- Email: [email protected]
Cyclotron resonance in lead —I
1. Phys. Chem. solids, 1977, Vol. 38, pp. 419-429.
Pergamon Press.
Printed in Great Britain
CYCLOTRON RESONANCE IN LEAD-I TIPPING EFFECT OF MAGNETIC FIELD YOSHICHIKA~NUKI, HIROYOSHISUEMATSUand SEI-ICHITANUMA The Institutefor Solid State Physics, The University of Tokyo, Roppongi,Minato-ku,Tokyo 106,Japan
(Received26 May 1976;accepted
in
revisedform 20 August 1976)
Abstract-The cyclotronresonancesin lead in fieldtippedgeometryhave been observedfor largetippinganglesup to 40”from the samplesurfacesof the (111)and (100)planes. In additionto the resonanceof the orbit5 on a cylindrical armof the electronFermisurface,two new series of resonances,lA and 5” have been found, of which the cyclotron massesare very close to thatof [ but whichdiier slightlyin tippingangledependence.The massof lA is dependenton microwavefrequency,so that5” is attributedto the Doppler-shiftedcyclotronresonance of a non-stationaryorbitnear[, whose locationis discussedin relationto the Fermisurfacemodelof Van Dyke. In fieldgeometrynormalto the (100) surface,two series of resonanceshave been observed,which are characterizedas the extinctionof even numbered harmonicresonances.Thiseffectarisesfromthe skipping orbit whichhas a trajectory topologicallysimilarto a baseball seam in real space. One of these series is attributedto the orbit Y aroundthe junctionof four electron arms.
1. lNlRODUcIlON
Many experimental works on Azbel’-Kaner cyclotron resonance (AKCR)[ 11 have been reported in various metals since the original theory[2] predicted the effect. The ordinary AKCR is observed when a steady magnetic field is applied exactly parallel to the surface of the specimen. The resonance occurs when the microwave frequency o is an integral multiple of the cyclotron frequency o,, i.e. w = nw,,
(1)
where n is an integer and wc = eH/m?c includes the cyclotron effective mass m?. The mass rn? is related to the electronic structure of the relevant metal in the following manner:
Kaner cyclotron resonance, and in general can be observed for a large tipping angle. The other comes from the non-stationary orbit which has a drift velocity component normal to the surface of the specimen and gives rise to Doppler effect with the incident microwave radiation. This type of resonance is observed for a small tipping angle of less than a few degrees. The cyclotron resonance of the non-stationary orbit has been studied in detail by Koch et al. [3]. If an orbit has the driit velocity uD along the magnetic field, the carrier feels effectively the rf field having a frequency dBerent from the incident microwave frequency o by an amount of uDsin a/S, where (Yis the tipping angle of the magnetic field from the metal surface and 6 is the skin depth in the anomalous limit. Hence, the resonance frequency is Doppler-shifted in the following manner: 0 ? v. sin aIS = no,,
(3)
or where EF is the Fermi energy, S(~H,E) is a crosssectional area of an equi-energy surface of energy E occupying the k space in the plane normal to the magnetic field H, and kH is the longitudinal component of the wave vector, i.e. k,, = k. H/IHI. In general, the cyclotron resonance occurs for the orbit, of which the cyclotron mass mf(k”) is extremal with respect to kH, i.e. am ?/ak,
= 0.
On the other hand, the cyclotron resonance in the magnetic field tipped with respect to the surface of the specimen has been observed in some metals [3-51 in spite of the fact that Azbel’ and Kaner have predicted the cyclotron resonance would vanish for a very small tipping angle (typically about 5’). The observed cyclotron resonances in field tipped geometry are classified as two types of resonances. One is attributed to the stationary orbit, of which the drift velocity is zero along kH corresponding to the tipped magnetic field. This type of resonance is essentially the same as the usual Azbel’-
~/~=(~)(l’(~)“‘(~)sina},
(4)
where p is the resistivity and I is the mean free path. The resonance field in eqn (4) varies according to the change of mf(kH) and uD(kH) which are functions of kH. The cyclotron resonance occurs therefore for the orbit, for which the right hand of eqn (4) is extremal with respect to k,,. If the cyclotron mass is assumed to be constant over the variation of kH under consideration, the cyclotron resonance is only expected for the orbit, whose drift velocity uD(kH) is extremal with respect to k,,. In addition to these resonances, distinct series of resonances have been observed in an extremely oblique configuration of the magnetic field, i.e. where the magnetic field is normal to the sample surface. When a cyclotron orbit has a non-flat and wavy closed trajectory in real space, the crests of waves may skip in the skin layer and feel the rf field. If such a trajectory has two 419
420
YOSHKHIKA ~NUKI et al.
crests and two troughs, i.e. has a shape similar to a baseball seam, an electron on the orbit interacts twice with the rf field at these crests during one cyclotron period. In this case, we can expect a cyclotron resonance similar to the ordinary AKCR type, but lacking the even numbered harmonics. In this paper, we describe observations of the cyclotron resonance in lead in various configurations including the field geometry of oblique angles up to 90”, and discuss the Fermi surface of lead. The Fermi surface of lead has been extensively investigated by Anderson and Gold[6] using the de Hass van Alphen (dHvA) effect. They have obtained a good agreement of experimentally determined Fermi surface geometry with the four orthogonalized plane waves representation. The Fermi surface of lead consists of two sheets, one of which is a large closed hole surface centered at F in the second zone and the other is a multiply connected electron surface in the third zone constructed by cylindrical arms in the [1lo] and its equivalent directions as shown in Fig. 1. Among many orbits describable as the peripheries of cross sections of a Fermi surface normal to the applied field, the following three orbits are treated in this report. These are shown in Fig. 1 as the orbits \y on the hole surface, 5 on the [llO] arm of electron surface and v around the junction of four [llO] arms; these orbits correspond to the dHvA oscillations (Y, y and p [6], respectively. As for the [ 1lo] arm of the electron surface, Anderson and Gold have proposed that this arm has a narrow minimum cross section at the point U (or K) and a coo11 I
region of inflection of the cross sectional area midway between U and W; this model predicts two kinds of dHvA frequencies differing from each other by about two percent. The recent studies of dHvA effect by Anderson et al.[7,8] and more recently by Ogawa et al.[9], and theoretical calculations by Van Dyke[lO] have revealed that the earlier model of Anderson and Gold for this arm should be modified, i.e. it should have the maximum cross section at U and the minimum somewhere between U and W to make a corrugated cylinder. However, this problem has not been solved. Previous works on cyclotron resonances [ 1l-131 in field geometry parallel to the surface give no information about the above mentioned fine structure in the [llO] arm, although almost all of the orbits found in dHvA effect have already been observed in the cyclotron resonance. One of our purposes is to investigate the problem by means of cyclotron resonance in field tipped geometry. In Section 2, we describe our experimental setup as well as the preparation of the samples. In Section 3.1, the cyclotron resonance data are described for large tipping angles of 0” to 40” from the (111) and (100) surfaces. To understand the tipping effect, we have studied the usual AKCR in the parallel field geometry using the (112) and (110) surfaces, respectively. Distinct series of resonances have been observed which were not reported in previous papers. One of them is explained in terms of the cyclotron resonance of the non-stationary orbit by the use of two different microwave frequencies. In Section 3.2, the cyclotron resonance in field geometry normal to the (100) surface is presented, and the phenomenon is attributed to the skipping resonance of the orbit v which has a trajectory somewhat similar to a baseball seam. In Section 4, discussions on the nature of relevant Fermi surface are presented. 2. EXPERIMENTAL. PROCEDURES
(a)
1
(b)
Fig. l(a). Hole Fermi surface in lead in the second Brillouin zone. (b) Electron Fermi surface in the third Brillouin zone.
single crystals were grown from material of 99.9999% purity (Osaka Asahi Metal Co.) in a graphite crucible by means of the Czochralski pulling method. The orientation of the sample was determined by X-ray within an accuracy of 1”.Thin parallelepiped specimens (8 x 8 x 1 mm’) were cut by a Servomet spark machine. The surface damage of the spark cut was removed by lapping the specimens on a cloth which was wetted with a solution of three parts hydrogen peroxide and seven parts acetic acid. The surface was then made microscopically smooth by electro-polishing with a solution of perchloric acid and acetic anhydride using the method reported by Tegart [ 141. The residual resistance ratio (~~~Jp~.~a)for such samples was (6-9) X lo-‘. The measurements were carried out at 1SK using a standard reflection type spectrometer of 23 GHz (TE, II mode) and a reaction type of 70GHz (TE,12mode). The magnetic field was modulated at an audio frequency and the field derivative of power absorption was detected by a lock-in amplifier. In order to permit an accurate and easy rotation of the magnetic field up to large tipping angles, we used an electromagnet whose field direction was rotatable in the horizontal plane, and the cylindrical cavity in which the rf electric field was directed vertically. The sample The
Cyclotron resonance in lead-l
was also attached vertically on the end wall of the cavity with silver cement.
Pb 11111 68 5L GHz 1.5LK
Jrt I H
3. EXPERIMENTAL RFSULTS ANDANALYSIS
3.1 Tipped field cyclotron resonance The cyclotron resonance in field tipped geometry have been obtained as a function of the tipping angle with respect to the sample surfaces of the (111) and (100) planes. These surfaces are used in order to simplify the effect of the field tipping. In these instances, at least one of the cylindrical [llO] arms is parallel to the sample surface; its resonance is not confused with those of the other equivalent arms, because the latter arms make a large angle with both the surface and the magnetic field. The rj current of the microwave is always transverse to the field direction, as shown in the inserts of Fig. 3 and Fig. 8. First, we present the results for the (111) surface. 3.1.1 The (111) surface (a) Tipping angle dependence of the cyclotron resonance. The typical recorder traces of the cyclotron resonance obtained at 70 GHz are shown in Fig. 2 for the (111) surface and for various angles (Yof the magnetic field tipped from the [ 1lo] axis in the (112) plane as shown in the insert of Fig. 3. The cyclotron masses are determined from the period of the resonance peaks, and their dependences on a are shown in Fig. 3. For a = O”,we observe two dominant series, one having a large amplitude and a cyclotron mass of rn?l= 0.54m0, and the other having a small amplitude and rn? = 1.14m0 which is shown not in Fig. 3 but in Fig. 4. These resonances are attributed to the 5 and \y orbits respectively. These cyclotron masses are in good agreement with those found by previous measurements[ll-131. In addition to these resonances, a third, previously unreported, series of weak resonances, labeled fA, is found as shown in Fig. 2. The cyclotron mass of 5” is 0.565m0, which is very close to that of & As the field is tipped from the surface, the resonances 5 and 5” retain appreciable amplitudes for tipping angles up to 30”. As shown in Fig. 2, for a = 10” there appears another kind of resonance, labeled tB, between 4’and g*. This is hidden for a 2 13”by virtue of the large amplitude of 5^. For tipping angles larger than 20”, two new series of resonances, labeled I’ and 5” appear. The amplitude of the resonance I” is very small in comparison with the others. The cyclotron masses of the series [, LA,4’”and 6’ increase with the increase of a, while that of 5” decreases. The branch 5 and g’ remain large in amplitude for tipping angles up to 38”,while the branch 5” decreases rapidly in amplitude beyond 28”. For the narrow a region of 33”-35”,where the dominant resonances are f and lc, the line shape changes drastically; the peaks become dull, while the troughs become sharp with increasing a as shown in Fig. 2. Similar phenomena have been reported in copper [3] and other metals[4, IS]. In the region of 35”-38”, the resonances become very weak and the line shapes are so complicated that we cannot assign them firmly. Beyond 38”, the line shape returns to that of the ordinary AKCR,
421
dR i-i
I Fig. 2. Typical recorder traces of cyclotron resonance at the frequency of 68.54GHz for various angles of field tipping. The notation Q is the angle of field tipping from the [I 101axis in the (111)surface toward the [ill] axis as shown in the insert of Fig. 3. The subscription number n for f; Vrand et al. stands for the nth harmonic of AKCR.
and a new series of resonance appears, which is labeled C”* The fact that the [ 1lo] arm is parallel to the surface of the (111) plane gives the cyclotron resonance of the
422
YOSHICHKA ~NUKIet al.
Pb I Ill t
10’ TIPPING
2 0” ANGLE t a
30’
40’-[Ill}
I
Fii. 3. Variation of the cyclotron masses measured at 68-70 GHz for the angles of field tipping ct.The co&oration is shown in the insert. The chain lines show mass anisotropy under the assumption that [llO] arms are perfect cylinders.
stationary orbit even for the field tipped geometry, as described in Section 1. It is important to first distinguish this type of reson~ce
from the above described branches
of resonances. When the sample surface of the (I 11) plane in the insert of Fig. 3 is replaced by the (112) surface as is shown in the insert of Fig. 4, the angle @is no longer a tipping angle from the surface, but becomes the angle in the (112) surface; the usual AKCR co~tion holds by varying B. In this case also the same mass branches will be obtained for the stationary orbits. In Fig. 4, thus obtained cyclotron masses are shown as functions of angle 0. In the figure, we can see the branches 5 of three equivalent [ 1lo] arms. The cyclotron mass of the orbit 1; varies appro~ately as a function of llcos 0, which is expected for a cylin~~ Fermi surface. A small deviation of the measured cyclotron mass from the function of llcos 6 can be explained by a slight swelling of the arm in the cross section of the (112) plane. From the comparison of cyclotron masses in these two geometries, it is concluded that the orbits i,[” and 6” in Figs. 3 and 4 are attributed to the stationary orbits of the Ill01 arms. The branch made by [” and {” in Fig. 3 corresponds to the branch having the minimum mass around 0 = 60” in Fig. 4. The lack of resonance point of this series between 30” and 38” in Fig. 3 is interpreted in terms of interference with the stronger resonances of [ and 5”. The branch 6 in Fig. 3 spreads over a wide range of tipping angles, because the orbit 5 is an orbit on an almost cylindrical arm, so that there is a large range of orbits with very small or zero drift velocities along the field direction. The branches t*, 5” and CCare found only in the field tipped configuration. The series 4” and f’ seems to form one branch, and the disappearance of the resonance between 12”and 18”may be due to interference with the series 6”. The branch formed by 5” and [”
probably corresponds to the cyclotron orbit on the [llO] arm in the electron Fermi surface, because the cyclotron masses of 5” and SCare very close to that of & However, the exact location of this orbit on the [l lo] arm is not yet determined. The series 4” is due to the Doppler-shifted cyclotron resonance mentioned in Section 1, because the cyclotron mass of this series is distinctive in its linear dependence on the tipping angle and its strong dependence on the microwave frequency as described in the next Section (b). (b) Frequency dependence. In order to distinguish whether the series of resonances CA,{” and 5’ is due to the stationary orbit or the non-stationary one, we have studied the results of similar experiments on the same surface using a frequency of 23 GHz instead of 70 GHz. As described in Section 1, the cyclotron resonance of the non-stationary orbit should have a frequency dependence. In Fig. 5, we show the typical recorder traces of the cyclotron resonance in field tipped geometry. There appear three series of resonances c, t” and Q. We show the dependence of the cyclotron masses in the (Ill) surface obtained at 23 GHz in Fig. 6. In the figure, the sotid line indicates the measured mass anisotropy at 23.19GHz, while the dashed line shows that at 68.54 GHz. The mass of the resonance 5 shows the same angular dependence for these two frequencies except in the region where cu= 0”. This is due to the fact that the orbit 6 is the s~tion~y orbit on the [IlO] arm. As for the mass of the resonance CA, the shit is linear and the slope of the branch at 23.19GHz is twice as large as the one at 68.98 GHz. The Doppler-shifted mechanism described in Section 1 predicts the frequency dependence of the resonance field as 1/0”~ in eqn (4). This predicted frequency dependence agrees well with the experiment on (J”. We fmd therefore that the orbit 5” is the nonstationary orbit on the [ 1lo] arm which has a drift velocity
Cyclotron resonance in lead-1
!1121 t
I
I
0'
30’ ANGLE
FROM
i
I
6 0’ CllOllN(
112)
90’ PLANE
( 0
)
Fig. 4. Variationof the cyclotronmassesin the (112)plane measured in parallel field geometry. The chain lines show mass anisotropy under the assumption that [ 1lo] arms are perfect cylinders. The angle 0 means the field angle in the (112) plane measured from the [110]axis, as shown in the insert. normal to the surface of the specimen and gives rise to the Doppler effect with incident microwave radiation. We must mention the apparent disagreement between the cyclotron mass 4 at 23.19GHz and the one at 68.54 GHz near a = 0”. For the fundamental resonance, the resonance field is not the peak field of the line shape. It is in fact somewhat lower than the peak field. On the other hand, for the higher harmonics, the resonance field coincides with the peak field, but the cyclotron mass in this case can not be obtained by the use of higher harmonics because the higher harmonics at 23.19 GHz are very small in number. Moreover, as the line shape of cyclotron resonance t at 23.19 GHz is broad in comparison with the one at 68.54 GHz and the resonance [” appears at the higher field side of the resonance l, the resonance field of 5 can not be. exactly determined and its
magnitude may be estimated as larger than the true value under the influence of the resonance lA. In the series IJ” and f’, we could find no resonance for the frequency 23 GHz. (c) Amplitude dependence. Using the frequency of 69GHz, we have studied the amplitude dependences of the resonances J; I” and Q on the tipping angle, as shown in Fig. 7. As the field is tipped from the surface, the resonance 5 decreases rapidly in amplitude in the following way: A a (sin (x)-‘**,where A is the amplitude. The amplitude of 5” does not depend very much on a. The resonance v’ having a small amplitude is hidden by the other resonances for the tipping angle beyond 7”. The amplitude dependence of the resonance on the tipping angle was examined previously in tin by Khaikin and Cheremisin[5]. Their results for the orbits E and 6 in the
YOSHICHIKA (~NUKI et al.
424
i
PCI
'i2
23.1Y GHz 1.51,K
(
Ill
’
Qz3.1”
I
2
/
3 MAGNETIC
I
1
L
5
FIELD
(
kC)e
/
)
Fig. 5. Typical recorder traces of cyclotron resonance at the frequency of 23.19GHz for two angles of field tipping. The angle LYhas the same meaning as in Fig. 2. The subscription number n for 5; Y and [” stands for the nth harmonic of AKCR.
-1”‘
0' TS PPING
20'
10’ ANGLE
f
a
30*41111
)
Fig. 6. Variation of the cyclotron mass measured at 23.19GHz for the angles of field tipping a. The angle a has the same meaning as in Fig. 2. The dashed line shows the variation of the masses 5 and 5” observed at 68.54GHz. The chain line shows mass anisotropy under the assumption that the [IlO] arm is a perfect cylinder.
third zone were A = (sin E)-‘~‘~~and for the orbit l in the fourth zone A 0~(sin a)~0.8r0.2. Our result for the resonance t is the same as that for the resonance of the orbit [ in tin 3.1.2 The (100) surface We have studied similar tipped field cyclotron resonance using the (100) surface. The mass dependence of the tipping angle is shown in Fig. 8. In order to distinguish resonance due to the stationary orbit from resonance due to the non-stationary ore, we have studied
the usual AKCR in the paraIle1field geometry with respect to the (110) surface as well as the (112) surface. The angular dependent cyclotron masses in the (110) plane are shown in Fig. 9. The cyclotron mass of the orbit 5 varies approximately as a function of l/cos 8. A small deviation of the measured mass from the function of l/cos B can be explained by a slight sing of the arm in the cross section of the (110) plane. The resonances 6” and 4” in Fig. 8 are not found in the parallel field geometry. Though the resonance 5” at the (100) surface shows a tipping angle dependence of cyclotron mass similar to the one at
425
Cyclotron resonance in lead--I -I
Pb
(111
and its cyclotron mass is smaller than that of the resonance 4’.As mentioned above, the location of the orbit I” is not yet clear.
)
3.2 Nonal field cyclotron resonance We have studied the cyclotron resonance in the magnetic field normal to the (100) surface of the crystal, on which the rf current is polarized in the [ 1lo] direction. In Fig. 10 are shown the recorder traces of the cyclotron resonance when the magnetic field is normal to the surface as well as when tipped a few degrees from the normal toward the [ 1lo] direction. We show in Fig. 11 the variation of measured cyclotron masses for the field angles a tipping from the [lOOIaxis on the (100) plane to the [ 1lo] axis. For the normal field geometry, we can find two series of resonances, of which the cyclotron masses are 1.25 m. and 1.30 mo. The resonance with the smaller mass is found only within Q < 5”,while the resonance with the larger mass has a larger amplitude and no mass anisotropy. Mina and Khaikin [ 131have reported that two resonances with the cyclotron masses of 1.25 mo and 1.30 m. are observed in lead in the parallel field geometry, and that the resonance giving the larger mass is independent of the orientation. They have suggested that the resonance with the mass 1.25 mo is due to the orbit v which surrounds the junction of four [IlO] arms as is shown in Fig. 12 (a). Our results in the field normal geometry agree well with the results obtained by Mina and Khaikin. Accordingly, the resonance with the smaller mass seems to arise from the orbit v. A peculiar feature of the line shape is observed for a very small value of a, as 0.4”in Fig. 10.That is, each of the even numbered harmonics is much reduced in amplitude, while odd numbered harmonics remain strong. The tipping of the magnetic field from the [NO] direction
0.82 ---Ac~(s~na)
_I 0'
10’ TIPPING
ANGLE
20’ CCX
--[Ill]
I
Fig. 7. Dependence of the amplitude A of resonances J, LAand I on the tipping angle a. The angle (I has the same meaning as in Fig. 2. The subscription number n for J, I” and \Y stands for the nth harmonic of AKCR.
the (111) surface, the branch of 4’”in Fig. 8 is not so linear as the one in Fig. 3. There is a large difference between the resonance 5” at the (100) surface and the one at the (111) surface. The resonance I” is found in the range a > 10” c7c
/-
1 Pb
(100 1
0.6 5
*v
E 0 60
0 52
1
0'
I
10’ TIPPING
I
I
I
20”
30’
40’
ANGLE
(
o(
+
[IO01
)
Fig. 8. Variation of the cyclotron masses measured at 6g-70 GHz for the angles of field tipping a from the [llO] axis in the (100) surface to the [lo01 axis as shown in the insert. The chain lines show mass anisotropy under the assumption that [llO] arms are perfect cylinders.
YOSHICHIKA
426
a* ANGLE
~NUKI
et at.
30’ FH’JM
Cl101
60’ IN
(110)
PLANE
90’
( @)
Fig. 9. Variation of the cyclotron masses in the (110) plane measured in parallel field geometry. The chain lines show mass anisotropy under the assumption that [I IO] arms are perfect cylinders.The angle 13is measured from the [I 101 axis in the (110) plane as shown in the insert.
within 2” more or less maintains this feature of the line shape. However, beyond 2” the difference between the amplitudes of the even and odd numbered harmonics is not appreciable. This feature can be interpreted in terms of the cyclotron resonance of the skipping orbit as follows. In this orientation of the magnetic field, the cyclotron resonance of the orbit v is expected to be a skipping orbit resonance. This is because in the presence of a magnetic field, an electron on the orbit v moves to make a wavy trajectory in real space, which is topologically similar to a baseball seam, as shown in Fig. 12(b). Therefore the electron near the surface comes into and then goes out the skin layer twice per revolution. The electron interacts twice with the radiation field in the course of one cyclotron period. In this case, the AKCR condition is also satisfied. But the resonance condition for the harmonics is as follows. Considering a closed trajectory of non-flat and wavy shape in which the number of waves is n and the waves are equispaced; if the median plane of the trajectory is parallel to the sample surface, there are n skipping-in traversals through the skin layer and the resonance condition becomes U/O, = 1+ qn,
(5)
where n is an integer. The resonance of q = 1, which has been observed in tin by Koch and Kip [ 161,is equivalent to
the usual AKCR for the tilted orbit. Walsh[17] has observed the resonance of r) = 3 for the octahedronal hole Fermi surface in tungsten. With lead, the orbit v corresponds to the case u = 2. According to eqn (5) in our case, i.e. W/W, = 1+2n, the odd numbered harmonic resonances are observable, while even numbered ones are not. The resonances observed for the (100) plane prove this characteristic in a distinct manner as is shown in Fig. 10. This type of resonance should be very sensitive to the angle of field tipping from the [ 1001axis on the (100)plane to the [ 1lo] axis. The condition, whereby the electron on the baseball seam trajectory can enter into the skin layer twice per cyclotron period is satisfied within an angle S/r, =4.5”, if the value for anomalous skin depth S at 1.5 K is taken as 0.2’76pm and the value for the cyclotron radius of the orbit v at fourth harmonic r, as 3.4 pm. Our experimental result is in approximate agreement with the above estimation. Another resonance series which is observed in the norma field geometry is one giving a larger mass, ml = 1.30mo. This resonance also has the characteristic of the skipping orbit. We believe that the orbit responsible for this resonance should have a non-flat and wavy trajectory. However, we can not yet assign this resonance to any orbit on the Fermi surface; it is neither the “doughnut electron” (orbit 7) nor “doughnut hole” (orbit 5) in the (100) plane. A possibility is that the responsible
Cyclotron
421
resonance in lead-1
Pb(100) 1~69.1 Gtiz 1. 1.5K
dF G
Jrf * H 5
median
pldne
[1101
Fig. 12(a). Orbit Y around the junction of four [110] and its equivalent arms. The orbit is flat in the k space. (b) Real space trajectory of the orbit v. The trajectory is similar to a baseball seam. An electron rises and falls twice per revolution along the trajectory.
0
I
I
5
10
HC kOe)
Fig. 10.Typicalrecordertracesofcyclotronresonancefortheangles of field tipping (I,where (I is measured from the [ loO]axis toward the [llO] axis in the (100) surface, as shown in the insert of Fig. 1 I.
0.8
’
I
-10' ANGLE
FR3h'
I IO"
I
0" [I251 of ~130
PLANE
(Q)
Fig. 11.Variation of the measured cyclotron masses for the angles of field tipping(I, where (xis measured from the [loO]axis toward the [110]axis in the (100) plane.
orbit is located also around the junction of four [llO] armst 4. CONSlDERATlON OF THE [llO] ARh4 FERhU SURFACE
4.1 Field tipping e#ect for a cylindtical Fermi surface We have mentioned in Section 3.1 that the resonance 5
tA heat pattern of about 600 cycle is observed by Anderson and Gold]61 in the dHvA oscillation for the orbit Y.
is found for the large tipping angle (0’40’) regardless of the surface of the sample. It arises from the fact that the [llO] arm, to which the orbit 5 is related, has an approximately cylindrical surface. Therefore an electron on the orbit 5 has no velocity component along the axis of the cylinder and does not drift even if the magnetic field is tilted from the cylindrical axis. For example, when the field is inclined 30” from the axis of the arm, the drift velocities of the orbits near C are evaluated from the corrugated surface of the [ 1lo] arm described in the next section as l/SCrl/lOO of the Fermi velocity VF.Thus, the tipping angle of 2” in copper corresponds to 20”-30” in lead. This is the reason why the cyclotron resonance of the stationary orbit 5 can be observed over a large range of tipping angles. Moreover, the fact that the resonance 5” is found for a large range of tipping angles (tY’-30”)will be also explained by this interpretation. 4.2 Doppler-shifted cyclotron resonance (5”) From the experiment of the frequency dependence of the cyclotron mass, it is quite clear that the resonance f” is the Doppler-shifted cyclotron resonance of the non-stationary orbit. Furthermore, it should have a close relation to the fine structure of the [ 1lo] arm, especially its cross sectional variation along k, in the [ 1lo] direction. As noted in Section 1, Anderson and Gold[6] have proposed a model of the Fermi surface of the [llO] arm based on observation of a long beat of the y oscillation in the dHvA effect; the y oscillation corresponds to the orbit 5 in the present cyclotron resonance experiment. However this model has been criticized by Van Dyke [ lo], who has proposed a modification for the Fermi surface of the [l lo] arm. This is characterized by a corrugated surface with one maximum cross section labeled CLat L
428
YOSHICHIKAGNUKI et al.
Fig. 13(a). Corrugated Fermi surface of the [llO] arm proposed by Van Dyke. (b) Variation of the drift velocities along the [I lo] direction.
(or K) and minimum cross sections labeled & somewhere nearer W in the both sides of lL as is shown in Fig. 13. The corrugation, i.e. the difference of the cross sectional areas between the orbit & and & is about two per cent. In other words, the calipers length of the orbit & is about one percent smaller than that of &, although it is shown four times larger in the figure. In our cyclotron resonance experiment carefully made in a parallel field configuration, we have found only a resonance of the orbit 5 on the [ 1lo] arm. The location of this orbit 5 (0.542 mo)must be the center of the [ 1lo] arm, i.e. at U (or K) for the following reason: its resonance is observed for angles within 50” from the [ 1lo] direction in the parallel field configuration with the use of the (100) plane, which is not shown in this paper, and for the large range of tipping angles up to 40” in the tipped field configuration. Therefore the orbit 5 corresponds to the orbit iL. Considering the experimental fact that the resonance l has a Chambers’ mass minimum line shape[l8], the cyclotron effective mass m?(k) of the [llO] arm possibly increases monotonously with the variation of kH from U to W and the cyclotron mass of the orbit cs does not become extremal with respect to kH. Next, we should like to determine the location of the orbit J”(O.565mo) on the [llO] arm on the basis of the corrugated Fermi surface. If small variations in cyclotron mass along the [ 1lo] axis can be neglected, the resonance occurs for the orbit with an extremal drift velocity, according to eqn (4). There should exist such an orbit between the orbit f; and the orbit &, because neither the orbit lL nor the orbit & has drift velocity along the [ 1101 direction, as shown in Fig. 13. Taking account of small variations of cyclotron mass in the real Fermi surface, the location of the orbit I” should be shifted slightly from the extremal point of the drift velocity. When the magnetic field is exactly parallel to the surface of the specimen, this orbit does not give rise to the resonance because it does not have the component of drift velocity normal to the surface of the specimen. When the field is tipped from the axis of the arm and from the surface of the specimen, we
can expect resonance. The location of 5” with respect to the tipping angle will vary in a complicated way by virtue of the change of the drift velocity and of the cyclotron mass. In fact, as shown in Fig. 3, the mass of 5” is proportional to the tipping angle within 20” and then increases rapidly beyond 20”. This feature is not interpreted by the angular dependence of the mass derived from eqn (4). As the resonance 5” is observed in a large tipping angle up to 30”, the orbit 5” must exist between U and C; C means the center point of the most inclined orbit (30”) as shown in Fig. 13. Ogawa and Aoki[9] have recently clarified from an analysis of the long beat in y oscillation of the dHvA effect that the orbits giving higher and lower frequency oscillations correspond to the maximum cross sectional area at U (or K) and the minimum one somewhere between U and W, respectively. They have also determined the cyclotron mass 0.55 ma for & and 0.57 m. for & from the temperature dependence of the amplitude of the dHvA oscillation. These values are in good agreement with our experimental results. Finally, we must mention the appearance of the resonance 5” at a = 0” as shown in Fig. 3 contrary to our expectation that the Doppler-shifted cyclotron resonance should not be observed in the parallel field configuration. We hypothesize two causes for it. The first is that the magnetic field was not directed exactly parallel to the surface of the specimen, i.e. tilted from the surface within 0.2”. The second is that the surface produced by electropolishing was microscopically smooth but was rather wavy on a macroscopic scale. This means that local regions of the surface are inclined to the magnetic field even when the average plane of the surface is perfectly aligned. In fact, by the use of a sample grown in a quarts mould with a very flat and polished surface, no resonance 5” has been reported at (Y= 00[19]. Acknowledgements--It is our pleasure to acknowledge the advice, suggestions and comments of Prof. M. Tsuji of the University of Kyushu. We should like to thank Dr. K. Ogawa and Mr. H. Aoki
Cyclotron resonance in lead-1 of the National Research Institute for Metals for informing us of the results of their experimental study prior to publication. RBFERENcEs
1. Walsh W. M., Jr., Solid State Physics (The Simon Fraser University Lecture) Electrons in Metals (Edited by J. F. Cockran and R. R. Haering), Vol. 1, p. 127. Gordon & Beach, New York (1968). 2. Azbel’ M. Ya. and Kaner E. A., Sou. Phys.-JETP 5, 730 (1957);J. Phys. Chem. Solids 6, 113 (1958). 3. Koch J. F., Stradling R. A. and Kip A. F., Phys. Rev. 6, A240 (1964). 4. Howard D. G., Phys. Reo. 29A, 1705(1965). 5. Khaikin M. S. and Cheremisin S. M., Sou. Phys.-JETP 27,38 (1968).
429
6. Anderson J. R. and Gold A. V., Phys. Rev. JOA, 1459(1%5). 7. Anderson J. R. and Hines D. C., Phys. Reu. B2,4752 (1970). 8. Anderson J. R. and Lee J. Y. M., Phys. Rev. Bll, 1308(1975). 9. Ogawa K. and Aoki H., private Communication. 10. Van Dyke J. P., Phys. Rev. B7, 2357 (1973). 11. Young R. C., Phil. &fag. 7, 2065 (1%2). 12. Khaikin M. S. and Mina R. T., Sou. Phys.-JETP 15,24 (1%2). 13. Mina R. T. and Khaikin M. S., Sou. Phys.-JEITP 18, 8% (1964). 14. Tegart W., ‘I’heElectrolytic and Chemical Polishing of Metals, p. iO1. Pergamon Press, London (1959). 15. Snona F. W. and KID A. F.. Phvs. Rev. 137A.431 (1%5). 16. I&&J. F. and Kip ‘A. F., Phys. Reu. Letters 8, 473 (1962). 17. Walsh M., Jr., Phys. Reu. Letters 12, 161 (1964). 18. Chambers R. G., Proc. Phys. Sot. 86, 305 (1%5). 19. Goy P. and Castaing B., Phys. Rev. Bl, 4409 (1973).
Pergamon Press.
Printed in Great Britain
CYCLOTRON RESONANCE IN LEAD-I TIPPING EFFECT OF MAGNETIC FIELD YOSHICHIKA~NUKI, HIROYOSHISUEMATSUand SEI-ICHITANUMA The Institutefor Solid State Physics, The University of Tokyo, Roppongi,Minato-ku,Tokyo 106,Japan
(Received26 May 1976;accepted
in
revisedform 20 August 1976)
Abstract-The cyclotronresonancesin lead in fieldtippedgeometryhave been observedfor largetippinganglesup to 40”from the samplesurfacesof the (111)and (100)planes. In additionto the resonanceof the orbit5 on a cylindrical armof the electronFermisurface,two new series of resonances,lA and 5” have been found, of which the cyclotron massesare very close to thatof [ but whichdiier slightlyin tippingangledependence.The massof lA is dependenton microwavefrequency,so that5” is attributedto the Doppler-shiftedcyclotronresonance of a non-stationaryorbitnear[, whose locationis discussedin relationto the Fermisurfacemodelof Van Dyke. In fieldgeometrynormalto the (100) surface,two series of resonanceshave been observed,which are characterizedas the extinctionof even numbered harmonicresonances.Thiseffectarisesfromthe skipping orbit whichhas a trajectory topologicallysimilarto a baseball seam in real space. One of these series is attributedto the orbit Y aroundthe junctionof four electron arms.
1. lNlRODUcIlON
Many experimental works on Azbel’-Kaner cyclotron resonance (AKCR)[ 11 have been reported in various metals since the original theory[2] predicted the effect. The ordinary AKCR is observed when a steady magnetic field is applied exactly parallel to the surface of the specimen. The resonance occurs when the microwave frequency o is an integral multiple of the cyclotron frequency o,, i.e. w = nw,,
(1)
where n is an integer and wc = eH/m?c includes the cyclotron effective mass m?. The mass rn? is related to the electronic structure of the relevant metal in the following manner:
Kaner cyclotron resonance, and in general can be observed for a large tipping angle. The other comes from the non-stationary orbit which has a drift velocity component normal to the surface of the specimen and gives rise to Doppler effect with the incident microwave radiation. This type of resonance is observed for a small tipping angle of less than a few degrees. The cyclotron resonance of the non-stationary orbit has been studied in detail by Koch et al. [3]. If an orbit has the driit velocity uD along the magnetic field, the carrier feels effectively the rf field having a frequency dBerent from the incident microwave frequency o by an amount of uDsin a/S, where (Yis the tipping angle of the magnetic field from the metal surface and 6 is the skin depth in the anomalous limit. Hence, the resonance frequency is Doppler-shifted in the following manner: 0 ? v. sin aIS = no,,
(3)
or where EF is the Fermi energy, S(~H,E) is a crosssectional area of an equi-energy surface of energy E occupying the k space in the plane normal to the magnetic field H, and kH is the longitudinal component of the wave vector, i.e. k,, = k. H/IHI. In general, the cyclotron resonance occurs for the orbit, of which the cyclotron mass mf(k”) is extremal with respect to kH, i.e. am ?/ak,
= 0.
On the other hand, the cyclotron resonance in the magnetic field tipped with respect to the surface of the specimen has been observed in some metals [3-51 in spite of the fact that Azbel’ and Kaner have predicted the cyclotron resonance would vanish for a very small tipping angle (typically about 5’). The observed cyclotron resonances in field tipped geometry are classified as two types of resonances. One is attributed to the stationary orbit, of which the drift velocity is zero along kH corresponding to the tipped magnetic field. This type of resonance is essentially the same as the usual Azbel’-
~/~=(~)(l’(~)“‘(~)sina},
(4)
where p is the resistivity and I is the mean free path. The resonance field in eqn (4) varies according to the change of mf(kH) and uD(kH) which are functions of kH. The cyclotron resonance occurs therefore for the orbit, for which the right hand of eqn (4) is extremal with respect to k,,. If the cyclotron mass is assumed to be constant over the variation of kH under consideration, the cyclotron resonance is only expected for the orbit, whose drift velocity uD(kH) is extremal with respect to k,,. In addition to these resonances, distinct series of resonances have been observed in an extremely oblique configuration of the magnetic field, i.e. where the magnetic field is normal to the sample surface. When a cyclotron orbit has a non-flat and wavy closed trajectory in real space, the crests of waves may skip in the skin layer and feel the rf field. If such a trajectory has two 419
420
YOSHKHIKA ~NUKI et al.
crests and two troughs, i.e. has a shape similar to a baseball seam, an electron on the orbit interacts twice with the rf field at these crests during one cyclotron period. In this case, we can expect a cyclotron resonance similar to the ordinary AKCR type, but lacking the even numbered harmonics. In this paper, we describe observations of the cyclotron resonance in lead in various configurations including the field geometry of oblique angles up to 90”, and discuss the Fermi surface of lead. The Fermi surface of lead has been extensively investigated by Anderson and Gold[6] using the de Hass van Alphen (dHvA) effect. They have obtained a good agreement of experimentally determined Fermi surface geometry with the four orthogonalized plane waves representation. The Fermi surface of lead consists of two sheets, one of which is a large closed hole surface centered at F in the second zone and the other is a multiply connected electron surface in the third zone constructed by cylindrical arms in the [1lo] and its equivalent directions as shown in Fig. 1. Among many orbits describable as the peripheries of cross sections of a Fermi surface normal to the applied field, the following three orbits are treated in this report. These are shown in Fig. 1 as the orbits \y on the hole surface, 5 on the [llO] arm of electron surface and v around the junction of four [llO] arms; these orbits correspond to the dHvA oscillations (Y, y and p [6], respectively. As for the [ 1lo] arm of the electron surface, Anderson and Gold have proposed that this arm has a narrow minimum cross section at the point U (or K) and a coo11 I
region of inflection of the cross sectional area midway between U and W; this model predicts two kinds of dHvA frequencies differing from each other by about two percent. The recent studies of dHvA effect by Anderson et al.[7,8] and more recently by Ogawa et al.[9], and theoretical calculations by Van Dyke[lO] have revealed that the earlier model of Anderson and Gold for this arm should be modified, i.e. it should have the maximum cross section at U and the minimum somewhere between U and W to make a corrugated cylinder. However, this problem has not been solved. Previous works on cyclotron resonances [ 1l-131 in field geometry parallel to the surface give no information about the above mentioned fine structure in the [llO] arm, although almost all of the orbits found in dHvA effect have already been observed in the cyclotron resonance. One of our purposes is to investigate the problem by means of cyclotron resonance in field tipped geometry. In Section 2, we describe our experimental setup as well as the preparation of the samples. In Section 3.1, the cyclotron resonance data are described for large tipping angles of 0” to 40” from the (111) and (100) surfaces. To understand the tipping effect, we have studied the usual AKCR in the parallel field geometry using the (112) and (110) surfaces, respectively. Distinct series of resonances have been observed which were not reported in previous papers. One of them is explained in terms of the cyclotron resonance of the non-stationary orbit by the use of two different microwave frequencies. In Section 3.2, the cyclotron resonance in field geometry normal to the (100) surface is presented, and the phenomenon is attributed to the skipping resonance of the orbit v which has a trajectory somewhat similar to a baseball seam. In Section 4, discussions on the nature of relevant Fermi surface are presented. 2. EXPERIMENTAL. PROCEDURES
(a)
1
(b)
Fig. l(a). Hole Fermi surface in lead in the second Brillouin zone. (b) Electron Fermi surface in the third Brillouin zone.
single crystals were grown from material of 99.9999% purity (Osaka Asahi Metal Co.) in a graphite crucible by means of the Czochralski pulling method. The orientation of the sample was determined by X-ray within an accuracy of 1”.Thin parallelepiped specimens (8 x 8 x 1 mm’) were cut by a Servomet spark machine. The surface damage of the spark cut was removed by lapping the specimens on a cloth which was wetted with a solution of three parts hydrogen peroxide and seven parts acetic acid. The surface was then made microscopically smooth by electro-polishing with a solution of perchloric acid and acetic anhydride using the method reported by Tegart [ 141. The residual resistance ratio (~~~Jp~.~a)for such samples was (6-9) X lo-‘. The measurements were carried out at 1SK using a standard reflection type spectrometer of 23 GHz (TE, II mode) and a reaction type of 70GHz (TE,12mode). The magnetic field was modulated at an audio frequency and the field derivative of power absorption was detected by a lock-in amplifier. In order to permit an accurate and easy rotation of the magnetic field up to large tipping angles, we used an electromagnet whose field direction was rotatable in the horizontal plane, and the cylindrical cavity in which the rf electric field was directed vertically. The sample The
Cyclotron resonance in lead-l
was also attached vertically on the end wall of the cavity with silver cement.
Pb 11111 68 5L GHz 1.5LK
Jrt I H
3. EXPERIMENTAL RFSULTS ANDANALYSIS
3.1 Tipped field cyclotron resonance The cyclotron resonance in field tipped geometry have been obtained as a function of the tipping angle with respect to the sample surfaces of the (111) and (100) planes. These surfaces are used in order to simplify the effect of the field tipping. In these instances, at least one of the cylindrical [llO] arms is parallel to the sample surface; its resonance is not confused with those of the other equivalent arms, because the latter arms make a large angle with both the surface and the magnetic field. The rj current of the microwave is always transverse to the field direction, as shown in the inserts of Fig. 3 and Fig. 8. First, we present the results for the (111) surface. 3.1.1 The (111) surface (a) Tipping angle dependence of the cyclotron resonance. The typical recorder traces of the cyclotron resonance obtained at 70 GHz are shown in Fig. 2 for the (111) surface and for various angles (Yof the magnetic field tipped from the [ 1lo] axis in the (112) plane as shown in the insert of Fig. 3. The cyclotron masses are determined from the period of the resonance peaks, and their dependences on a are shown in Fig. 3. For a = O”,we observe two dominant series, one having a large amplitude and a cyclotron mass of rn?l= 0.54m0, and the other having a small amplitude and rn? = 1.14m0 which is shown not in Fig. 3 but in Fig. 4. These resonances are attributed to the 5 and \y orbits respectively. These cyclotron masses are in good agreement with those found by previous measurements[ll-131. In addition to these resonances, a third, previously unreported, series of weak resonances, labeled fA, is found as shown in Fig. 2. The cyclotron mass of 5” is 0.565m0, which is very close to that of & As the field is tipped from the surface, the resonances 5 and 5” retain appreciable amplitudes for tipping angles up to 30”. As shown in Fig. 2, for a = 10” there appears another kind of resonance, labeled tB, between 4’and g*. This is hidden for a 2 13”by virtue of the large amplitude of 5^. For tipping angles larger than 20”, two new series of resonances, labeled I’ and 5” appear. The amplitude of the resonance I” is very small in comparison with the others. The cyclotron masses of the series [, LA,4’”and 6’ increase with the increase of a, while that of 5” decreases. The branch 5 and g’ remain large in amplitude for tipping angles up to 38”,while the branch 5” decreases rapidly in amplitude beyond 28”. For the narrow a region of 33”-35”,where the dominant resonances are f and lc, the line shape changes drastically; the peaks become dull, while the troughs become sharp with increasing a as shown in Fig. 2. Similar phenomena have been reported in copper [3] and other metals[4, IS]. In the region of 35”-38”, the resonances become very weak and the line shapes are so complicated that we cannot assign them firmly. Beyond 38”, the line shape returns to that of the ordinary AKCR,
421
dR i-i
I Fig. 2. Typical recorder traces of cyclotron resonance at the frequency of 68.54GHz for various angles of field tipping. The notation Q is the angle of field tipping from the [I 101axis in the (111)surface toward the [ill] axis as shown in the insert of Fig. 3. The subscription number n for f; Vrand et al. stands for the nth harmonic of AKCR.
and a new series of resonance appears, which is labeled C”* The fact that the [ 1lo] arm is parallel to the surface of the (111) plane gives the cyclotron resonance of the
422
YOSHICHKA ~NUKIet al.
Pb I Ill t
10’ TIPPING
2 0” ANGLE t a
30’
40’-[Ill}
I
Fii. 3. Variation of the cyclotron masses measured at 68-70 GHz for the angles of field tipping ct.The co&oration is shown in the insert. The chain lines show mass anisotropy under the assumption that [llO] arms are perfect cylinders.
stationary orbit even for the field tipped geometry, as described in Section 1. It is important to first distinguish this type of reson~ce
from the above described branches
of resonances. When the sample surface of the (I 11) plane in the insert of Fig. 3 is replaced by the (112) surface as is shown in the insert of Fig. 4, the angle @is no longer a tipping angle from the surface, but becomes the angle in the (112) surface; the usual AKCR co~tion holds by varying B. In this case also the same mass branches will be obtained for the stationary orbits. In Fig. 4, thus obtained cyclotron masses are shown as functions of angle 0. In the figure, we can see the branches 5 of three equivalent [ 1lo] arms. The cyclotron mass of the orbit 1; varies appro~ately as a function of llcos 0, which is expected for a cylin~~ Fermi surface. A small deviation of the measured cyclotron mass from the function of llcos 6 can be explained by a slight swelling of the arm in the cross section of the (112) plane. From the comparison of cyclotron masses in these two geometries, it is concluded that the orbits i,[” and 6” in Figs. 3 and 4 are attributed to the stationary orbits of the Ill01 arms. The branch made by [” and {” in Fig. 3 corresponds to the branch having the minimum mass around 0 = 60” in Fig. 4. The lack of resonance point of this series between 30” and 38” in Fig. 3 is interpreted in terms of interference with the stronger resonances of [ and 5”. The branch 6 in Fig. 3 spreads over a wide range of tipping angles, because the orbit 5 is an orbit on an almost cylindrical arm, so that there is a large range of orbits with very small or zero drift velocities along the field direction. The branches t*, 5” and CCare found only in the field tipped configuration. The series 4” and f’ seems to form one branch, and the disappearance of the resonance between 12”and 18”may be due to interference with the series 6”. The branch formed by 5” and [”
probably corresponds to the cyclotron orbit on the [llO] arm in the electron Fermi surface, because the cyclotron masses of 5” and SCare very close to that of & However, the exact location of this orbit on the [l lo] arm is not yet determined. The series 4” is due to the Doppler-shifted cyclotron resonance mentioned in Section 1, because the cyclotron mass of this series is distinctive in its linear dependence on the tipping angle and its strong dependence on the microwave frequency as described in the next Section (b). (b) Frequency dependence. In order to distinguish whether the series of resonances CA,{” and 5’ is due to the stationary orbit or the non-stationary one, we have studied the results of similar experiments on the same surface using a frequency of 23 GHz instead of 70 GHz. As described in Section 1, the cyclotron resonance of the non-stationary orbit should have a frequency dependence. In Fig. 5, we show the typical recorder traces of the cyclotron resonance in field tipped geometry. There appear three series of resonances c, t” and Q. We show the dependence of the cyclotron masses in the (Ill) surface obtained at 23 GHz in Fig. 6. In the figure, the sotid line indicates the measured mass anisotropy at 23.19GHz, while the dashed line shows that at 68.54 GHz. The mass of the resonance 5 shows the same angular dependence for these two frequencies except in the region where cu= 0”. This is due to the fact that the orbit 6 is the s~tion~y orbit on the [IlO] arm. As for the mass of the resonance CA, the shit is linear and the slope of the branch at 23.19GHz is twice as large as the one at 68.98 GHz. The Doppler-shifted mechanism described in Section 1 predicts the frequency dependence of the resonance field as 1/0”~ in eqn (4). This predicted frequency dependence agrees well with the experiment on (J”. We fmd therefore that the orbit 5” is the nonstationary orbit on the [ 1lo] arm which has a drift velocity
Cyclotron resonance in lead-1
!1121 t
I
I
0'
30’ ANGLE
FROM
i
I
6 0’ CllOllN(
112)
90’ PLANE
( 0
)
Fig. 4. Variationof the cyclotronmassesin the (112)plane measured in parallel field geometry. The chain lines show mass anisotropy under the assumption that [ 1lo] arms are perfect cylinders. The angle 0 means the field angle in the (112) plane measured from the [110]axis, as shown in the insert. normal to the surface of the specimen and gives rise to the Doppler effect with incident microwave radiation. We must mention the apparent disagreement between the cyclotron mass 4 at 23.19GHz and the one at 68.54 GHz near a = 0”. For the fundamental resonance, the resonance field is not the peak field of the line shape. It is in fact somewhat lower than the peak field. On the other hand, for the higher harmonics, the resonance field coincides with the peak field, but the cyclotron mass in this case can not be obtained by the use of higher harmonics because the higher harmonics at 23.19 GHz are very small in number. Moreover, as the line shape of cyclotron resonance t at 23.19 GHz is broad in comparison with the one at 68.54 GHz and the resonance [” appears at the higher field side of the resonance l, the resonance field of 5 can not be. exactly determined and its
magnitude may be estimated as larger than the true value under the influence of the resonance lA. In the series IJ” and f’, we could find no resonance for the frequency 23 GHz. (c) Amplitude dependence. Using the frequency of 69GHz, we have studied the amplitude dependences of the resonances J; I” and Q on the tipping angle, as shown in Fig. 7. As the field is tipped from the surface, the resonance 5 decreases rapidly in amplitude in the following way: A a (sin (x)-‘**,where A is the amplitude. The amplitude of 5” does not depend very much on a. The resonance v’ having a small amplitude is hidden by the other resonances for the tipping angle beyond 7”. The amplitude dependence of the resonance on the tipping angle was examined previously in tin by Khaikin and Cheremisin[5]. Their results for the orbits E and 6 in the
YOSHICHIKA (~NUKI et al.
424
i
PCI
'i2
23.1Y GHz 1.51,K
(
Ill
’
Qz3.1”
I
2
/
3 MAGNETIC
I
1
L
5
FIELD
(
kC)e
/
)
Fig. 5. Typical recorder traces of cyclotron resonance at the frequency of 23.19GHz for two angles of field tipping. The angle LYhas the same meaning as in Fig. 2. The subscription number n for 5; Y and [” stands for the nth harmonic of AKCR.
-1”‘
0' TS PPING
20'
10’ ANGLE
f
a
30*41111
)
Fig. 6. Variation of the cyclotron mass measured at 23.19GHz for the angles of field tipping a. The angle a has the same meaning as in Fig. 2. The dashed line shows the variation of the masses 5 and 5” observed at 68.54GHz. The chain line shows mass anisotropy under the assumption that the [IlO] arm is a perfect cylinder.
third zone were A = (sin E)-‘~‘~~and for the orbit l in the fourth zone A 0~(sin a)~0.8r0.2. Our result for the resonance t is the same as that for the resonance of the orbit [ in tin 3.1.2 The (100) surface We have studied similar tipped field cyclotron resonance using the (100) surface. The mass dependence of the tipping angle is shown in Fig. 8. In order to distinguish resonance due to the stationary orbit from resonance due to the non-stationary ore, we have studied
the usual AKCR in the paraIle1field geometry with respect to the (110) surface as well as the (112) surface. The angular dependent cyclotron masses in the (110) plane are shown in Fig. 9. The cyclotron mass of the orbit 5 varies approximately as a function of l/cos 8. A small deviation of the measured mass from the function of l/cos B can be explained by a slight sing of the arm in the cross section of the (110) plane. The resonances 6” and 4” in Fig. 8 are not found in the parallel field geometry. Though the resonance 5” at the (100) surface shows a tipping angle dependence of cyclotron mass similar to the one at
425
Cyclotron resonance in lead--I -I
Pb
(111
and its cyclotron mass is smaller than that of the resonance 4’.As mentioned above, the location of the orbit I” is not yet clear.
)
3.2 Nonal field cyclotron resonance We have studied the cyclotron resonance in the magnetic field normal to the (100) surface of the crystal, on which the rf current is polarized in the [ 1lo] direction. In Fig. 10 are shown the recorder traces of the cyclotron resonance when the magnetic field is normal to the surface as well as when tipped a few degrees from the normal toward the [ 1lo] direction. We show in Fig. 11 the variation of measured cyclotron masses for the field angles a tipping from the [lOOIaxis on the (100) plane to the [ 1lo] axis. For the normal field geometry, we can find two series of resonances, of which the cyclotron masses are 1.25 m. and 1.30 mo. The resonance with the smaller mass is found only within Q < 5”,while the resonance with the larger mass has a larger amplitude and no mass anisotropy. Mina and Khaikin [ 131have reported that two resonances with the cyclotron masses of 1.25 mo and 1.30 m. are observed in lead in the parallel field geometry, and that the resonance giving the larger mass is independent of the orientation. They have suggested that the resonance with the mass 1.25 mo is due to the orbit v which surrounds the junction of four [IlO] arms as is shown in Fig. 12 (a). Our results in the field normal geometry agree well with the results obtained by Mina and Khaikin. Accordingly, the resonance with the smaller mass seems to arise from the orbit v. A peculiar feature of the line shape is observed for a very small value of a, as 0.4”in Fig. 10.That is, each of the even numbered harmonics is much reduced in amplitude, while odd numbered harmonics remain strong. The tipping of the magnetic field from the [NO] direction
0.82 ---Ac~(s~na)
_I 0'
10’ TIPPING
ANGLE
20’ CCX
--[Ill]
I
Fig. 7. Dependence of the amplitude A of resonances J, LAand I on the tipping angle a. The angle (I has the same meaning as in Fig. 2. The subscription number n for J, I” and \Y stands for the nth harmonic of AKCR.
the (111) surface, the branch of 4’”in Fig. 8 is not so linear as the one in Fig. 3. There is a large difference between the resonance 5” at the (100) surface and the one at the (111) surface. The resonance I” is found in the range a > 10” c7c
/-
1 Pb
(100 1
0.6 5
*v
E 0 60
0 52
1
0'
I
10’ TIPPING
I
I
I
20”
30’
40’
ANGLE
(
o(
+
[IO01
)
Fig. 8. Variation of the cyclotron masses measured at 6g-70 GHz for the angles of field tipping a from the [llO] axis in the (100) surface to the [lo01 axis as shown in the insert. The chain lines show mass anisotropy under the assumption that [llO] arms are perfect cylinders.
YOSHICHIKA
426
a* ANGLE
~NUKI
et at.
30’ FH’JM
Cl101
60’ IN
(110)
PLANE
90’
( @)
Fig. 9. Variation of the cyclotron masses in the (110) plane measured in parallel field geometry. The chain lines show mass anisotropy under the assumption that [I IO] arms are perfect cylinders.The angle 13is measured from the [I 101 axis in the (110) plane as shown in the insert.
within 2” more or less maintains this feature of the line shape. However, beyond 2” the difference between the amplitudes of the even and odd numbered harmonics is not appreciable. This feature can be interpreted in terms of the cyclotron resonance of the skipping orbit as follows. In this orientation of the magnetic field, the cyclotron resonance of the orbit v is expected to be a skipping orbit resonance. This is because in the presence of a magnetic field, an electron on the orbit v moves to make a wavy trajectory in real space, which is topologically similar to a baseball seam, as shown in Fig. 12(b). Therefore the electron near the surface comes into and then goes out the skin layer twice per revolution. The electron interacts twice with the radiation field in the course of one cyclotron period. In this case, the AKCR condition is also satisfied. But the resonance condition for the harmonics is as follows. Considering a closed trajectory of non-flat and wavy shape in which the number of waves is n and the waves are equispaced; if the median plane of the trajectory is parallel to the sample surface, there are n skipping-in traversals through the skin layer and the resonance condition becomes U/O, = 1+ qn,
(5)
where n is an integer. The resonance of q = 1, which has been observed in tin by Koch and Kip [ 161,is equivalent to
the usual AKCR for the tilted orbit. Walsh[17] has observed the resonance of r) = 3 for the octahedronal hole Fermi surface in tungsten. With lead, the orbit v corresponds to the case u = 2. According to eqn (5) in our case, i.e. W/W, = 1+2n, the odd numbered harmonic resonances are observable, while even numbered ones are not. The resonances observed for the (100) plane prove this characteristic in a distinct manner as is shown in Fig. 10. This type of resonance should be very sensitive to the angle of field tipping from the [ 1001axis on the (100)plane to the [ 1lo] axis. The condition, whereby the electron on the baseball seam trajectory can enter into the skin layer twice per cyclotron period is satisfied within an angle S/r, =4.5”, if the value for anomalous skin depth S at 1.5 K is taken as 0.2’76pm and the value for the cyclotron radius of the orbit v at fourth harmonic r, as 3.4 pm. Our experimental result is in approximate agreement with the above estimation. Another resonance series which is observed in the norma field geometry is one giving a larger mass, ml = 1.30mo. This resonance also has the characteristic of the skipping orbit. We believe that the orbit responsible for this resonance should have a non-flat and wavy trajectory. However, we can not yet assign this resonance to any orbit on the Fermi surface; it is neither the “doughnut electron” (orbit 7) nor “doughnut hole” (orbit 5) in the (100) plane. A possibility is that the responsible
Cyclotron
421
resonance in lead-1
Pb(100) 1~69.1 Gtiz 1. 1.5K
dF G
Jrf * H 5
median
pldne
[1101
Fig. 12(a). Orbit Y around the junction of four [110] and its equivalent arms. The orbit is flat in the k space. (b) Real space trajectory of the orbit v. The trajectory is similar to a baseball seam. An electron rises and falls twice per revolution along the trajectory.
0
I
I
5
10
HC kOe)
Fig. 10.Typicalrecordertracesofcyclotronresonancefortheangles of field tipping (I,where (I is measured from the [ loO]axis toward the [llO] axis in the (100) surface, as shown in the insert of Fig. 1 I.
0.8
’
I
-10' ANGLE
FR3h'
I IO"
I
0" [I251 of ~130
PLANE
(Q)
Fig. 11.Variation of the measured cyclotron masses for the angles of field tipping(I, where (xis measured from the [loO]axis toward the [110]axis in the (100) plane.
orbit is located also around the junction of four [llO] armst 4. CONSlDERATlON OF THE [llO] ARh4 FERhU SURFACE
4.1 Field tipping e#ect for a cylindtical Fermi surface We have mentioned in Section 3.1 that the resonance 5
tA heat pattern of about 600 cycle is observed by Anderson and Gold]61 in the dHvA oscillation for the orbit Y.
is found for the large tipping angle (0’40’) regardless of the surface of the sample. It arises from the fact that the [llO] arm, to which the orbit 5 is related, has an approximately cylindrical surface. Therefore an electron on the orbit 5 has no velocity component along the axis of the cylinder and does not drift even if the magnetic field is tilted from the cylindrical axis. For example, when the field is inclined 30” from the axis of the arm, the drift velocities of the orbits near C are evaluated from the corrugated surface of the [ 1lo] arm described in the next section as l/SCrl/lOO of the Fermi velocity VF.Thus, the tipping angle of 2” in copper corresponds to 20”-30” in lead. This is the reason why the cyclotron resonance of the stationary orbit 5 can be observed over a large range of tipping angles. Moreover, the fact that the resonance 5” is found for a large range of tipping angles (tY’-30”)will be also explained by this interpretation. 4.2 Doppler-shifted cyclotron resonance (5”) From the experiment of the frequency dependence of the cyclotron mass, it is quite clear that the resonance f” is the Doppler-shifted cyclotron resonance of the non-stationary orbit. Furthermore, it should have a close relation to the fine structure of the [ 1lo] arm, especially its cross sectional variation along k, in the [ 1lo] direction. As noted in Section 1, Anderson and Gold[6] have proposed a model of the Fermi surface of the [llO] arm based on observation of a long beat of the y oscillation in the dHvA effect; the y oscillation corresponds to the orbit 5 in the present cyclotron resonance experiment. However this model has been criticized by Van Dyke [ lo], who has proposed a modification for the Fermi surface of the [l lo] arm. This is characterized by a corrugated surface with one maximum cross section labeled CLat L
428
YOSHICHIKAGNUKI et al.
Fig. 13(a). Corrugated Fermi surface of the [llO] arm proposed by Van Dyke. (b) Variation of the drift velocities along the [I lo] direction.
(or K) and minimum cross sections labeled & somewhere nearer W in the both sides of lL as is shown in Fig. 13. The corrugation, i.e. the difference of the cross sectional areas between the orbit & and & is about two per cent. In other words, the calipers length of the orbit & is about one percent smaller than that of &, although it is shown four times larger in the figure. In our cyclotron resonance experiment carefully made in a parallel field configuration, we have found only a resonance of the orbit 5 on the [ 1lo] arm. The location of this orbit 5 (0.542 mo)must be the center of the [ 1lo] arm, i.e. at U (or K) for the following reason: its resonance is observed for angles within 50” from the [ 1lo] direction in the parallel field configuration with the use of the (100) plane, which is not shown in this paper, and for the large range of tipping angles up to 40” in the tipped field configuration. Therefore the orbit 5 corresponds to the orbit iL. Considering the experimental fact that the resonance l has a Chambers’ mass minimum line shape[l8], the cyclotron effective mass m?(k) of the [llO] arm possibly increases monotonously with the variation of kH from U to W and the cyclotron mass of the orbit cs does not become extremal with respect to kH. Next, we should like to determine the location of the orbit J”(O.565mo) on the [llO] arm on the basis of the corrugated Fermi surface. If small variations in cyclotron mass along the [ 1lo] axis can be neglected, the resonance occurs for the orbit with an extremal drift velocity, according to eqn (4). There should exist such an orbit between the orbit f; and the orbit &, because neither the orbit lL nor the orbit & has drift velocity along the [ 1101 direction, as shown in Fig. 13. Taking account of small variations of cyclotron mass in the real Fermi surface, the location of the orbit I” should be shifted slightly from the extremal point of the drift velocity. When the magnetic field is exactly parallel to the surface of the specimen, this orbit does not give rise to the resonance because it does not have the component of drift velocity normal to the surface of the specimen. When the field is tipped from the axis of the arm and from the surface of the specimen, we
can expect resonance. The location of 5” with respect to the tipping angle will vary in a complicated way by virtue of the change of the drift velocity and of the cyclotron mass. In fact, as shown in Fig. 3, the mass of 5” is proportional to the tipping angle within 20” and then increases rapidly beyond 20”. This feature is not interpreted by the angular dependence of the mass derived from eqn (4). As the resonance 5” is observed in a large tipping angle up to 30”, the orbit 5” must exist between U and C; C means the center point of the most inclined orbit (30”) as shown in Fig. 13. Ogawa and Aoki[9] have recently clarified from an analysis of the long beat in y oscillation of the dHvA effect that the orbits giving higher and lower frequency oscillations correspond to the maximum cross sectional area at U (or K) and the minimum one somewhere between U and W, respectively. They have also determined the cyclotron mass 0.55 ma for & and 0.57 m. for & from the temperature dependence of the amplitude of the dHvA oscillation. These values are in good agreement with our experimental results. Finally, we must mention the appearance of the resonance 5” at a = 0” as shown in Fig. 3 contrary to our expectation that the Doppler-shifted cyclotron resonance should not be observed in the parallel field configuration. We hypothesize two causes for it. The first is that the magnetic field was not directed exactly parallel to the surface of the specimen, i.e. tilted from the surface within 0.2”. The second is that the surface produced by electropolishing was microscopically smooth but was rather wavy on a macroscopic scale. This means that local regions of the surface are inclined to the magnetic field even when the average plane of the surface is perfectly aligned. In fact, by the use of a sample grown in a quarts mould with a very flat and polished surface, no resonance 5” has been reported at (Y= 00[19]. Acknowledgements--It is our pleasure to acknowledge the advice, suggestions and comments of Prof. M. Tsuji of the University of Kyushu. We should like to thank Dr. K. Ogawa and Mr. H. Aoki
Cyclotron resonance in lead-1 of the National Research Institute for Metals for informing us of the results of their experimental study prior to publication. RBFERENcEs
1. Walsh W. M., Jr., Solid State Physics (The Simon Fraser University Lecture) Electrons in Metals (Edited by J. F. Cockran and R. R. Haering), Vol. 1, p. 127. Gordon & Beach, New York (1968). 2. Azbel’ M. Ya. and Kaner E. A., Sou. Phys.-JETP 5, 730 (1957);J. Phys. Chem. Solids 6, 113 (1958). 3. Koch J. F., Stradling R. A. and Kip A. F., Phys. Rev. 6, A240 (1964). 4. Howard D. G., Phys. Reo. 29A, 1705(1965). 5. Khaikin M. S. and Cheremisin S. M., Sou. Phys.-JETP 27,38 (1968).
429
6. Anderson J. R. and Gold A. V., Phys. Rev. JOA, 1459(1%5). 7. Anderson J. R. and Hines D. C., Phys. Reu. B2,4752 (1970). 8. Anderson J. R. and Lee J. Y. M., Phys. Rev. Bll, 1308(1975). 9. Ogawa K. and Aoki H., private Communication. 10. Van Dyke J. P., Phys. Rev. B7, 2357 (1973). 11. Young R. C., Phil. &fag. 7, 2065 (1%2). 12. Khaikin M. S. and Mina R. T., Sou. Phys.-JETP 15,24 (1%2). 13. Mina R. T. and Khaikin M. S., Sou. Phys.-JEITP 18, 8% (1964). 14. Tegart W., ‘I’heElectrolytic and Chemical Polishing of Metals, p. iO1. Pergamon Press, London (1959). 15. Snona F. W. and KID A. F.. Phvs. Rev. 137A.431 (1%5). 16. I&&J. F. and Kip ‘A. F., Phys. Reu. Letters 8, 473 (1962). 17. Walsh M., Jr., Phys. Reu. Letters 12, 161 (1964). 18. Chambers R. G., Proc. Phys. Sot. 86, 305 (1%5). 19. Goy P. and Castaing B., Phys. Rev. Bl, 4409 (1973).