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Mechanical stress influence on effective masses in Si inversion layers
Surface Science 58 (1976) 169-177 0 North-Holland Publishing Company
MECHANICALSTRESSINFLUENCEONEFFECTIVEMASSESIN Si INVERSIONLAYERS I. EISELE, H. GESCH * and G. DORDA Forschungslaboratorien
der Siemens AG, Balanstr. 73, D-8 Miinchen.
West-Germany
Shubnikov-De Haas oscillations and Hall mobility have been measured under uniaxial stress for (100) and (110) n-type silicon inversion layers. For both cases no population effect into valleys with different masses could be seen. Intravalley mass increases up to 25% have been observed for uniaxial compression of P = 230 N/mm2. Under stress the occupied degenerate valleys shift relative to each other and cause changes of the SdH amplitudes. For the (110) surface orientation a preexisting uniaxial stress in the surface plane has been detected.
In the inversion layer of a silicon MOS transistor the charge carriers are confined to a narrow potential well near the surface, and at low temperatures a two-dimensional electron or hole gas exists [ I] . Application of a large magnetic field leads to a Landau quantization which can be demonstrated by magnetoresistance oscillations of the Shubnikov-De Haas (SdH) type [2] . These surface quantum oscillations yield valuable information about the magnitude of the effective mass, the LandC g-factor, the degeneracy of the subbands, and the energy splitting between various subbands. For n-type inversion layers theory and experiment do not agree very well in some parts and the origin of these discrepancies is not known [3]. However, it has been suggested that the mechanical stress which is built up during the preparation of the Si-Sio2 interface plays an important role. To obtain more information on this subject studies of SdH oscillations as a function of uniaxial stress are a useful tool. The resulting cyclotron masses and relaxation times are closely connected to the transport properties in the inversion layer, and therefore it seems obvious to carry out simultaneous mobility measurements under identical stress conditions. For the sample geometry a Hall bar with a channel length of 400 m and a width of 40 fl was selected. A 120 nm oxide was grown on 10 52 cm (100) and (110) ptype substrates. The quality of the samples can be described best by the mobility maximum [4] at 4.2 K, which appeared between 6200 and 12700 cm2/V-s for (100) samples and about 3000 cm2/V-s for (110) samples. For both cases the scattering time r is sufficiently large to study SdH oscillations. The periodicity of the magneto-
* Physics Department, Technical University Munich, D-8046 Garching, West Germany. 169
170
I. Eisele et al. f Mechanical stress influence on effective masses
resistance oscillations
~=ACOS P
for one subband with effective mass m, is given by [S] :
2n(EF
-
EO)mc
fieB
where A is the amplitude, E. the energy edge of the subband for a magnetic field and EF is the Fermi level. For the bulk where the density of states is still a function of energy, Aq appears to be n/4, whereas at the surface Ap becomes zero because the density of states is limited to discrete energy levels [6]. According to eq. (1) the periodicity for one valley with a given m, should only depend on the position of the Fermi level. If by stress or any other means different valleys in the band structure of silicon are shifted relative to each other, a transfer of electrons can occur [6]. In this case the Fermi level in one particular valley changes and consequently the periodicity of the oscillations. Measurements of SdH oscillations at 2.0 K for (100) and (110) n-channel inversion layers are shown in the first three figures. Longitudinal tension as well as compression was applied to the samples, and in the measured range the periodicity of the oscillations was not altered significantly. It should be pointed out that no periodicity change has been observed even at the lowest carrier concentrations. In this case the energy splitting of the subband systems is small and according to theory the mechanical stress should be sufficient to cause a complete transfer of electrons into empty valleys with a different cyclotron mass. The lack of such a population effect is most surprising for the (110) surface orientation, where the theoretical calculations for N,v = 1 X 1016 rnd2 predict a splitting energy of only several meV between the lowest subband levels of different valleys [8]. This is about the same order of magnitude as can be shifted by mechanical stress [9]. From these results we must state that for a (110) surface the lowest electric subband of the two fold valleys is much further separated from the lowest electric subband of the four fold valleys than was predicted. At the moment none of the theories accounts for this behaviour. For a (100) surface a similar energy splitting problem has been solved by considering the effect of electron-electron interaction [ 10,l l] . At first glance the drastic amplitude variations with mechanical stress are astonishing. The amplitude is a function of magnetic field B, temperature T, effective cyclotron mass m,, Fermi level EF, and the Dingle temperature TD [5] : B = 0,
1
112
A = const. X (EF )7110P
T
sinh(um ,T/B)
exp(-v)
.
This equation suggests that an amplitude variation can be attributed to a change of which is inversely proportional to a collision time, 7,. The Dingle tempem,orTJj, rature can be calculated from the ratio of two amplitudes at different magnetic fields. From the analysis of the data it turns out that for (100) and (110) surface orientations TD is independent of temperature and therefore the effective mass m, can be
I. Eisele et al. /Mechanical
stress influence on effective masses
111
evaluated from the temperature dependence of the SdH amplitudes. In fig. 4, me is plotted versus concentration of mobile electrons for a (100) surface. Without mechanical stress, the results agree quite well with other authors [ 121. For a tension P = 14.5 N/mm2 = 14.5 X lo* dyne/cm 2, almost no deviation from the unstressed values occurred. Compression, however, leads to a considerable increase of the effective mass, with a maximum variation Am, = 0.25m,. The only explanation for this behaviour is preexisting lattice distortion near the Si-Sio, interface, which increases with compression and decreases with tension. The mass increase explains partially the damping of the oscillations, but for a complete fitting of the profiles an additional effect has to be considered. We postulate a stress dependent lifting of the degeneracy of the two identical valleys [ 131, which seems the only possible way to explain the oscillating modulation of a given amplitude (see fig. 4 in ref. [ 131). The other possible description of the damping is due to a stress dependent change of the Dingle temperature, TD. For the analysis both effects have been considered with the result that the stress has a diminishing influence on TD. The lifting of the degeneracy of the two identical valleys by an energy AE causes two different sets of oscillations. Their superposition usually modulates the amplitudes and even cancellation can occur. From earlier measurements [ 131 we obtained AE = 1.6 meV for a compression P = 230 N/mm 2. Independent of the carrier concentration, the beating effect should be a periodical function of the magnetic field and A,!?. However, from fig. 1 it is evident that this effect is not pronounced at high carrier concentrations. This could be due to the exponential damping of the amplitude, exp(-am,TD/B), and the observed magnetic field dependence of AE. For low carrier concentrations where spin splitting effects of the order of AE occur, the modulation of the amplitudes becomes more pronounced. At first the results for a (110) surface (figs. 2 and 3) seem to be contrary to the (100) surface because compression in fig. 2 yields a huge increase of the amplitudes. In the following, however, we will show that this behaviour can also be explained on the basis of a superposition model. For a (110) surface in a transverse magnetic field it has been observed that the spin splitting can be detected over the whole concentration range. The Landau splitting depends on the magnetic field component normal to the surface, whereas for the spin splitting the total B is responsible. Tilting the magnetic field B leads to an overlap of oscillations from Landau splitting and spin splitting [3]. A similar effect can be achieved by applying mechanical stress without tilting B. This proves again the lifting of the valley degeneracy by mechanical stress. in addition the tilted sample (fig. 2) exhibits a huge influence of compression in the [OOl] direction on the amplitudes of the oscillations. The temperature dependence of the oscillations yields me = 0.46mu for a compression P = 180 N/mm2 compared to m, = 0.39mo for P = 0. This mass increase under compression should lead to a damping effect and cannot be responsible at all for the increase of the amplitudes. Therefore only a superposition model remains. Because the spin overlap is already achieved by tilting the sample, we assume a superposition of the Landau levels of the remaining two-fold degenerate valleys. At this point it should be noted that in
172
I. Eisele et al. /Mechanical
stress influence
of ejyective
masses
Si n-Channel (100) T=Z.OK N I"Y =3.85~10'~ me2
ension P-145Nmm
Compression P=230Nmm‘*
-7
----
2
--
4
6
T
B
B-
Fig. 1. (100) surface: electron concentration direction.
relativeresistance
Ninv = 3.85
X
change AR/R versus magnetic field B at 2.0 K for an lOI m-*. Parameter is longitudinal stressP in [OOl]
accordance with the literature [3] we only see a valley degeneracy factor n, = 2, instead of the theoretically expected n, = 4. The superposition of two different OSciliations gives an estimate of the splitting energy of the two remaining valleys, for which we obtain Al? = 1.5 me’?. AE refers to a preexisting stress which is compensated by the applied uniaxial compression. This leads to a summation of the two sets of amplitudes and explains the huge increase. Tension does not alter the oscillations at all, i.e., the preexisting lifting of the valley degeneracy is not changed. Quite different results are obtained if the uniaxial stress is applied perpendicular to [OOl] , i.e., in [l^TO] direction (fig. 3). The periodicity and amplitudes of the OScillations remain almost unch~ged for compression as well as tension. This hints again that the preexisting stress must have a preferred direction. The superposition model and effective mass changes can be confmned with Hall mobility measurements. The results are shown in figs. 5 and 6. For the interpretation it is important to point out that the Hall mobility contains only the mass value in the current direction, whereas SdH oscillations yield a measure of the cyclotron
I. .!%elc et al. /Mechanical
stress influence of effective masses
173
T=2.0 K N ,“v=3.4~10’~ m -2 J.P in [OOl]
Compression P=lBONmm -2
0
2
4
T
6
61-
Fig. 2. (110) dinal stressP
surface: AR/R versusB at 2.0 K for Ninv = 3.4 X 1016 m-*. Parameter in [OOl] direction. The angle between B and the (110) surface is 25”.
is longitu-
mass m, = (mplm,2)1/2. Here, mpl and mp2 are the principal masses. At first view two things are surprising: the large mobility variation with stress and the inconsistent mobility changes with stress direction. The experimental data cannot be understood in terms of a change in relaxation time 7. This is in accordance with the analysis of SdH oscillations with respect to the Dingle temperature, r,. Proof of this suggestion is the comparison between the strong mobility decrease and the simultaneous large amplitude increase of the SdH oscillations for a (110) surface under compression in the [OOl] direction (see figs. 2 and 5). For the further discussion we suppose for , Si n-Channel
I
(110)
Tension: P=160Nmm-2
T=2.05K N I”” =3.4x10’” m -2 J.P in [liO] 4 B, (110)=0°
Compression: P=195Nmm-2 \/ I
0
I
2
I
I
4
6
1
0
B-
pig. 3. (110) surface: AR/R versus B at 2.0 K for Ninv = 3.4 X 1016 m-*. dinal stressP in [ 1101 direction.
Parameter
is longitu-
174
I. Eisele et al. /Mechanical
stress influence of effective masses
Sin-CHANNEL
I
(1001
0.28 -
9 mo
_ 0.24
o:P=
0
..P=-145 N/mm2 I.P=+230N/mm2 0.16 I 0
I 2
I 4
I -2 Ill
6~10'~
Fig. 4. (100) surface: cyclotron mass of electrons as a function ofNinv. (0) without mechanical stress; cn) CompressionP = 230 N/mm’ in [OOl] direction; (x) tensionp = 145 N/mm2 in [OOl] direction.
simplicity that mainly changes of the masses are responsible for the stress dependent mobility deviations. Combining mobilities of the two stress directions in fig. 5, it is possible to get an idea about the variation of the cyclotron mass with stress. For the evaluation it has to be taken into account that the ratio of longitudinal and transverse stiffness coefficients is about 3, in other words, the transverse mobility was taken at a stress which is l/3 of the longitudinal value. A qualitative estimate only yields a change of m, if compression is applied in the [OOl] direction. In this case the mass value should increase, which is in good agreement with the experimental value m, = 0.46~~~. Compression in [ 1101 and tension in [OOl] as well as [ 1101 direction show almost no deviation from the unstressed case with m, = 0.39~2,. Agreement between mobility measurements (fig. 6) and SdH amplitudes (fig. 1) is also obtained for the (100) surface. For tension only a small mobility increase has been detected, whereas compression lowers the mobility significantly. The cyclotron mass m, (fig. 4) deviates under stress in the same manner and confirms the relation between stress and effective mass or, in other words, the importance of the lattice distortion. Piezomeasurements at low temperatures were reported earlier [93 and correspond well to the Hall mobility data. On the basis of the present paper the piezoresistance results for (100) as well as (110) surface orientations have to be interpreted in terms of an effective mass change inside of one particular subband system. Evidence for the theoretically
I. Eisele et al. /Mechanical
stress injluence of effective masses
,
O.B-
JJ’inWl
SildlWIdm
_!!I
vs
175
T=4.2 K
0.4. I WI 0.2. A: P=o 0: Compression P-75 X: Compression
o : Tension
P=210
P-155 Nmm
0 J,P in liO]
0.6 m2 vs 0.4 t IQ 0.2 A:
P=o
x : Compression P=200Nmm-2 o
0
: Tension
P=170 N mm-’
6
10 N I””
Fig. 5. (110) surface: Hall mobility /.JHversusNin, P in [ 0011 and [ 1101 direction, respectively.
20
me2
5OhO’”
-
for B = 1 T. Parameter is longitudinal stress
expected population effect between different subbands could not be seen at all, This is especially surprising for the (I 10) surface where supposedly the energy splitting is very small [8]. Adding up all the experimental details we come to the conclusion that the effective mass values depend on the Si-SK?, interface and are closely related to the technological processing. Especially for the (1 IO) surface we could show a preexisting stress which is uniaxial along the surface. For all n- and p-type surface orientations stress might be an explanation for the discrepancies of the mass values in the literature. The Hall mobility results give evidence for large changes of the intravalley masses in the current direction. This can only be explained if the constant energy surfaces for the electrons are remarkably distorted and the parabolic E(k) relation becomes non-quadratic.
176
I. Eisele et al. /Mechanical stress influence of effective masses
Si n-Chad nool
0.6 4
m= vs
0.4 t PH
0.2~
q:
Tension
0
5
T
5
10
P=ZlO N my -2
: o : P=150 mm2
N mm -2 5oi10’5
N I”” -
Fig. 6. (100) direction.
surface:
pi
versusNinv
for B = 1 T. Parameter
is longitudinal
stress P in [OOl]
Acknowledgement The authors would like to thank E. Doering for the device fabrication and Z. Cehovec for sample preparation and technical assistance. This work comprises part of the Diplom-thesis of H.G. to be submitted to the Physics Department of the Technical University Munich.
References [l]
For a review see: G. Dorda, in: Festkarperprobleme, Vol. 13 (Pergamon-Vieweg, p. 215. [2] G. Landwehr, in: FestkGrperprobleme, Vol. 15 (Pergamon-Vieweg, 1975) p.49. [3] T. Neugebauer, K. von Klitzing, G. Landwehr and G. Dorda, Solid State Commun. (1975) 295.
1973)
17
I. Eisele et al. /Mechanical
stress influence of effective masses
[4] I. Eisele and G. Dorda, 3. Appl. Phys., submitted. [5] L.M. Roth and P.N. Argyres, in: Semiconductors and Semimetals, Vol. 1 (Academic New York, 1966) p. 159. [6] E. Banger& private communication. [7] C.S. Smith, Phys. Rev. 94 (1954) 42. [S] F. Stern, private communication. [9] G. Dorda and 1. Eisele, Phys. Status Solidi (a) 20 (1973) 263. [lo] B. Vinter and F. Stern, Surface Sci. 58 (1976) 141. [l l] T. Ando, Surface Sci. 58 (1976) 128. [12] J.L. Smith and P.J. Stiles, Phys. Rev. Letters 29 (1972) 102. [ 131 I. Eisele, H. Gesch and G. Dorda, Solid State Commun., 18 (1976) 743.
171
Press,
MECHANICALSTRESSINFLUENCEONEFFECTIVEMASSESIN Si INVERSIONLAYERS I. EISELE, H. GESCH * and G. DORDA Forschungslaboratorien
der Siemens AG, Balanstr. 73, D-8 Miinchen.
West-Germany
Shubnikov-De Haas oscillations and Hall mobility have been measured under uniaxial stress for (100) and (110) n-type silicon inversion layers. For both cases no population effect into valleys with different masses could be seen. Intravalley mass increases up to 25% have been observed for uniaxial compression of P = 230 N/mm2. Under stress the occupied degenerate valleys shift relative to each other and cause changes of the SdH amplitudes. For the (110) surface orientation a preexisting uniaxial stress in the surface plane has been detected.
In the inversion layer of a silicon MOS transistor the charge carriers are confined to a narrow potential well near the surface, and at low temperatures a two-dimensional electron or hole gas exists [ I] . Application of a large magnetic field leads to a Landau quantization which can be demonstrated by magnetoresistance oscillations of the Shubnikov-De Haas (SdH) type [2] . These surface quantum oscillations yield valuable information about the magnitude of the effective mass, the LandC g-factor, the degeneracy of the subbands, and the energy splitting between various subbands. For n-type inversion layers theory and experiment do not agree very well in some parts and the origin of these discrepancies is not known [3]. However, it has been suggested that the mechanical stress which is built up during the preparation of the Si-Sio2 interface plays an important role. To obtain more information on this subject studies of SdH oscillations as a function of uniaxial stress are a useful tool. The resulting cyclotron masses and relaxation times are closely connected to the transport properties in the inversion layer, and therefore it seems obvious to carry out simultaneous mobility measurements under identical stress conditions. For the sample geometry a Hall bar with a channel length of 400 m and a width of 40 fl was selected. A 120 nm oxide was grown on 10 52 cm (100) and (110) ptype substrates. The quality of the samples can be described best by the mobility maximum [4] at 4.2 K, which appeared between 6200 and 12700 cm2/V-s for (100) samples and about 3000 cm2/V-s for (110) samples. For both cases the scattering time r is sufficiently large to study SdH oscillations. The periodicity of the magneto-
* Physics Department, Technical University Munich, D-8046 Garching, West Germany. 169
170
I. Eisele et al. f Mechanical stress influence on effective masses
resistance oscillations
~=ACOS P
for one subband with effective mass m, is given by [S] :
2n(EF
-
EO)mc
fieB
where A is the amplitude, E. the energy edge of the subband for a magnetic field and EF is the Fermi level. For the bulk where the density of states is still a function of energy, Aq appears to be n/4, whereas at the surface Ap becomes zero because the density of states is limited to discrete energy levels [6]. According to eq. (1) the periodicity for one valley with a given m, should only depend on the position of the Fermi level. If by stress or any other means different valleys in the band structure of silicon are shifted relative to each other, a transfer of electrons can occur [6]. In this case the Fermi level in one particular valley changes and consequently the periodicity of the oscillations. Measurements of SdH oscillations at 2.0 K for (100) and (110) n-channel inversion layers are shown in the first three figures. Longitudinal tension as well as compression was applied to the samples, and in the measured range the periodicity of the oscillations was not altered significantly. It should be pointed out that no periodicity change has been observed even at the lowest carrier concentrations. In this case the energy splitting of the subband systems is small and according to theory the mechanical stress should be sufficient to cause a complete transfer of electrons into empty valleys with a different cyclotron mass. The lack of such a population effect is most surprising for the (110) surface orientation, where the theoretical calculations for N,v = 1 X 1016 rnd2 predict a splitting energy of only several meV between the lowest subband levels of different valleys [8]. This is about the same order of magnitude as can be shifted by mechanical stress [9]. From these results we must state that for a (110) surface the lowest electric subband of the two fold valleys is much further separated from the lowest electric subband of the four fold valleys than was predicted. At the moment none of the theories accounts for this behaviour. For a (100) surface a similar energy splitting problem has been solved by considering the effect of electron-electron interaction [ 10,l l] . At first glance the drastic amplitude variations with mechanical stress are astonishing. The amplitude is a function of magnetic field B, temperature T, effective cyclotron mass m,, Fermi level EF, and the Dingle temperature TD [5] : B = 0,
1
112
A = const. X (EF )7110P
T
sinh(um ,T/B)
exp(-v)
.
This equation suggests that an amplitude variation can be attributed to a change of which is inversely proportional to a collision time, 7,. The Dingle tempem,orTJj, rature can be calculated from the ratio of two amplitudes at different magnetic fields. From the analysis of the data it turns out that for (100) and (110) surface orientations TD is independent of temperature and therefore the effective mass m, can be
I. Eisele et al. /Mechanical
stress influence on effective masses
111
evaluated from the temperature dependence of the SdH amplitudes. In fig. 4, me is plotted versus concentration of mobile electrons for a (100) surface. Without mechanical stress, the results agree quite well with other authors [ 121. For a tension P = 14.5 N/mm2 = 14.5 X lo* dyne/cm 2, almost no deviation from the unstressed values occurred. Compression, however, leads to a considerable increase of the effective mass, with a maximum variation Am, = 0.25m,. The only explanation for this behaviour is preexisting lattice distortion near the Si-Sio, interface, which increases with compression and decreases with tension. The mass increase explains partially the damping of the oscillations, but for a complete fitting of the profiles an additional effect has to be considered. We postulate a stress dependent lifting of the degeneracy of the two identical valleys [ 131, which seems the only possible way to explain the oscillating modulation of a given amplitude (see fig. 4 in ref. [ 131). The other possible description of the damping is due to a stress dependent change of the Dingle temperature, TD. For the analysis both effects have been considered with the result that the stress has a diminishing influence on TD. The lifting of the degeneracy of the two identical valleys by an energy AE causes two different sets of oscillations. Their superposition usually modulates the amplitudes and even cancellation can occur. From earlier measurements [ 131 we obtained AE = 1.6 meV for a compression P = 230 N/mm 2. Independent of the carrier concentration, the beating effect should be a periodical function of the magnetic field and A,!?. However, from fig. 1 it is evident that this effect is not pronounced at high carrier concentrations. This could be due to the exponential damping of the amplitude, exp(-am,TD/B), and the observed magnetic field dependence of AE. For low carrier concentrations where spin splitting effects of the order of AE occur, the modulation of the amplitudes becomes more pronounced. At first the results for a (110) surface (figs. 2 and 3) seem to be contrary to the (100) surface because compression in fig. 2 yields a huge increase of the amplitudes. In the following, however, we will show that this behaviour can also be explained on the basis of a superposition model. For a (110) surface in a transverse magnetic field it has been observed that the spin splitting can be detected over the whole concentration range. The Landau splitting depends on the magnetic field component normal to the surface, whereas for the spin splitting the total B is responsible. Tilting the magnetic field B leads to an overlap of oscillations from Landau splitting and spin splitting [3]. A similar effect can be achieved by applying mechanical stress without tilting B. This proves again the lifting of the valley degeneracy by mechanical stress. in addition the tilted sample (fig. 2) exhibits a huge influence of compression in the [OOl] direction on the amplitudes of the oscillations. The temperature dependence of the oscillations yields me = 0.46mu for a compression P = 180 N/mm2 compared to m, = 0.39mo for P = 0. This mass increase under compression should lead to a damping effect and cannot be responsible at all for the increase of the amplitudes. Therefore only a superposition model remains. Because the spin overlap is already achieved by tilting the sample, we assume a superposition of the Landau levels of the remaining two-fold degenerate valleys. At this point it should be noted that in
172
I. Eisele et al. /Mechanical
stress influence
of ejyective
masses
Si n-Channel (100) T=Z.OK N I"Y =3.85~10'~ me2
ension P-145Nmm
Compression P=230Nmm‘*
-7
----
2
--
4
6
T
B
B-
Fig. 1. (100) surface: electron concentration direction.
relativeresistance
Ninv = 3.85
X
change AR/R versus magnetic field B at 2.0 K for an lOI m-*. Parameter is longitudinal stressP in [OOl]
accordance with the literature [3] we only see a valley degeneracy factor n, = 2, instead of the theoretically expected n, = 4. The superposition of two different OSciliations gives an estimate of the splitting energy of the two remaining valleys, for which we obtain Al? = 1.5 me’?. AE refers to a preexisting stress which is compensated by the applied uniaxial compression. This leads to a summation of the two sets of amplitudes and explains the huge increase. Tension does not alter the oscillations at all, i.e., the preexisting lifting of the valley degeneracy is not changed. Quite different results are obtained if the uniaxial stress is applied perpendicular to [OOl] , i.e., in [l^TO] direction (fig. 3). The periodicity and amplitudes of the OScillations remain almost unch~ged for compression as well as tension. This hints again that the preexisting stress must have a preferred direction. The superposition model and effective mass changes can be confmned with Hall mobility measurements. The results are shown in figs. 5 and 6. For the interpretation it is important to point out that the Hall mobility contains only the mass value in the current direction, whereas SdH oscillations yield a measure of the cyclotron
I. .!%elc et al. /Mechanical
stress influence of effective masses
173
T=2.0 K N ,“v=3.4~10’~ m -2 J.P in [OOl]
Compression P=lBONmm -2
0
2
4
T
6
61-
Fig. 2. (110) dinal stressP
surface: AR/R versusB at 2.0 K for Ninv = 3.4 X 1016 m-*. Parameter in [OOl] direction. The angle between B and the (110) surface is 25”.
is longitu-
mass m, = (mplm,2)1/2. Here, mpl and mp2 are the principal masses. At first view two things are surprising: the large mobility variation with stress and the inconsistent mobility changes with stress direction. The experimental data cannot be understood in terms of a change in relaxation time 7. This is in accordance with the analysis of SdH oscillations with respect to the Dingle temperature, r,. Proof of this suggestion is the comparison between the strong mobility decrease and the simultaneous large amplitude increase of the SdH oscillations for a (110) surface under compression in the [OOl] direction (see figs. 2 and 5). For the further discussion we suppose for , Si n-Channel
I
(110)
Tension: P=160Nmm-2
T=2.05K N I”” =3.4x10’” m -2 J.P in [liO] 4 B, (110)=0°
Compression: P=195Nmm-2 \/ I
0
I
2
I
I
4
6
1
0
B-
pig. 3. (110) surface: AR/R versus B at 2.0 K for Ninv = 3.4 X 1016 m-*. dinal stressP in [ 1101 direction.
Parameter
is longitu-
174
I. Eisele et al. /Mechanical
stress influence of effective masses
Sin-CHANNEL
I
(1001
0.28 -
9 mo
_ 0.24
o:P=
0
..P=-145 N/mm2 I.P=+230N/mm2 0.16 I 0
I 2
I 4
I -2 Ill
6~10'~
Fig. 4. (100) surface: cyclotron mass of electrons as a function ofNinv. (0) without mechanical stress; cn) CompressionP = 230 N/mm’ in [OOl] direction; (x) tensionp = 145 N/mm2 in [OOl] direction.
simplicity that mainly changes of the masses are responsible for the stress dependent mobility deviations. Combining mobilities of the two stress directions in fig. 5, it is possible to get an idea about the variation of the cyclotron mass with stress. For the evaluation it has to be taken into account that the ratio of longitudinal and transverse stiffness coefficients is about 3, in other words, the transverse mobility was taken at a stress which is l/3 of the longitudinal value. A qualitative estimate only yields a change of m, if compression is applied in the [OOl] direction. In this case the mass value should increase, which is in good agreement with the experimental value m, = 0.46~~~. Compression in [ 1101 and tension in [OOl] as well as [ 1101 direction show almost no deviation from the unstressed case with m, = 0.39~2,. Agreement between mobility measurements (fig. 6) and SdH amplitudes (fig. 1) is also obtained for the (100) surface. For tension only a small mobility increase has been detected, whereas compression lowers the mobility significantly. The cyclotron mass m, (fig. 4) deviates under stress in the same manner and confirms the relation between stress and effective mass or, in other words, the importance of the lattice distortion. Piezomeasurements at low temperatures were reported earlier [93 and correspond well to the Hall mobility data. On the basis of the present paper the piezoresistance results for (100) as well as (110) surface orientations have to be interpreted in terms of an effective mass change inside of one particular subband system. Evidence for the theoretically
I. Eisele et al. /Mechanical
stress injluence of effective masses
,
O.B-
JJ’inWl
SildlWIdm
_!!I
vs
175
T=4.2 K
0.4. I WI 0.2. A: P=o 0: Compression P-75 X: Compression
o : Tension
P=210
P-155 Nmm
0 J,P in liO]
0.6 m2 vs 0.4 t IQ 0.2 A:
P=o
x : Compression P=200Nmm-2 o
0
: Tension
P=170 N mm-’
6
10 N I””
Fig. 5. (110) surface: Hall mobility /.JHversusNin, P in [ 0011 and [ 1101 direction, respectively.
20
me2
5OhO’”
-
for B = 1 T. Parameter is longitudinal stress
expected population effect between different subbands could not be seen at all, This is especially surprising for the (I 10) surface where supposedly the energy splitting is very small [8]. Adding up all the experimental details we come to the conclusion that the effective mass values depend on the Si-SK?, interface and are closely related to the technological processing. Especially for the (1 IO) surface we could show a preexisting stress which is uniaxial along the surface. For all n- and p-type surface orientations stress might be an explanation for the discrepancies of the mass values in the literature. The Hall mobility results give evidence for large changes of the intravalley masses in the current direction. This can only be explained if the constant energy surfaces for the electrons are remarkably distorted and the parabolic E(k) relation becomes non-quadratic.
176
I. Eisele et al. /Mechanical stress influence of effective masses
Si n-Chad nool
0.6 4
m= vs
0.4 t PH
0.2~
q:
Tension
0
5
T
5
10
P=ZlO N my -2
: o : P=150 mm2
N mm -2 5oi10’5
N I”” -
Fig. 6. (100) direction.
surface:
pi
versusNinv
for B = 1 T. Parameter
is longitudinal
stress P in [OOl]
Acknowledgement The authors would like to thank E. Doering for the device fabrication and Z. Cehovec for sample preparation and technical assistance. This work comprises part of the Diplom-thesis of H.G. to be submitted to the Physics Department of the Technical University Munich.
References [l]
For a review see: G. Dorda, in: Festkarperprobleme, Vol. 13 (Pergamon-Vieweg, p. 215. [2] G. Landwehr, in: FestkGrperprobleme, Vol. 15 (Pergamon-Vieweg, 1975) p.49. [3] T. Neugebauer, K. von Klitzing, G. Landwehr and G. Dorda, Solid State Commun. (1975) 295.
1973)
17
I. Eisele et al. /Mechanical
stress influence of effective masses
[4] I. Eisele and G. Dorda, 3. Appl. Phys., submitted. [5] L.M. Roth and P.N. Argyres, in: Semiconductors and Semimetals, Vol. 1 (Academic New York, 1966) p. 159. [6] E. Banger& private communication. [7] C.S. Smith, Phys. Rev. 94 (1954) 42. [S] F. Stern, private communication. [9] G. Dorda and 1. Eisele, Phys. Status Solidi (a) 20 (1973) 263. [lo] B. Vinter and F. Stern, Surface Sci. 58 (1976) 141. [l l] T. Ando, Surface Sci. 58 (1976) 128. [12] J.L. Smith and P.J. Stiles, Phys. Rev. Letters 29 (1972) 102. [ 131 I. Eisele, H. Gesch and G. Dorda, Solid State Commun., 18 (1976) 743.
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Press,