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Magnetoabsorption of the indirect transition in germanium
J. Phys. Chem. Sol&z3 Pergamon Press 1965. Vol. 26, pp. 911-919.
MAGNETOABSORPTION TRANSITION
Printed in Great Britain.
OF THE INDIRECT
IN GERMANIUM
JOHN HALPERN* Lincoln Laboratory,?
Massachusetts
Institute of Technology,
Lexington,
Massachusetts
BENJAMIN LAX National Magnet LaboratoryJ
Lincoln Laboratory,?
Massachusetts
Massachusetts
Institute of Technology,
Institute of Technology,
Cambridge,
Lexington,
Massachusetts
Massachusetts
(Received 6 November 1964) Abstract-The indirect transition magnetoabsorption in germanium at 1*7OK has been studied in the Faraday configuration with the magnetic field perpendicular to the (loo), (111) and (110) faces respectively and at fields of up to 74 kG. The main features in the Landau transition spectrum are due to transitions from the states at the top of the valence bands to the electronic ladders in the conduction band. The periodicities obtained in the spectra enabled us to determine the effective masses of the conduction band electrons; the values obtained are in good agreement with those determined from cyclotron resonance. A fine structure corresponding to the spacings of the ladders in the valence bands was observed superposed on the gross structure due to transitions to successive electronic levels; this fine structure was analyzed and a number of valence band levels were identified. Values of the free electron g-factor were deduced and correlated with theory. INTRODUCTION transition at 1*7”K and in the Faraday configuraTHE MAGNETOABSORPTION of the indirect transition tion with the magnetic field perpendicular to the in germanium has been investigated by ZWERDLING (loo), (111) and (110) faces respectively. Through et al.(l) Using the Voigt geometry, they showed the use of fields a factor of two greater than those that not only could the effect be observed with high available previously and by immersing our samples resolution equipment but also that it gave rise to in a helium bath rather than heat sinking them we line shapes which were in agreement with the have been able to increase the magnitude of the theoretical predictions.(Q) However, although this effects to the point where very specific information earlier work was able to display the qualitative about the nature of the transitions and the effective aspects of the indirect transition, a truly quantita- masses of the electrons in the conduction band tive analysis was not carried out for a number of could be obtained. This is discussed in the followreasons; primarily that the amplitude of the effect ing sections. was not as large as desired at the magnetic fields then available. We have studied the magnetoabsorption indirect BxPEImIENTAL ARRANGEMBNTSAND TECHNIQUES
* Visiting scientist, National Magnet Laboratory. t Operated with support from the U.S. Air Force. $ Supported by the U.S. Air Force Office Scientific Research.
of 911
The optical system used was very similar to the one described by ZWERDLING et al.(l) The main difference was in our use of a Bitter solenoid to
912
JUHN
HALPERN
and BENJAMIN
obtain the higher magnetic fields. The ~gnet used had a 2 in. bore and a l+ in. radial access hole to accommodate the helium dewar. Peak fields obtainable were (73*8&O-4) kG. The data were taken in the Faraday co~~ration (d-c. magnetic field and propagation vector parallel) ; the results described in this paper were obtained using unpolarized light. The magnetoabsorption measurements reported here are on the indirect transition in germanium. The solution of the ‘cold window’ problem* enabled us to make all measurements with the samples immersed in liquid helium; the helium was pumped on in order to take it below the X-point. Although it is certainly desirable to pump on the helium to reduce temperature broadening, the main reason it is necessary to go below the h-point is to eliminate the bubbling which both introduces noise and attenuates the signal by 50 per cent or more. Once pumping equilibrium is reached, the temperature during a run can be maintained within 0.05°K. The data reported here were taken at 1-7°K. The experiments were performed on high purity single crystal germanium samples of thickness 0.200 in. and whose carrier concen~tion at 77°K ranged from 4-6 x 101s to 2.4 x 101s cm-s. There were no heat sinking problems since the samples were directly immersed in the helium bath ; hence they needed only to be held mechanically. The main factor limiting the resolution obtainable was the variation in magnetic field in the Bitter magnet along both the axial and radial directions since it deviates from a true solenoid because of the radial access hole; this had not been compensated for. Over the thickness of a sample used in the indirect transition (a in.) there was a measured variation of ~1 per cent along tlie axis of the solenoid and a similar variation in the radial direction over the height of the focussed beam. The magnetic broadening at a given value of field will depend on the particular transition involved but at, say, 50 kG a representative value would be -0.2 millielectron volts (meV). This is a good deal larger than the energy resolution of the optical system which at a slit width of 60 p is less than 0.1 meV.
* As of this writing the dewar has been cycled 35 times.
LAX
A typical experimental curve is shown in Fig. 1 for the propagation pe~nd~cular to the (110) face and at a magnetic field of 738 kG. The energy range covered is approximately 40 meV and more than fifteen separate transitions are observed. The theoretically expected(W) staircase-like behavior is readily apparent.
WA’ELENGTH
MUM EACSffi
FIG. 1. Indirect transition transmission trace (110) orientation at 73-8 kG.
for the
The gross characteristics of the spectrum can much more readify be seen if one plots [~~~~~~~)J as a function of energy; here Z(H>is the transmitted intensity at a given magnetic field and I(0) that at zero field. Such a normafized plot will weight the stronger transitions more heavily. This has been done in Figs. 2, 3 and 4 for the (loo), (111) and (110) orientations respectively; H in all three cases was 73.8 kG. In all three cases the data show strong transitions having a periodic energy spacing. For the [IOO] direction there is a single periodicity
MAGNETOABSORPTION
OF
THE
INDIRECT
while for the [ 11 l] and [ 1lo] directions one obtains a double periodicity; these are indicated by the spacings A through E in the figures. When one divides the characteristic energy (eHZ/m = 0.85 meV at 73.8 kG) by the periodic energy spacings one obtains effective masses that very clearly correspond to the electron effective masses in the
GERMANIUM H=73.8
TRANSITION
GERMANIUM
(100)
790 Energy
FIG. 2. Plot of [I(H)/I(O)]
GERMANIUM
(millislectron
800
810
volts)
as a function of energy for the (100) orientation.
(111)
KILOGAUSS
0~“““““““““’ 790
FIG. 3. Plot of [I(H)/1(0)]
790 Energy
913
conduction band for the various orientations of the magnetic field. The main features in the indirect transition magnetoabsorption spectrum are therefore due to transitions from the top of the valence band to the electronic ladders in the conduction band. Superposed on this primary spectrum is a fine structure corresponding to the various hole
KILOGAUSS
780
H=738
IN
800
910
(millialectronvolts)
as a function of energy for the (111) orientation.
914
JOHN
and
HALPERN
GERMANIUM H -73%
1
(ItO)
KILOGAUSS
I
I
I 780
I
I1
1 Enarqy
FIG. 4. Plot of [I(H)/1(0)]
I1 790
m*]m 0~131~0~003 o-197 _t O-005 0.079 &O-O02 O-340 + o-014 o-101 50.002
r-
with that obtained by LAX, DEXTER and ZHGER@) and by DRESSELHAUS,K,IP and KITTEL(~) using cyclotron resonance measurements. From our results we can obtain values for the effective masses rnt and ml associated with the individual ellipsoids in the conduction band. We make use of the expression @.I = f.IJt
ml co&V + mt sins8 mr
II
I 800
I
I.1
1
f 810
volts)
as a function of energy for the (110) orientation.
Table 1. Effective mass for electrons in germanium at 1*7”K at 73.8 kG for three principal directions in the crystal ---
(110)
t
~millielaclron
ladders in the valence bands and, in certain cases, the spin splitting in the conduction band. Fitting of the spacings for the three orientations give for the effective electron masses the values showninTablel.Thisdataisinverygoodagreement
::z;
LAX
t------E_
OGO-
direction
BENJAMIN
where CZJ~ = eH/mt and & is the angle the magnetic field makes with the particular ellipsoid under consideration. We obtain mt = 0_079m, ml = 1*74m and the mass ratio rna[rnt = 22-O. As noted above, we have interpreted the fine structure superposed on the periodic spectra as being due to a combination of two effects; the transitions from other than the topmost states in the valence bands and the spin splitting in the electronic conduction band ladders. Since there are no selection rules for the indirect case we would expect that there would be a huge number of such transitions and they would in general have different intensities. Using the data of GOODY who calculated numerical values for the main crystallographic directions for the valence band ladders of germanium we have, in Figs. 2 and 3, indicated where one would expect transitions for the (100) and (111) orientations respectively. (For the (110) orientation one obtains an even more densely populated series of lines.) The spin splitting in the conduction band has not been included because it was, in general, not observed (except for two cases which will subsequently be discussed). The theoretical transitions indicated in Figs. 2 and 3 are assumed to be to the centers of the spin split conduction band levels. No attempt was made to caIculate the relative intensities; this is an involved task which
would be more appropriate to do when better resolved data are obtained through the use of more homog~e~us magnetic fields. ft is &ar that a ~~c~~ation of the matrix eiemente and hence the relative intensities is ultimately necessary as thii would permit us to discard a great many of the lines and would correspondingly simplify the identification of the fine structure. We have from the present data, however$ been able to identify a number of fine structure lines observed for the direction of highest symmetry in germanium; (propagation vector and magnetic field perpendicular to the (100) face). These have been labeled 3 through 6 in Fig. 2,” For this dire&on the spin splitting of the conduction band states was observed only at the highest field values used (73.8 kG) and then only for the first step in the transmission curve. It is therefore assumed that the fine structure represents the structure in the valence bands snd that the conduction band states can be characterized by the magnetic quantum number only. Xn Figs. S,6 and 7 we show the energy level diagrams as determined by Goodman for the valence bands of germanium for H1in the [lOO& [III], and fllO”j directions respectively (in units of eH,/mc). Line 1 in Fig. 2 corresponds to the group of transitions from the valence band energies 2*34,2*59 and 2.76 and the n =r 0 conduction oand level; these transitions were not resolved. Lines Z and 3 are respectively the transitions from the 5.11 and 6.53 valence band levels to the n = 0 conduction baud level. Line 4 is again the group of 2.34, 2.59 and ‘2.76 valence band to the it = 1 conduction band level with the possibility of sm& unr~olved ~on~butions from the (8*32,% = 0) and (9-17, pz= 0) transitions. L,mes 5 and fi are due to the S-11 and 6.53 valence to the 9t = 1 conduction band level although here again there are possible contributions from the (X&61, n = 0) and (13.92, n = 0) transitions respectively. Beyond this point it is dif&ult to make assignments in more than a quslitative manner since there are so many overlapping transitions. However, the periodic&y representing the electron ladder can still be clearly seen as well as a set of groupings which correlates quite well with the observed structure.
755I
5b48
II=4
n=iQ
32.36
#I=*
323i
It=10
25-65
ns8
2&62
n-9
24.86
n=f
24e31
n-g
2os7
n=e
21.16
n=7
.lcw4
nq_
17-32
n=6
w4.
9.79
I%=3
S+l
n=Z?
nq
4w5
wp
n=3
3W8
13.92
64-21
II.46
WE
12-61 IO-79
II=3 nr4
6.53
n=3
2-39
n=D
2 -34
WC?
F‘IG. 5. Energy beep diagrsm for the tieme bands of germmium for H in the [lOO] direction (in units of
The zero field edge was obtained by fitting the calculated transitions to the observed structurepe. Specifically, lines 2 and 3 were fitted because each of these is due to only a single transition and these could both be obtained to 20.1 meV from the tr~~ssion trace. The edge was thus dete~ned as f771*4+ 0.1) meV at 1*7”K. This can be compared with the data of Z~DLING et aL@f who determined a zero field edge for the indirect transition of (771~3rt:O-4) meV which was quoted at helium temperature. The first step on the QOO) transmission trace exhibits a spitted of fO%9 +O*IO) meV at 73.8 kG; it is not observed at lower fields. When one takes into account the selection rule prohibiting spin flip one sees (Fig. 5) that the separation of the valence band levels contributing to this line is much less than the observed splitting. This splitting is therefore interpreted as being due to spin splitting in the conduction band. This sphtting corresponds to an ehxtronic g-factor of (l-8 A 0.3)
916
JOHN
HALPERN
and BENJAMIN
for the [loo] direction. This result is in good agreement with the theoretical calculations of ROTH and LAX(~)who obtained a value g = 1.7 for a free electron in the conduction band of germanium for H in the [loo] direction. It is also in agreement with the work of FEHER et d.(Q who, using much
n-3 n=*
22.33 2 w&3
n:7
16.32
z
.
nz8
83’16
“=5
12aa
“r-i
1035
n=4
9.96
n=6
266
n*6
4.73
n=2
4.33
n=2
2.47
n=3
51.61
a=4
4514
n=2
26.91
LAX
transitions respectively in the energy range of 40 meV above the zero field gap. Unless some information about the relative intensity of each transition is included in the calculations it would be difficult to compare experiment with theory since the entire theoretical spectrum is densely populated. Experimen~lly~ however, there appear dominant lines with characteristic spacings as has been previously indicated. In principle, the best way to analyze the transitions is to plot [I(H)/1(0)] vs. energy curves for a given orientation for the complete range of W values and to follow a given transition as a function of field; in such a display both the transition energies and the lineshapes can be studied. Unfortunately, such a set of point by point plots to the accuracy in which we are interested represents an enormous amount of data reduction and is therefore impracticable. Instead, we have located from the transmission traces such as Fig. 1, the points which correspond to the observed transition
31.62
n-3
44.6
n--z
22-33
ll=l
9.48
n= 1
6.93
II=3
yyf
FIG. 6. Energy level diagram for the valence bands of
3435
germanium for H in the [ill J direction (in units of eH/mc) .
more accurate spin resonance techniques, obtained an electronicg-factor, g = 1.57. These latter results were for bound donors; however, according to Roth and Lax, there is but a small difference in the g-factor for the case of free and bound electrons. (For the (100) orientation their results give g = 1.7 for free electrons and g = 1.6 for bound electrons.) It is not feasible to analyze the data for the other orientations in as detailed a manner as was done for the (100) orientation. As a consequence of having two sets of electronic levels each, the (111) and (110) orientations have, according to the level scheme calculated by Goodman, 168 and 318
2623
II=10
2362
n-9
ll=2
2081 n=s rso7 n=7
23.69
n=10
2096
n=9
LB27
.I6
ISZI II=6
IS*55
l?=7
12.97
ll=5
12.81
n=6
9%
n-4
9.69
ll=5
7m
a=4
4.74
n=3
1.70
n=2
1091
“=I
5.75 n=3 3-35 “=*
3-19 I.10
n=o
n-0
FIG. 7. Energy level diagram for the valence bands of germanium for H in the [ilO] direction (in units of eH/mc) .
MAGNETOABSORPTION
OF
THE
INDIRECT
TRANSITION
energies and plotted these energies as a function of magnetic field. Figures 8, 9 and 10 show the plots which are obtained for the [loo], [ill] and [llO] directions respectively. Since the conduction band has been determined to be parabolic (quadratic energy-momentum relation) to the highest field strengths used in the experiment and since only those states at or near the top of the valence bands enter into the transitions, the transition energies will vary linearly with the magnetic field. The data
I 620
I
I
GERMANIUM
IN
GERMANIUM
91’7
(III)
I
-
GERMANlUMtlOO)
610
-
3 =p 600
-
z z Y ,’ .z E &
t
0
790
-
FIG.
l% z
780
-
0
I
I
I
20
40
60
1 60
H (kilogauss) FIG.
Positions of indirect transitions for (100) orientation at 1.7”K as a function of magnetic field.
8.
were therefore fitted with straight lines obtained on the basis of least mean square calculations. The data for the various directions extrapolate to the zero field value of (771.2 + 0.4) meV, corresponding to the indirect energy gap at 1.7”K plus the
I
I
20
I
I I 40 H (Kilogauss)
I 60
I
I 80
9. Positions of indirect transitions at 1.7OK for (111) orientation as a function of magnetic field.
energy of the emitted phonon. This value is in agreement with that obtained by fitting the valence band ladders to the observed structure as was done in Fig. 2. Considering the (111) orientation and referring to Fig. 9, the lines labeled 1, 3, 4 and 5 exhibit spacings at 73.8 kG characteristic of the ladder in the conduction band corresponding to the electronic effective mass of m* = 0.20. A calculation from the slope of line 1 gives a value that the level involved in the series of transitions is the n = 2, 2.47 (in units of eH/mc) in the valence band. There is, however, a systematic discrepancy here; all the experimental levels in question (2, 3, 4, 5) are approximately O-7 meV higher (at 73~8 kG) than the values obtained from the theoretically calculated values given in Fig. 6. These results, which were found to be reproducible, would indicate either that the set of lines observed were due to transitions from more than one valence band
918
JOHN
HALPERN
and
level, or that the calculated value is in error. This discrepancy, however, is quite small. Line 2 agrees very closely with the energy difference between the 1.75 and 1.83 levels in the valence band and the rzz = 0 conduction band level (where nl and ~2sare respectively the magnetic quantum numbers of the larger and smaller electronic masses observed in the particular orientation). As can be seen from the energy spectrum plotted in Fig. 3 any interpretation of the higher lying lines of Fig, 9 would
t
GERMANIUM
(110)
1
770
k
L I
0
I 20
I 40
I H
I 60
I
I 80
I Kilogauss)
FIG. 10. Positions of indirect transitions at 1*7’K for (110) orientation as a function of magnetic field.
only be conjecture without a detailed knowledge of the strength of the various transitions. At the values of fields used no spin splitting was observed in the transmission curves for the (111) orientation. The (110) data is somewhat easier to analyze than that for the (111) orientation because the electron masses differ by the fairly large ratio of 3.4. Hence the lowest lines in Fig. 10 are due to transitions only to them* = 0 *34m electronic ladder.
BENJAMIN
LAX
Levels 1,3,5, 6 and 7 have the spacing characteristic of this ladder; the slope of line I indicates that it is the nr = 0 ‘light hole’ state of energy 3.19 (in units eZZ/mc) that is the main contributor to this series of lines. Furthermore, at 73.8 kG the nr = 1 ‘heavy’ electron state and as = 0 ‘light’ electron state fall approximately at the same energy. This energy corresponds to the point where there is the pronounced splitting of the second step of the transmission trace of Fig. 1. These are lines 3 and 4 in Fig. 10; at lower fields they cannot be resolved. Good agreement between theory and experiment is obtained if it is assumed that the 3.95, n = 2 ‘heavy hole’ valence band state is responsible for line 4. Line 2 is observed as a small hump on the transmission curve after the first step, at the two highest values of magnetic field (see Fig. 1). Its energy corresponds to a transition from the 4.74, n = 3 ‘heavy hole’ level to the nl = 0 state in the conduction band. Above line 7 it becomes difficult to make assignments. Referring to Fig. 7, it is not clear why there should not be strong transitions involving one or both of the 1.70 valence band levels; however, none are observed. As in the (100) orientation, a splitting is observed in the first step of the transmission trace as far down as to fields of 48 kG; at 73.8 kG this splitting is (0.58+0*1) meV which corresponds to a free electron g-factor of (1.5 kO.3). This result can be compared with the theoretically calculated value of g = 1.3 obtained by ROTH and LAX(~) for a free electron with the magnetic field in the [llO] direction. CONCLUSIONS
We have examined the magnetoabsorption indirect transition in germanium at 1~7°K in magnetic fields of up to 74 kG. The experiments were performed in the Faraday configuration with the magnetic field perpendicular to the (loo), (111) and (110) faces respectively. The experiments were carried out at lower temperatures and higher magnetic fields than those of previous investigators. The solution of the ‘cold window’ problem enabled us to work with immersed, rather than heat sunk, samples and the Bitter magnets at the National Magnet Laboratory gave us magnetic fields twice as large as previously used in similar investigations. This represents
MAGNETOABSORPTION
OF
THE
INDIRECT
more than just an extension of previously performed experiments; the magnitudes of the effects increase rapidly with field and our results have, for the first time, been sufficiently pronounced to enable a quantitative analysis to be made and to permit an evaluation of the electronic masses in the conduction band for the various orientations. The main disadvantage of the magnet we used as compared with one with flat pole pieces is the inhomogeneity in field introduced by the presence of the radial access hole. It is expected that the even higher field magnets which are now being constructed should have much better homogeneity particularly where the spacing is adjustable near to the Helmholtz value. Under thesecircumstances it would then be worthwhile to go into a detailed quantitative analysis of the intensities to be expected for the entire fine structure spectrum of transitions between the valence and conduction band ladders. The main advantage in the use of the indirect transition stems from the fact that the selection rules break down and hence one can obtain hole and electron parameters separately. The significance of the present work lies not so much in the experimental results obtained on germa~um {because here the electron masses, electron g-factors, and valence band ladders are known), but on the possible application of the experimental methods and the analysis to materials for which these quantities are not yet known. In materials such as Sic, AgBr and AgCl people are beginning
TRANSITION
IN
GERMANIUM
919
to see indirect transitions and for these materials quantities such as the electronic effective masses are not known; one of the problems is that they are apparently large effective masses. Here again one would need even larger magnetic fields and high resolution, The present results in germanium indicate the feasibility of carrying out similar investigations on these other materials. AcknowZedgements-The authors are particularly indebted to J. STEZLA of the National Magnet Laboratory for his skillful help in the maintenance of the optical system, the solution of the ‘cold window’ problem, and the reduction of the experimental data. They are also mateful to Dr. A. 1. STRAUSS for nrovidina Hall measurements on the samples and to J. W. SA~HEZ, Mrs. K. M. NEAREN and W. H. LASWELL, respectively, for the sample orientation, cutting and polishing. REFERENCES 1. ZWBRDLING S., LAX B., ROTH L. M. and BUTTON K. J., Phys. Rev. 114, 80 (1959). 2. ELLIOTT R. T.. MCLEAN T. P. and MACFARLANE G. G., Proi:Roy. Sot. 72, 5.53 (1958). 3. ROTH L. M., LAX B. and ZWERDLINGS., Phys. Reo. t14,QO (1959). 4. LAX B., ZEIGERH. J. and DBXTER R. N., Physica 20, sis (1954). 5. D~~~SELHAUSG., KIP A. F. and KITTEL C., Phys. Rev. 98, 368 (1955). 6. GOODMAN R. R., Doctor’s Thesis, University of Michigan (1958). 7. ROTH L. M. and LAX B., Phys. Rev. Lett. 3, 217 (1959). 8. F~HIZR G., WILSON D. K. and GERE E. A., Phys. Rev. Lett. 3, 25 (1959).
MAGNETOABSORPTION TRANSITION
Printed in Great Britain.
OF THE INDIRECT
IN GERMANIUM
JOHN HALPERN* Lincoln Laboratory,?
Massachusetts
Institute of Technology,
Lexington,
Massachusetts
BENJAMIN LAX National Magnet LaboratoryJ
Lincoln Laboratory,?
Massachusetts
Massachusetts
Institute of Technology,
Institute of Technology,
Cambridge,
Lexington,
Massachusetts
Massachusetts
(Received 6 November 1964) Abstract-The indirect transition magnetoabsorption in germanium at 1*7OK has been studied in the Faraday configuration with the magnetic field perpendicular to the (loo), (111) and (110) faces respectively and at fields of up to 74 kG. The main features in the Landau transition spectrum are due to transitions from the states at the top of the valence bands to the electronic ladders in the conduction band. The periodicities obtained in the spectra enabled us to determine the effective masses of the conduction band electrons; the values obtained are in good agreement with those determined from cyclotron resonance. A fine structure corresponding to the spacings of the ladders in the valence bands was observed superposed on the gross structure due to transitions to successive electronic levels; this fine structure was analyzed and a number of valence band levels were identified. Values of the free electron g-factor were deduced and correlated with theory. INTRODUCTION transition at 1*7”K and in the Faraday configuraTHE MAGNETOABSORPTION of the indirect transition tion with the magnetic field perpendicular to the in germanium has been investigated by ZWERDLING (loo), (111) and (110) faces respectively. Through et al.(l) Using the Voigt geometry, they showed the use of fields a factor of two greater than those that not only could the effect be observed with high available previously and by immersing our samples resolution equipment but also that it gave rise to in a helium bath rather than heat sinking them we line shapes which were in agreement with the have been able to increase the magnitude of the theoretical predictions.(Q) However, although this effects to the point where very specific information earlier work was able to display the qualitative about the nature of the transitions and the effective aspects of the indirect transition, a truly quantita- masses of the electrons in the conduction band tive analysis was not carried out for a number of could be obtained. This is discussed in the followreasons; primarily that the amplitude of the effect ing sections. was not as large as desired at the magnetic fields then available. We have studied the magnetoabsorption indirect BxPEImIENTAL ARRANGEMBNTSAND TECHNIQUES
* Visiting scientist, National Magnet Laboratory. t Operated with support from the U.S. Air Force. $ Supported by the U.S. Air Force Office Scientific Research.
of 911
The optical system used was very similar to the one described by ZWERDLING et al.(l) The main difference was in our use of a Bitter solenoid to
912
JUHN
HALPERN
and BENJAMIN
obtain the higher magnetic fields. The ~gnet used had a 2 in. bore and a l+ in. radial access hole to accommodate the helium dewar. Peak fields obtainable were (73*8&O-4) kG. The data were taken in the Faraday co~~ration (d-c. magnetic field and propagation vector parallel) ; the results described in this paper were obtained using unpolarized light. The magnetoabsorption measurements reported here are on the indirect transition in germanium. The solution of the ‘cold window’ problem* enabled us to make all measurements with the samples immersed in liquid helium; the helium was pumped on in order to take it below the X-point. Although it is certainly desirable to pump on the helium to reduce temperature broadening, the main reason it is necessary to go below the h-point is to eliminate the bubbling which both introduces noise and attenuates the signal by 50 per cent or more. Once pumping equilibrium is reached, the temperature during a run can be maintained within 0.05°K. The data reported here were taken at 1-7°K. The experiments were performed on high purity single crystal germanium samples of thickness 0.200 in. and whose carrier concen~tion at 77°K ranged from 4-6 x 101s to 2.4 x 101s cm-s. There were no heat sinking problems since the samples were directly immersed in the helium bath ; hence they needed only to be held mechanically. The main factor limiting the resolution obtainable was the variation in magnetic field in the Bitter magnet along both the axial and radial directions since it deviates from a true solenoid because of the radial access hole; this had not been compensated for. Over the thickness of a sample used in the indirect transition (a in.) there was a measured variation of ~1 per cent along tlie axis of the solenoid and a similar variation in the radial direction over the height of the focussed beam. The magnetic broadening at a given value of field will depend on the particular transition involved but at, say, 50 kG a representative value would be -0.2 millielectron volts (meV). This is a good deal larger than the energy resolution of the optical system which at a slit width of 60 p is less than 0.1 meV.
* As of this writing the dewar has been cycled 35 times.
LAX
A typical experimental curve is shown in Fig. 1 for the propagation pe~nd~cular to the (110) face and at a magnetic field of 738 kG. The energy range covered is approximately 40 meV and more than fifteen separate transitions are observed. The theoretically expected(W) staircase-like behavior is readily apparent.
WA’ELENGTH
MUM EACSffi
FIG. 1. Indirect transition transmission trace (110) orientation at 73-8 kG.
for the
The gross characteristics of the spectrum can much more readify be seen if one plots [~~~~~~~)J as a function of energy; here Z(H>is the transmitted intensity at a given magnetic field and I(0) that at zero field. Such a normafized plot will weight the stronger transitions more heavily. This has been done in Figs. 2, 3 and 4 for the (loo), (111) and (110) orientations respectively; H in all three cases was 73.8 kG. In all three cases the data show strong transitions having a periodic energy spacing. For the [IOO] direction there is a single periodicity
MAGNETOABSORPTION
OF
THE
INDIRECT
while for the [ 11 l] and [ 1lo] directions one obtains a double periodicity; these are indicated by the spacings A through E in the figures. When one divides the characteristic energy (eHZ/m = 0.85 meV at 73.8 kG) by the periodic energy spacings one obtains effective masses that very clearly correspond to the electron effective masses in the
GERMANIUM H=73.8
TRANSITION
GERMANIUM
(100)
790 Energy
FIG. 2. Plot of [I(H)/I(O)]
GERMANIUM
(millislectron
800
810
volts)
as a function of energy for the (100) orientation.
(111)
KILOGAUSS
0~“““““““““’ 790
FIG. 3. Plot of [I(H)/1(0)]
790 Energy
913
conduction band for the various orientations of the magnetic field. The main features in the indirect transition magnetoabsorption spectrum are therefore due to transitions from the top of the valence band to the electronic ladders in the conduction band. Superposed on this primary spectrum is a fine structure corresponding to the various hole
KILOGAUSS
780
H=738
IN
800
910
(millialectronvolts)
as a function of energy for the (111) orientation.
914
JOHN
and
HALPERN
GERMANIUM H -73%
1
(ItO)
KILOGAUSS
I
I
I 780
I
I1
1 Enarqy
FIG. 4. Plot of [I(H)/1(0)]
I1 790
m*]m 0~131~0~003 o-197 _t O-005 0.079 &O-O02 O-340 + o-014 o-101 50.002
r-
with that obtained by LAX, DEXTER and ZHGER@) and by DRESSELHAUS,K,IP and KITTEL(~) using cyclotron resonance measurements. From our results we can obtain values for the effective masses rnt and ml associated with the individual ellipsoids in the conduction band. We make use of the expression @.I = f.IJt
ml co&V + mt sins8 mr
II
I 800
I
I.1
1
f 810
volts)
as a function of energy for the (110) orientation.
Table 1. Effective mass for electrons in germanium at 1*7”K at 73.8 kG for three principal directions in the crystal ---
(110)
t
~millielaclron
ladders in the valence bands and, in certain cases, the spin splitting in the conduction band. Fitting of the spacings for the three orientations give for the effective electron masses the values showninTablel.Thisdataisinverygoodagreement
::z;
LAX
t------E_
OGO-
direction
BENJAMIN
where CZJ~ = eH/mt and & is the angle the magnetic field makes with the particular ellipsoid under consideration. We obtain mt = 0_079m, ml = 1*74m and the mass ratio rna[rnt = 22-O. As noted above, we have interpreted the fine structure superposed on the periodic spectra as being due to a combination of two effects; the transitions from other than the topmost states in the valence bands and the spin splitting in the electronic conduction band ladders. Since there are no selection rules for the indirect case we would expect that there would be a huge number of such transitions and they would in general have different intensities. Using the data of GOODY who calculated numerical values for the main crystallographic directions for the valence band ladders of germanium we have, in Figs. 2 and 3, indicated where one would expect transitions for the (100) and (111) orientations respectively. (For the (110) orientation one obtains an even more densely populated series of lines.) The spin splitting in the conduction band has not been included because it was, in general, not observed (except for two cases which will subsequently be discussed). The theoretical transitions indicated in Figs. 2 and 3 are assumed to be to the centers of the spin split conduction band levels. No attempt was made to caIculate the relative intensities; this is an involved task which
would be more appropriate to do when better resolved data are obtained through the use of more homog~e~us magnetic fields. ft is &ar that a ~~c~~ation of the matrix eiemente and hence the relative intensities is ultimately necessary as thii would permit us to discard a great many of the lines and would correspondingly simplify the identification of the fine structure. We have from the present data, however$ been able to identify a number of fine structure lines observed for the direction of highest symmetry in germanium; (propagation vector and magnetic field perpendicular to the (100) face). These have been labeled 3 through 6 in Fig. 2,” For this dire&on the spin splitting of the conduction band states was observed only at the highest field values used (73.8 kG) and then only for the first step in the transmission curve. It is therefore assumed that the fine structure represents the structure in the valence bands snd that the conduction band states can be characterized by the magnetic quantum number only. Xn Figs. S,6 and 7 we show the energy level diagrams as determined by Goodman for the valence bands of germanium for H1in the [lOO& [III], and fllO”j directions respectively (in units of eH,/mc). Line 1 in Fig. 2 corresponds to the group of transitions from the valence band energies 2*34,2*59 and 2.76 and the n =r 0 conduction oand level; these transitions were not resolved. Lines Z and 3 are respectively the transitions from the 5.11 and 6.53 valence band levels to the n = 0 conduction baud level. Line 4 is again the group of 2.34, 2.59 and ‘2.76 valence band to the it = 1 conduction band level with the possibility of sm& unr~olved ~on~butions from the (8*32,% = 0) and (9-17, pz= 0) transitions. L,mes 5 and fi are due to the S-11 and 6.53 valence to the 9t = 1 conduction band level although here again there are possible contributions from the (X&61, n = 0) and (13.92, n = 0) transitions respectively. Beyond this point it is dif&ult to make assignments in more than a quslitative manner since there are so many overlapping transitions. However, the periodic&y representing the electron ladder can still be clearly seen as well as a set of groupings which correlates quite well with the observed structure.
755I
5b48
II=4
n=iQ
32.36
#I=*
323i
It=10
25-65
ns8
2&62
n-9
24.86
n=f
24e31
n-g
2os7
n=e
21.16
n=7
.lcw4
nq_
17-32
n=6
w4.
9.79
I%=3
S+l
n=Z?
nq
4w5
wp
n=3
3W8
13.92
64-21
II.46
WE
12-61 IO-79
II=3 nr4
6.53
n=3
2-39
n=D
2 -34
WC?
F‘IG. 5. Energy beep diagrsm for the tieme bands of germmium for H in the [lOO] direction (in units of
The zero field edge was obtained by fitting the calculated transitions to the observed structurepe. Specifically, lines 2 and 3 were fitted because each of these is due to only a single transition and these could both be obtained to 20.1 meV from the tr~~ssion trace. The edge was thus dete~ned as f771*4+ 0.1) meV at 1*7”K. This can be compared with the data of Z~DLING et aL@f who determined a zero field edge for the indirect transition of (771~3rt:O-4) meV which was quoted at helium temperature. The first step on the QOO) transmission trace exhibits a spitted of fO%9 +O*IO) meV at 73.8 kG; it is not observed at lower fields. When one takes into account the selection rule prohibiting spin flip one sees (Fig. 5) that the separation of the valence band levels contributing to this line is much less than the observed splitting. This splitting is therefore interpreted as being due to spin splitting in the conduction band. This sphtting corresponds to an ehxtronic g-factor of (l-8 A 0.3)
916
JOHN
HALPERN
and BENJAMIN
for the [loo] direction. This result is in good agreement with the theoretical calculations of ROTH and LAX(~)who obtained a value g = 1.7 for a free electron in the conduction band of germanium for H in the [loo] direction. It is also in agreement with the work of FEHER et d.(Q who, using much
n-3 n=*
22.33 2 w&3
n:7
16.32
z
.
nz8
83’16
“=5
12aa
“r-i
1035
n=4
9.96
n=6
266
n*6
4.73
n=2
4.33
n=2
2.47
n=3
51.61
a=4
4514
n=2
26.91
LAX
transitions respectively in the energy range of 40 meV above the zero field gap. Unless some information about the relative intensity of each transition is included in the calculations it would be difficult to compare experiment with theory since the entire theoretical spectrum is densely populated. Experimen~lly~ however, there appear dominant lines with characteristic spacings as has been previously indicated. In principle, the best way to analyze the transitions is to plot [I(H)/1(0)] vs. energy curves for a given orientation for the complete range of W values and to follow a given transition as a function of field; in such a display both the transition energies and the lineshapes can be studied. Unfortunately, such a set of point by point plots to the accuracy in which we are interested represents an enormous amount of data reduction and is therefore impracticable. Instead, we have located from the transmission traces such as Fig. 1, the points which correspond to the observed transition
31.62
n-3
44.6
n--z
22-33
ll=l
9.48
n= 1
6.93
II=3
yyf
FIG. 6. Energy level diagram for the valence bands of
3435
germanium for H in the [ill J direction (in units of eH/mc) .
more accurate spin resonance techniques, obtained an electronicg-factor, g = 1.57. These latter results were for bound donors; however, according to Roth and Lax, there is but a small difference in the g-factor for the case of free and bound electrons. (For the (100) orientation their results give g = 1.7 for free electrons and g = 1.6 for bound electrons.) It is not feasible to analyze the data for the other orientations in as detailed a manner as was done for the (100) orientation. As a consequence of having two sets of electronic levels each, the (111) and (110) orientations have, according to the level scheme calculated by Goodman, 168 and 318
2623
II=10
2362
n-9
ll=2
2081 n=s rso7 n=7
23.69
n=10
2096
n=9
LB27
.I6
ISZI II=6
IS*55
l?=7
12.97
ll=5
12.81
n=6
9%
n-4
9.69
ll=5
7m
a=4
4.74
n=3
1.70
n=2
1091
“=I
5.75 n=3 3-35 “=*
3-19 I.10
n=o
n-0
FIG. 7. Energy level diagram for the valence bands of germanium for H in the [ilO] direction (in units of eH/mc) .
MAGNETOABSORPTION
OF
THE
INDIRECT
TRANSITION
energies and plotted these energies as a function of magnetic field. Figures 8, 9 and 10 show the plots which are obtained for the [loo], [ill] and [llO] directions respectively. Since the conduction band has been determined to be parabolic (quadratic energy-momentum relation) to the highest field strengths used in the experiment and since only those states at or near the top of the valence bands enter into the transitions, the transition energies will vary linearly with the magnetic field. The data
I 620
I
I
GERMANIUM
IN
GERMANIUM
91’7
(III)
I
-
GERMANlUMtlOO)
610
-
3 =p 600
-
z z Y ,’ .z E &
t
0
790
-
FIG.
l% z
780
-
0
I
I
I
20
40
60
1 60
H (kilogauss) FIG.
Positions of indirect transitions for (100) orientation at 1.7”K as a function of magnetic field.
8.
were therefore fitted with straight lines obtained on the basis of least mean square calculations. The data for the various directions extrapolate to the zero field value of (771.2 + 0.4) meV, corresponding to the indirect energy gap at 1.7”K plus the
I
I
20
I
I I 40 H (Kilogauss)
I 60
I
I 80
9. Positions of indirect transitions at 1.7OK for (111) orientation as a function of magnetic field.
energy of the emitted phonon. This value is in agreement with that obtained by fitting the valence band ladders to the observed structure as was done in Fig. 2. Considering the (111) orientation and referring to Fig. 9, the lines labeled 1, 3, 4 and 5 exhibit spacings at 73.8 kG characteristic of the ladder in the conduction band corresponding to the electronic effective mass of m* = 0.20. A calculation from the slope of line 1 gives a value that the level involved in the series of transitions is the n = 2, 2.47 (in units of eH/mc) in the valence band. There is, however, a systematic discrepancy here; all the experimental levels in question (2, 3, 4, 5) are approximately O-7 meV higher (at 73~8 kG) than the values obtained from the theoretically calculated values given in Fig. 6. These results, which were found to be reproducible, would indicate either that the set of lines observed were due to transitions from more than one valence band
918
JOHN
HALPERN
and
level, or that the calculated value is in error. This discrepancy, however, is quite small. Line 2 agrees very closely with the energy difference between the 1.75 and 1.83 levels in the valence band and the rzz = 0 conduction band level (where nl and ~2sare respectively the magnetic quantum numbers of the larger and smaller electronic masses observed in the particular orientation). As can be seen from the energy spectrum plotted in Fig. 3 any interpretation of the higher lying lines of Fig, 9 would
t
GERMANIUM
(110)
1
770
k
L I
0
I 20
I 40
I H
I 60
I
I 80
I Kilogauss)
FIG. 10. Positions of indirect transitions at 1*7’K for (110) orientation as a function of magnetic field.
only be conjecture without a detailed knowledge of the strength of the various transitions. At the values of fields used no spin splitting was observed in the transmission curves for the (111) orientation. The (110) data is somewhat easier to analyze than that for the (111) orientation because the electron masses differ by the fairly large ratio of 3.4. Hence the lowest lines in Fig. 10 are due to transitions only to them* = 0 *34m electronic ladder.
BENJAMIN
LAX
Levels 1,3,5, 6 and 7 have the spacing characteristic of this ladder; the slope of line I indicates that it is the nr = 0 ‘light hole’ state of energy 3.19 (in units eZZ/mc) that is the main contributor to this series of lines. Furthermore, at 73.8 kG the nr = 1 ‘heavy’ electron state and as = 0 ‘light’ electron state fall approximately at the same energy. This energy corresponds to the point where there is the pronounced splitting of the second step of the transmission trace of Fig. 1. These are lines 3 and 4 in Fig. 10; at lower fields they cannot be resolved. Good agreement between theory and experiment is obtained if it is assumed that the 3.95, n = 2 ‘heavy hole’ valence band state is responsible for line 4. Line 2 is observed as a small hump on the transmission curve after the first step, at the two highest values of magnetic field (see Fig. 1). Its energy corresponds to a transition from the 4.74, n = 3 ‘heavy hole’ level to the nl = 0 state in the conduction band. Above line 7 it becomes difficult to make assignments. Referring to Fig. 7, it is not clear why there should not be strong transitions involving one or both of the 1.70 valence band levels; however, none are observed. As in the (100) orientation, a splitting is observed in the first step of the transmission trace as far down as to fields of 48 kG; at 73.8 kG this splitting is (0.58+0*1) meV which corresponds to a free electron g-factor of (1.5 kO.3). This result can be compared with the theoretically calculated value of g = 1.3 obtained by ROTH and LAX(~) for a free electron with the magnetic field in the [llO] direction. CONCLUSIONS
We have examined the magnetoabsorption indirect transition in germanium at 1~7°K in magnetic fields of up to 74 kG. The experiments were performed in the Faraday configuration with the magnetic field perpendicular to the (loo), (111) and (110) faces respectively. The experiments were carried out at lower temperatures and higher magnetic fields than those of previous investigators. The solution of the ‘cold window’ problem enabled us to work with immersed, rather than heat sunk, samples and the Bitter magnets at the National Magnet Laboratory gave us magnetic fields twice as large as previously used in similar investigations. This represents
MAGNETOABSORPTION
OF
THE
INDIRECT
more than just an extension of previously performed experiments; the magnitudes of the effects increase rapidly with field and our results have, for the first time, been sufficiently pronounced to enable a quantitative analysis to be made and to permit an evaluation of the electronic masses in the conduction band for the various orientations. The main disadvantage of the magnet we used as compared with one with flat pole pieces is the inhomogeneity in field introduced by the presence of the radial access hole. It is expected that the even higher field magnets which are now being constructed should have much better homogeneity particularly where the spacing is adjustable near to the Helmholtz value. Under thesecircumstances it would then be worthwhile to go into a detailed quantitative analysis of the intensities to be expected for the entire fine structure spectrum of transitions between the valence and conduction band ladders. The main advantage in the use of the indirect transition stems from the fact that the selection rules break down and hence one can obtain hole and electron parameters separately. The significance of the present work lies not so much in the experimental results obtained on germa~um {because here the electron masses, electron g-factors, and valence band ladders are known), but on the possible application of the experimental methods and the analysis to materials for which these quantities are not yet known. In materials such as Sic, AgBr and AgCl people are beginning
TRANSITION
IN
GERMANIUM
919
to see indirect transitions and for these materials quantities such as the electronic effective masses are not known; one of the problems is that they are apparently large effective masses. Here again one would need even larger magnetic fields and high resolution, The present results in germanium indicate the feasibility of carrying out similar investigations on these other materials. AcknowZedgements-The authors are particularly indebted to J. STEZLA of the National Magnet Laboratory for his skillful help in the maintenance of the optical system, the solution of the ‘cold window’ problem, and the reduction of the experimental data. They are also mateful to Dr. A. 1. STRAUSS for nrovidina Hall measurements on the samples and to J. W. SA~HEZ, Mrs. K. M. NEAREN and W. H. LASWELL, respectively, for the sample orientation, cutting and polishing. REFERENCES 1. ZWBRDLING S., LAX B., ROTH L. M. and BUTTON K. J., Phys. Rev. 114, 80 (1959). 2. ELLIOTT R. T.. MCLEAN T. P. and MACFARLANE G. G., Proi:Roy. Sot. 72, 5.53 (1958). 3. ROTH L. M., LAX B. and ZWERDLINGS., Phys. Reo. t14,QO (1959). 4. LAX B., ZEIGERH. J. and DBXTER R. N., Physica 20, sis (1954). 5. D~~~SELHAUSG., KIP A. F. and KITTEL C., Phys. Rev. 98, 368 (1955). 6. GOODMAN R. R., Doctor’s Thesis, University of Michigan (1958). 7. ROTH L. M. and LAX B., Phys. Rev. Lett. 3, 217 (1959). 8. F~HIZR G., WILSON D. K. and GERE E. A., Phys. Rev. Lett. 3, 25 (1959).