Foundation), for supporting this work and for use of their X-ray and crystal growth facilities. High magnetic field measurements were made at both the National Magnet Laboratory and at the Lewis Research Center. One of us (I.L.S.) also wishes to thank Dr. R. Allgaier for discussions on the interpretation of the Hall constant. —
REFERENCES 1.
GOBRECHT H. and SEECK S., Z. Phys. 222,93 (1969).
2. 3.
CAYWOOD L.P. and MILLER G.R., Phys. Rev. B2, 3209 (1970). KOHLER H. and LANDWEHRG.,Phys. Status Solidi 45, Kl09 (1971).
4.
HASHIMOTO K.,J. Phys. Soc. Japan 16, 1970 (1961).
5. 6.
ALLGAIER R.S.,Phys. Rev. 165, 775 (1968). MALLINSON R.B., RAYNE J.R. and URE R.W.,Phys. Rev. 175, 1049 (1968).
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SHUBNIKOV—DE HAAS EFFECTS IN Bi2Se3
7.
SMITH R.A., Semiconductors, chapter 5, Cambridge University Press (1959).
8.
HERRING C. and VOGT E.,Phys. Rev. 101,944(1956).
Die in einem höchst degenerierten n-Typ Bi2Se3 gemessenen Shubnikov— als Frequenzen vorher berichtet wurde. Man fand, dass man die Fermi(‘-9 Oberfiache de 3) Haas ergaben em höheres spezifIsches Gewicht X 1025 m elne Ausdehnung entlang det Trigonalachse einem sphäroidischen durch Modell mit einem Achsenverhãltnis von 4.8 für Winkel bis zu 0 = 450 anpassen kann. Em Vergleich, der durch Halls Messungen erhaltene Trãger. anzahl mit den de Haasischen Trägeranzahlmessungen unterstUtzt die Behauptung, dass das niedrigste Leistungsbandminimum eine einzige Oberfiache ist, die im Mittelpunkt der Brillouin zone liegt. Die höhere effektive Masse (0.25 m 0), die für these Trägerkonzentrierung gefunden wurde, lässt erkennen, dass das Band nicht parabolisch ist.
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