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Influence of a parallel magnetic field on subband energies in Si(001) electron inversion layers
Surface Science 170 (1986) 353-358 North-Holland, Amsterdam
353
INFLUENCE OF A PARALLEL MAGNETIC FIELD ON SUBBAND E N E R G I E S IN Si(001) E L E C T R O N I N V E R S I O N L A Y E R S U. K U N Z E Institut fur Elektrophysik, Technische Universiti~t Braunschweig, and Hochmagnetfeldanlage der Technischen Universit~t Braunschweig, D-3300 Braunschweig, Fed Rep. of Germany
Received 1 July 1985; accepted for publication 13 September 1985
Using electron tunneling, the energies of subbands of the anisotropic system in Si(001) electron inversion layers are investigated under high magnetic fields parallel to the surface. The wave functions' spread perpendicular to the surface, and its dependence on the depletion field, is determined from the diamagnetic rise in energy of the subband minima, which is measured with magnetic fields along [110]. Measurement with arbitrary magnetic field directions parallel to the surface, where the [100] and [010] valley pairs are not equivalent, show. for the first time, the splitting of each of the primed levels into two levels.
1. Introduction W h e n a strong magnetic field is applied parallel to a semiconductor surface space charge layer, the quantized subbands are raised in their m i n i m u m energies, and the centers of the corresponding E ( k ) parabolas are displaced with respect to kll = 0 [1]. These effects depend on the spatial extent of the b o u n d state wave functions perpendicular to the interface, which is larger for excited subbands than for the ground subband, and which is a function of the surface field as well as of the depletion field. This has been demonstrated in infrared absorption experiments on Si inversion layers by the shifting and broadening of the resonance line for an intersubband transition [2]. In a tunneling experiment the parallel wave vector is insignificant. Hence one directly obtains the m i n i m u m energy of a subband relative to the semiconductor Fermi level and its change due to a parallel magnetic field, as shown by the work of Tsui [3] on tunneling spectroscopy of accumulation layers on degenerate InAs. Tunneling spectroscopy has also been employed for extensive investigations of the subband structure in electron inversion layers at Si/SiO2 interfaces [4]. Subsequently, the effect of high magnetic fields perpendicular to the (001) surface on the subband structure was studied by tunneling [5]. In the course of this work, the magnetic field has also been applied parallel to the surface along 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) and Y a m a d a Science F o u n d a t i o n
354
U. Kunze / Parallel magnetic field and subband energies
the [110] direction, in order to demonstrate the absence of those structures in the tunneling characteristics dZI/dV 2 which are due to Landau quantization. A comparison of this recording with one taken at zero magnetic field showed for the first time the diamagnetic shift of those subbands, which arise from the anisotropic valleys along the [100] and [010] directions. Since these primed subhands cause much larger structures in the second derivative characteristics d Z I / d V 2 than the unprimed subbands of the [001] valleys [4], they are particularly well suited for a more detailed study of the influence of high parallel magnetic fields on their minimum energies.
2. Theoretical background The energy shift of the ith subband minimum of the [100] valleys in a parallel magnetic field is approximately given by [1] e2B2 ( c°s2~ + sin2~ ) 8,2,
E,(B)-E,(O)=AE,(B)=~---
m,
m~
(1)
where the angle ~ is defined as given in fig. 1, m t and m~ are the principal masses of the constant-energy ellipses in the k-plane, and 6, = ( ( z 2) - (z)2) I/2 is the spread of the wave function perpendicular to the surface in the absence of magnetic field. The corresponding expression for the diamagnetic rise of subbands which belong to the [010] valleys is obtained by interchanging m, and m~ in eq. (1). Evidently, the two valley pairs of primed subbands are equivalent only for B][[110] (g, = 45°), whereas the fourfold valley degeneracy is lifted for arbitrary directions of B. The perturbation result eq. (1) is valid if the magnetic field is low enough that the radius of the cyclotron orbit in the ith subband q. = [(2i + 1 ) h / e B ] ~/2 is large compared to the spread 3, [6]. Since tunneling spectroscopy gives the subband energies E, relative to the Fermi level E v, we have to examine the change of EF: with magnetic field, which can be written in the electric quantum limit as
E v ( B ) - Ev(O ) = AEo( B) + A( E v - E,,)( B).
(2)
The diamagnetic rise of the lowest subband, A E o, is partially canceled by the
[°~°]
O••s
~ _- [~oo] i(001l
Fig. 1. Schematic diagram of the constant-energy lines of Si(001) surfaces. The magnetic field direction is indicated.
U. Kunze / Parallel magnetic field and subband energies
355
second term A(E F - E 0 ) , which reflects the increase of the density of states A D - B 2 [7], when the surface electron density remains unaffected by the magnetic field. The magnetic field dependence of the density of states is of particular importance in semicondcutors which have small effective masses. In ref. [3], e.g., the decrease in the energy difference E F - E 0 is most easily explained by the density-of-states effect rather than the diamagnetic rise, because the surface electron density, influenced by the work function difference of the electrodes, can be assumed as constant. However, in Si both the corrections given in eq. (2) are still below the detection limit even at magnetic fields as high as 15 T, hence the shift in the bias positions of subband-edge induced structures in d2I/dV 2 directly reflects the diamagnetic term of eq. (1).
3. Experimental details The samples were MOS tunnel junctions prepared on p-type Si with 3.5 f2 cm resistivity at 300 K. Their geometry was similar to a field-effect transistor with channel dimensions of 33 × 900 /tm 2 length by width. Ti was used as counterelectrode which creates a typical built-in surface electron density of 2.8 × 1012 cm -2. The subband energies were determined from the dip positions in the second derivative characteristics d2I/dV 2 taken at 4.2 K [4]. Different surface densities of depletion charge have been established by the application of a bias voltage at the substrate. We have verified by three-terminal measurements, as described recently in ref. [8], that the voltage drop across the channel series resistance was always negligible. Due to the influence of the applied tunnel voltage on the surface electron density, a systematic error may affect the determination of the change of subband energies from the bias voltage shift of the corresponding structures in d2I/dV 2. However, previous investigations [8] have proved this error to amount to less than 1%.
4. Results and discussion Fig. 2 shows the diamagnetic shift of E v to E 5, as a function of magnetic field applied in the [110] direction. The positions of dips in d2I/dV 2 were obtained by fitting the lower part of a dip by a parabola. The error in their center position is estimated to about _+ 0.1 mV. The E 0, dip has not been evaluated because it coincides with the E~ dip. Structures of higher unprimed subbands coinciding with those of primed subbands, e.g. E 3 with El,, E s with E2,, etc. [4,8], are assumed to increase the error in the positions only slightly, because these dips are much weaker than those of primed levels.
U. Kunze / Parallel magnetic fieM and subband energies
356 i
6
i
i
i
TF 138
5
Si
/..
BII[IIO]
/ /~ 5'
(001)
.~- 3 r-
~2 ~
2'
t'-
0
0
50
100
150
200
B 2 {T 2 ) Fig. 2. Diamagnetic energy shift of primed subbands versus square of magnetic field. Straight lines are fits to experimental points. Depletion charge density Nd~v~=- 2.4× l0 II cm 2 T = 4.2 K.
The straight lines in the AE-versus-B 2 plot of fig. 2 show that in this case the perturbation result given in eq. (1) holds up to B = 15 T. A significant deviation is expected when Ndep~ is of the order of 10 ]° c m - 2 o r less, where the spread of the wave function ~, is comparable to the cyclotron radius. According to eq. (1), we can determine ~, from the slope of the linear fits. Fig. 3 presents data points obtained at different depletion charge densities, which have been calculated from the applied substrate voltage and the bulk doping level [4]. Since the spatial extent of the wave functions of the higherlying subbands is mainly determined by the depletion-layer potential, a simple calculation of 8g can be performed where a constant surface field F = eNdepl/e is assumed. Within this triangular-well approximation the average value of the separation z of an electron in the ith-subband from the interface is z, = 2Ei/3eF, and the average value of z= is 6zf/5 [9]. Thus the spread 6, can be calculated from 1
1 (
h2
6i----~zi-~-~ 12mzeF
) l/3 4j , ( i + !~a/3
(3)
where rn. = 0.19m 0. Though this simple model neglects the inversion layer potential and further corrections arising from, e.g. the curvature of the depletion potential and the penetration into the oxide barrier, the results of eq. (3) agree reasonably well with the experimental data. Fig. 4 shows recordings of tunneling characteristics at a high magnetic field applied at different angles 4'. At q, = 45 °, each dip reflects the minimum of one
U. Kunze / Parallel magnetic fieM and subband energies i
12
i
i
i
5'o" ~
357
i
TF 136
E 10 r"
N
--
8
N
v ,
6
A
% v
4 I
I
I
i
I
2
3
4
5
Ndep[ (1011/cm2) Fig. 3. Spread of the wave function of primed subbands as a function of depletion charge density. Solid lines are calculated from the triangular potential model.
1
i
i
TF 138
1'//~
Vs=0.5V
45 °
.2'f~
/
A i
i
i
3'
7'6'5'4'
,I ,I
,
2
', --
45
"
"
i
2'
1'
,, "
230 -4 15
~
II
en
0 I
-120 -100 -80 -60 bias (rnV)
-40
-20
I
I
-120 -100 -80 bias (mY) -
60
I -
40
-20
Fig. 4. Second derivative characteristics d e l / d V 2 at fixed magnetic field, applied at various angles q~. V~ denotes the substrate voltage. Ndepn-= 1.8 X l0 II cm 2 T = 4.2 K. Fig. 5. Calculated positions of the subband levels at fixed magnetic field as a function of ~.
U. Kunze / Parallel magnetic field and subband energies
358
of the primed subbands. The splitting of the subband levels at 4) < 45 ° leads to a phase change in the oscillatory characteristics, which occurs when the splitting exceeds the separation of consecutive subbands. It is not resolved as single structures because the half-width of the dips of about 7 mV is c o m p a r a ble to the level separation. According to eq. (1) the split energy levels can be expressed as
Ei( B, 4,) = E , ( B , 45 °) + AE, -ml --- mt COS 24), m I+ m t
(4)
where A E i is the magnetic field induced shift of the ith m i n i m u m at 4, = 45 °, and the positive and negative signs refer respectively to the [100] and [010] valleys. Fig. 5 shows the positions of s u b b a n d dips calculated from eq. (4) using bulk mass parameters and the measured shifts A E,. As long as the splitting of a given subband level is too small to resolve it, we expect the formation of a single dip of reduced amplitude, whose position is the same as for 4, = 45 °. This is exactly what is observed in fig. 4 for subbands of low index i. If the splitting E, I)°°1 - E~I°1°] exceeds the separation E~I°1°1 - Ei I1°°1, a new dip is formed by the latter two levels, whose position is about midway between the dips of E,(B, 45°). The result is oscillations of opposite phase as observed. The position of the phase change determined from fig. 5, e.g. between the levels 5' and 6' at 4, = 30 °, is in agreement with the recording in fig. 4.
Acknowledgement The author would like to express his sincere thanks to Professor G. Lautz for his p e r m a n e n t interest in this work.
References [1] [2] [3] [4] [5] [6] [7] [8] [91
F. Stern and W.E. Howard, Phys. Rev. 163 (1967) 816. W. Beinvogl, A. Kamgar and J.F. Koch, Phys. Rev. B14 (1976) 4274. D.C. Tsui, Solid State Commun. 9 (1971) 1789. U. Kunze, J. Phys. C17 (1984) 5677. U. Kunze and G. Lautz, Surface Sci. 142 (1984) 314. T. Ando, J. Phys. Soc. Japan 44 (1978) 475. J.H. Crasemann, U. Merkt and J.P. Kotthaus, Phys. Rev. B28 (1983) 2271. U. Kunze, Phys. Rev. B32 (1985) 5328. F. Stem, Phys. Rev. B5 (1972) 4891.
353
INFLUENCE OF A PARALLEL MAGNETIC FIELD ON SUBBAND E N E R G I E S IN Si(001) E L E C T R O N I N V E R S I O N L A Y E R S U. K U N Z E Institut fur Elektrophysik, Technische Universiti~t Braunschweig, and Hochmagnetfeldanlage der Technischen Universit~t Braunschweig, D-3300 Braunschweig, Fed Rep. of Germany
Received 1 July 1985; accepted for publication 13 September 1985
Using electron tunneling, the energies of subbands of the anisotropic system in Si(001) electron inversion layers are investigated under high magnetic fields parallel to the surface. The wave functions' spread perpendicular to the surface, and its dependence on the depletion field, is determined from the diamagnetic rise in energy of the subband minima, which is measured with magnetic fields along [110]. Measurement with arbitrary magnetic field directions parallel to the surface, where the [100] and [010] valley pairs are not equivalent, show. for the first time, the splitting of each of the primed levels into two levels.
1. Introduction W h e n a strong magnetic field is applied parallel to a semiconductor surface space charge layer, the quantized subbands are raised in their m i n i m u m energies, and the centers of the corresponding E ( k ) parabolas are displaced with respect to kll = 0 [1]. These effects depend on the spatial extent of the b o u n d state wave functions perpendicular to the interface, which is larger for excited subbands than for the ground subband, and which is a function of the surface field as well as of the depletion field. This has been demonstrated in infrared absorption experiments on Si inversion layers by the shifting and broadening of the resonance line for an intersubband transition [2]. In a tunneling experiment the parallel wave vector is insignificant. Hence one directly obtains the m i n i m u m energy of a subband relative to the semiconductor Fermi level and its change due to a parallel magnetic field, as shown by the work of Tsui [3] on tunneling spectroscopy of accumulation layers on degenerate InAs. Tunneling spectroscopy has also been employed for extensive investigations of the subband structure in electron inversion layers at Si/SiO2 interfaces [4]. Subsequently, the effect of high magnetic fields perpendicular to the (001) surface on the subband structure was studied by tunneling [5]. In the course of this work, the magnetic field has also been applied parallel to the surface along 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) and Y a m a d a Science F o u n d a t i o n
354
U. Kunze / Parallel magnetic field and subband energies
the [110] direction, in order to demonstrate the absence of those structures in the tunneling characteristics dZI/dV 2 which are due to Landau quantization. A comparison of this recording with one taken at zero magnetic field showed for the first time the diamagnetic shift of those subbands, which arise from the anisotropic valleys along the [100] and [010] directions. Since these primed subhands cause much larger structures in the second derivative characteristics d Z I / d V 2 than the unprimed subbands of the [001] valleys [4], they are particularly well suited for a more detailed study of the influence of high parallel magnetic fields on their minimum energies.
2. Theoretical background The energy shift of the ith subband minimum of the [100] valleys in a parallel magnetic field is approximately given by [1] e2B2 ( c°s2~ + sin2~ ) 8,2,
E,(B)-E,(O)=AE,(B)=~---
m,
m~
(1)
where the angle ~ is defined as given in fig. 1, m t and m~ are the principal masses of the constant-energy ellipses in the k-plane, and 6, = ( ( z 2) - (z)2) I/2 is the spread of the wave function perpendicular to the surface in the absence of magnetic field. The corresponding expression for the diamagnetic rise of subbands which belong to the [010] valleys is obtained by interchanging m, and m~ in eq. (1). Evidently, the two valley pairs of primed subbands are equivalent only for B][[110] (g, = 45°), whereas the fourfold valley degeneracy is lifted for arbitrary directions of B. The perturbation result eq. (1) is valid if the magnetic field is low enough that the radius of the cyclotron orbit in the ith subband q. = [(2i + 1 ) h / e B ] ~/2 is large compared to the spread 3, [6]. Since tunneling spectroscopy gives the subband energies E, relative to the Fermi level E v, we have to examine the change of EF: with magnetic field, which can be written in the electric quantum limit as
E v ( B ) - Ev(O ) = AEo( B) + A( E v - E,,)( B).
(2)
The diamagnetic rise of the lowest subband, A E o, is partially canceled by the
[°~°]
O••s
~ _- [~oo] i(001l
Fig. 1. Schematic diagram of the constant-energy lines of Si(001) surfaces. The magnetic field direction is indicated.
U. Kunze / Parallel magnetic field and subband energies
355
second term A(E F - E 0 ) , which reflects the increase of the density of states A D - B 2 [7], when the surface electron density remains unaffected by the magnetic field. The magnetic field dependence of the density of states is of particular importance in semicondcutors which have small effective masses. In ref. [3], e.g., the decrease in the energy difference E F - E 0 is most easily explained by the density-of-states effect rather than the diamagnetic rise, because the surface electron density, influenced by the work function difference of the electrodes, can be assumed as constant. However, in Si both the corrections given in eq. (2) are still below the detection limit even at magnetic fields as high as 15 T, hence the shift in the bias positions of subband-edge induced structures in d2I/dV 2 directly reflects the diamagnetic term of eq. (1).
3. Experimental details The samples were MOS tunnel junctions prepared on p-type Si with 3.5 f2 cm resistivity at 300 K. Their geometry was similar to a field-effect transistor with channel dimensions of 33 × 900 /tm 2 length by width. Ti was used as counterelectrode which creates a typical built-in surface electron density of 2.8 × 1012 cm -2. The subband energies were determined from the dip positions in the second derivative characteristics d2I/dV 2 taken at 4.2 K [4]. Different surface densities of depletion charge have been established by the application of a bias voltage at the substrate. We have verified by three-terminal measurements, as described recently in ref. [8], that the voltage drop across the channel series resistance was always negligible. Due to the influence of the applied tunnel voltage on the surface electron density, a systematic error may affect the determination of the change of subband energies from the bias voltage shift of the corresponding structures in d2I/dV 2. However, previous investigations [8] have proved this error to amount to less than 1%.
4. Results and discussion Fig. 2 shows the diamagnetic shift of E v to E 5, as a function of magnetic field applied in the [110] direction. The positions of dips in d2I/dV 2 were obtained by fitting the lower part of a dip by a parabola. The error in their center position is estimated to about _+ 0.1 mV. The E 0, dip has not been evaluated because it coincides with the E~ dip. Structures of higher unprimed subbands coinciding with those of primed subbands, e.g. E 3 with El,, E s with E2,, etc. [4,8], are assumed to increase the error in the positions only slightly, because these dips are much weaker than those of primed levels.
U. Kunze / Parallel magnetic fieM and subband energies
356 i
6
i
i
i
TF 138
5
Si
/..
BII[IIO]
/ /~ 5'
(001)
.~- 3 r-
~2 ~
2'
t'-
0
0
50
100
150
200
B 2 {T 2 ) Fig. 2. Diamagnetic energy shift of primed subbands versus square of magnetic field. Straight lines are fits to experimental points. Depletion charge density Nd~v~=- 2.4× l0 II cm 2 T = 4.2 K.
The straight lines in the AE-versus-B 2 plot of fig. 2 show that in this case the perturbation result given in eq. (1) holds up to B = 15 T. A significant deviation is expected when Ndep~ is of the order of 10 ]° c m - 2 o r less, where the spread of the wave function ~, is comparable to the cyclotron radius. According to eq. (1), we can determine ~, from the slope of the linear fits. Fig. 3 presents data points obtained at different depletion charge densities, which have been calculated from the applied substrate voltage and the bulk doping level [4]. Since the spatial extent of the wave functions of the higherlying subbands is mainly determined by the depletion-layer potential, a simple calculation of 8g can be performed where a constant surface field F = eNdepl/e is assumed. Within this triangular-well approximation the average value of the separation z of an electron in the ith-subband from the interface is z, = 2Ei/3eF, and the average value of z= is 6zf/5 [9]. Thus the spread 6, can be calculated from 1
1 (
h2
6i----~zi-~-~ 12mzeF
) l/3 4j , ( i + !~a/3
(3)
where rn. = 0.19m 0. Though this simple model neglects the inversion layer potential and further corrections arising from, e.g. the curvature of the depletion potential and the penetration into the oxide barrier, the results of eq. (3) agree reasonably well with the experimental data. Fig. 4 shows recordings of tunneling characteristics at a high magnetic field applied at different angles 4'. At q, = 45 °, each dip reflects the minimum of one
U. Kunze / Parallel magnetic fieM and subband energies i
12
i
i
i
5'o" ~
357
i
TF 136
E 10 r"
N
--
8
N
v ,
6
A
% v
4 I
I
I
i
I
2
3
4
5
Ndep[ (1011/cm2) Fig. 3. Spread of the wave function of primed subbands as a function of depletion charge density. Solid lines are calculated from the triangular potential model.
1
i
i
TF 138
1'//~
Vs=0.5V
45 °
.2'f~
/
A i
i
i
3'
7'6'5'4'
,I ,I
,
2
', --
45
"
"
i
2'
1'
,, "
230 -4 15
~
II
en
0 I
-120 -100 -80 -60 bias (rnV)
-40
-20
I
I
-120 -100 -80 bias (mY) -
60
I -
40
-20
Fig. 4. Second derivative characteristics d e l / d V 2 at fixed magnetic field, applied at various angles q~. V~ denotes the substrate voltage. Ndepn-= 1.8 X l0 II cm 2 T = 4.2 K. Fig. 5. Calculated positions of the subband levels at fixed magnetic field as a function of ~.
U. Kunze / Parallel magnetic field and subband energies
358
of the primed subbands. The splitting of the subband levels at 4) < 45 ° leads to a phase change in the oscillatory characteristics, which occurs when the splitting exceeds the separation of consecutive subbands. It is not resolved as single structures because the half-width of the dips of about 7 mV is c o m p a r a ble to the level separation. According to eq. (1) the split energy levels can be expressed as
Ei( B, 4,) = E , ( B , 45 °) + AE, -ml --- mt COS 24), m I+ m t
(4)
where A E i is the magnetic field induced shift of the ith m i n i m u m at 4, = 45 °, and the positive and negative signs refer respectively to the [100] and [010] valleys. Fig. 5 shows the positions of s u b b a n d dips calculated from eq. (4) using bulk mass parameters and the measured shifts A E,. As long as the splitting of a given subband level is too small to resolve it, we expect the formation of a single dip of reduced amplitude, whose position is the same as for 4, = 45 °. This is exactly what is observed in fig. 4 for subbands of low index i. If the splitting E, I)°°1 - E~I°1°] exceeds the separation E~I°1°1 - Ei I1°°1, a new dip is formed by the latter two levels, whose position is about midway between the dips of E,(B, 45°). The result is oscillations of opposite phase as observed. The position of the phase change determined from fig. 5, e.g. between the levels 5' and 6' at 4, = 30 °, is in agreement with the recording in fig. 4.
Acknowledgement The author would like to express his sincere thanks to Professor G. Lautz for his p e r m a n e n t interest in this work.
References [1] [2] [3] [4] [5] [6] [7] [8] [91
F. Stern and W.E. Howard, Phys. Rev. 163 (1967) 816. W. Beinvogl, A. Kamgar and J.F. Koch, Phys. Rev. B14 (1976) 4274. D.C. Tsui, Solid State Commun. 9 (1971) 1789. U. Kunze, J. Phys. C17 (1984) 5677. U. Kunze and G. Lautz, Surface Sci. 142 (1984) 314. T. Ando, J. Phys. Soc. Japan 44 (1978) 475. J.H. Crasemann, U. Merkt and J.P. Kotthaus, Phys. Rev. B28 (1983) 2271. U. Kunze, Phys. Rev. B32 (1985) 5328. F. Stem, Phys. Rev. B5 (1972) 4891.