- Email: [email protected]
AlAs quantum-wire superlattices
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999 Article No. spmi.1998.0640 Available online at http://www.idealibrary.com on
Study and characterization by magnetophonon resonance of the energy structuring in GaAs/AlAs quantum-wire superlattices T. F ERRUS† , B. G OUTIERS , L. R ESSIER , J. P. P EYRADE Laboratoire de Physique de la Mati`ere Condens´ee UMR 58-30, INSA, 31077 Toulouse Cedex, France
J. G ALIBERT Service National des Champs Puls´es UMR 58-30, INSA, 31077 Toulouse Cedex, France ´ J. A. P ORTO , J. S ANCHEZ - DEHESA Departamento de F´ıs´ıca Te´orica de la Materia Condensada, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain (Received 26 October 1998) We present the characterization of the band structure of GaAs/AlAs quantum-wire 1D superlattices performed by magnetophonon resonance with pulsed magnetic fields up to 35 T. The samples, generated by the ‘atomic saw method’ from original quantum-well 2D superlattices, underwent substantial modifications of their energy bands built up on the X-states of the bulk. We have calculated the band structure by a finite element method and we have studied the various miniband structures built up of the masses m t and m l of GaAs and AlAs at the point X. From an experimental point of view, the main result is that in the 2D case we observe only resonances when the magnetic field B is applied along the growth axis whereas in the 1D case we obtain resonances in all magnetic field configurations. The analysis of the maxima (or minima for B//E) in the resistivity ρx y as a function of B allows us to account, qualitatively and semi-quantitatively, for the band structure theoretically expected. c 1999 Academic Press
Key words: GaAs/AlAs 1D superlattices, quantum wires, miniband structure, magnetophonon resonance.
1. Introduction In practice, the study of the electronic properties of quantum-wire 1D superlattices, except T-wires, relies mainly on two costly modes of sample generation, those using growth vicinal steps [1] and those generated by controlled dislocation slipping from quantum-plane 2D superlattices [2]. Due to the thickness of the steps, the first mode induces an important wire overlapping whereas the second allows the best separation of the wires. We propose in this paper to analyse the restructuring of the energies in k-space at † E-mail: [email protected]
0749–6036/99/010213 + 07
$30.00/0
c 1999 Academic Press
214
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
z
z
x
y
x
A
y
B
Fig. 1. Representation of the space restructuring by the ‘atomic saw method’: A, 2D superlattice; B, 1D quantum-wire superlattice.
x (001) ml mt mt
(010) mt ml mt
z (110)
mt mt
ml (100)
y (110) Fig. 2. Isoenergy surfaces in the bulk sample.
the time of the transition from 2D to 1D induced by the ‘atomic saw method’ [2] and to experimentally check these modifications by magnetophonon resonance on GaAs/AlAs:Si.
2. Samples and miniband characteristics The initial 2D sample is a GaAs/AlAs:Si superlattice with AlAs wells and GaAs barriers both 20 Å wide, doped at 1 × 1016 cm−3 . The direction of the initial growth layers is (001). The final 1D sample is a wire superlattice with mean wire width of 20 Å along (001), a height of 150 Å along (110) and infinite length along (110) (Fig. 1). Both samples are indirect, the lower minibands being built on the bulk X-states because of the thickness of the wells and barriers. The energy shift between wells and barriers is taken to be approximately equal to 260 meV. For the 2D sample we have two families of minibands: those built with the bulk mass m l of the X-states along x and m t along y and z; and the doubly-degenerate minibands, built with m t along x, m l and m t or m t and m l along y and z, respectively (Fig. 2). Similarly, for the 1D sample, we have a first family of minibands generated by m l and m t , respectively, along x and y and a second family with m t along x and m l and m t or m t and m l along y and z respectively. The first Brillouin zone of the last sample is represented in Fig. 3.
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ky M2
K2 M3 K1 kx
G M1
K3 Fig. 3. Representation of the first Brillouin zone and characteristic points K1 , K2 , K3 , M1 , M2 and M3 .
500 B
A
400
E(k) (meV)
E(k) (meV)
500
300 200 100 0
400 300 200 100
0
1
2
3
0
0
1
kδ
2
3
kδ
Fig. 4. 2D band structures calculated with A, m t = 0.19m e− and B, m l = 1.4m e− .
3. Band structure calculation In both cases, the two miniband structures of the 2D sample are trivial and are represented in Fig. 4A and B. As for the 1D sample, we have to write the Hamiltonian in relation to the natural axes x (001), y (110) and z (110). These axes have been chosen so as to ensure the best homogeneity of the wire dimensions and an optimization of the width of the sliding steps. For the first family (m l along x) we have H =−
~2 2 ~2 2 kx • − (k • − k z2 ) + V (x, y), 2m l 2m t y
(1)
and for the second one (m t along x) H =−
~2 2 ~2 2 ~2 2 kx • − kY • − k + V (x, y). 2m t 2m Y 2m z z
(2)
k x • , k y • and kY • being the k operator component in the reference (001) and (110) and k z its scalar value along (110). Taking the wavefunction φ = exp(ik z z)ϕ(x, y), we find the following equalities: −1 1 1 1 1 1 1 1 = + − + , , α = m −1 Y 2 mt ml mt ml mt ml (3) −1 −1 2 = m Y (1 − α ) kY • = k y • + αk z . mz
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100 50 0
E(kx, ky) (meV)
E(kx, ky) (meV)
150
A G
M2
K2
G
200 E(kx, ky) (meV)
E(kx,ky) (meV)
200
150 100 50 0
C G
M3
K1
G
300 250 200 150 100 50 B 0 G 300 250 200 150 100 50 0
K2
M2
G
D G
M3
K1
G
Fig. 5. 1D band structures calculated along the path G–M2 –K2 –G for A, the nondegenerated family and B, the twice degenerated family and along G–M3 –K1 –G for C, the first family and D, the second one.
For the calculations, we have replaced the GaAs and AlAs longitudinal mass by their mean value m l = 1.4m e− according to Guzzi and Staehli [3]; as for the transverse mass, it is taken to be m t = 0.19m e− for the two materials. Concerning the second family, it is adequate to calculate the dispersion equation of the energy for k z = 0 and move the origin of kY for k z 6= 0. The band structures calculated by the spline-function method [4] are represented in Fig. 5A and B for a mean wire section of approximately 20 Å × 150 Å.
4. Magnetic field effect in the 1D case In the absence of a magnetic field, all band structures could be approximated by two half parabolas: E i (k x , k y , k z ) =
~2 2 ~2 2 ~2 2 kx + ky + k + Ctei . 2m xi 2m yi 2m zi z
(4)
We have used i as the index characterizing the miniband number for each miniband family. We assign positive values (respectively, negative values) for the masses if the half parabola is set at the first Brillouin zone centre (respectively, at the first Brillouin zone boundary), the two half parabolas being tangentially connected in the inflexion points. In such conditions, in the presence of a magnetic field, Landau levels and cyclotron frequencies could appear only if, for example, in the case of B along x: 1 1 ~eB 1xi =√ ≤ (5) ni + ~ωci n i + 2 m yi m zi 2 2 where 1xi is the smallest width of the miniband i profile along y or z, and n i the index of the Landau level which is considered. We remark that m yi and m zi are necessarily of the same sign to obtain Landau levels. For the other field configurations, it suffices to cyclically permute x, y and z. The relation of type (5) allows to define
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the transitions between two Landau levels, intra- or inter-minibands, that would be authorized or, on the contrary, would not exist, for a given magnetic field B configuration and for each miniband family.
5. Analysis of the allowed magnetophonon resonance in the 1D case We will use magnetotransport measurement to analyse them, the electric field being applied along y with a low level so that the only the transitions of a single phonon would be considered. The drift mobility is µd ≈ 6000 cm2 V−1 s−1 . The constraints are eBm −1 c τ > 1 and ~ω > kT where m c is the cyclotron mass of the transition under consideration, τ being given by µd . Considering the values of the masses calculated theoretically, we ought to observe resonance only for B higher than 13 T. Thus, the maximum experimental field being 30 T because of field return, we will explore the allowed resonances between 13 and 30 T. The energy of the phonon responsible for the transition is taken at ~ωLO = 36 meV [5, 6]. In the case of m l along x, because of the weak miniband widths and the energy distance, any inter- or intra-band transition is possible in the range of the magnetic field B we consider. On the contrary, for the second family (m t along x) various transitions are allowed. (a) B along x: the calculations give m yi = 0.2m e− ≈ m y , m z ≈ 0.795m e− and m c = (m yi m zi )1/2 ≈ 0.399m e− . The energy is very nearly parabolic along y and z. We also have to deal with classical intra-band resonance. The resonance fields would be 30.8, 24.7 and 20.6 T, and would correspond to transitions induced by a phonon of 36 meV between, respectively, n = 0 and n = 4, n = 0 and n = 5, n = 0 and n = 6. (b) B along y: the values of the cyclotron masses calculated for the first three minibands likely to interact are: m c1 = 0.373m e− , m c2 = 0.357m e− , m c3 = 0.347m e− . As for the distance between the first and the second miniband, the first and the third and finally the second and the third (between two successive miniband minima) are, respectively, 6.5, 22.5 and 16 meV. The only allowed transitions are all inter-miniband transitions, between the Landau level n 1 = 0 of the first miniband and the level n 3 = 2 of the third miniband which corresponds to a 20 T field, between n 2 = 0 and n 3 = 2 with a 29 T field, between n 2 = 0 and n 3 = 3 with a 20 T field and between n 2 = 0 and n 3 = 4 with a 14 T field. (c) B along z: the cyclotron masses are m c1 = 0.179m e− , m c2 = 0.171m e− and m c3 = 0.167m e− . The energy distance between the first and the third miniband and between the second and the third are, respectively, 22.5 and 16 meV. The only allowed transitions are all inter-miniband transitions between n 1 = 0 of the first miniband and n 3 = 1 with a 18.8 T field, between n 2 = 0 and n 3 = 1 with a 28.5 T field and between n 2 = 0 and n 3 = 2 with a 14 T field.
6. Analysis of the allowed magnetophonon resonance in the 2D case Taking into account the miniband widths along x and the distance between minibands, we cannot have magnetophonon resonance for B in the plane (yz), whichever family we consider. For B along x, there are two cases: (a) family with m l along x: in the range of 13–30 T, the resonance fields corresponding to a cyclotron mass equal to 0.19m e− are 29.6, 19.8 and 14.8 T; (b) family with m t along x: the cyclotron mass is equal to 0.516m e− . No intra-band transitions ought to be observed below 30 T which is the upper limit of our experimental field of investigation.
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0.50 0.25
0.1
d2I/dB2
d2I/dB2
0.2 0.0 –0.1 –0.2 A 0 5
0.00 –0.25 –0.50
10 15 20 25 30 35 B (T) 1.5
–0.75
B 0
5
10 15 20 25 30 35 B (T)
d2I/dB2
1.0 0.5 0.0
–0.5 –1.0
C 0
5
10 15 20 25 30 35 B (T)
Fig. 6. Magnetophonon oscillations obtained on 1D samples at T = 100 K in the configuration E parallel to y and B parallel to A, x, B, y and C, z.
7. Experimental results Magnetophonon resonance shots have been performed with pulsed magnetic fields upto 35 T at S.N.C.M.P. of Toulouse on the two types of sample (1D and 2D) described previously. The curves I (B) exhibit a monotonic behaviour, so we use the current second derivatives in terms of the magnetic field obtained through a signal treatment by a double derivative amplifier to analyse oscillations. The resistivity maxima or minima for B along y due to the absorption of a phonon correspond to the maxima of the second derivatives and are defined at 0.5 T. Figures 6A, B and C represent some typical characteristics of d 2 I /d B 2 obtained for the three field configurations on the 1D sample at 100 K which is the optimal temperature for the observation of these resonances. The series we observe for this sample are: (a) B along x: 30, 23 and 20 T; (b) B along y: 29, 19 and 12 T; (c) B along z: 28 and 18 T. For the 2D sample, no magnetoresistance oscillations have been observed in the configuration B ⊥ x. On the contrary, when B is parallel to x, resonances are apparent at 22 and 13.5 T. We have to compare these values to the 19.8 and 14.8 T previously calculated, which suggests that the real masses are slightly heavier than those calculated in the 2D case.
8. Conclusion The first remark to make is that there is relatively good agreement between the measured resonance fields and the calculated ones, in all cases (1D or 2D) and for the various field configurations. The second remark is that the modelling in the mean slipping-step approximation for the miniband calculations, although rough, is a good enough approach for the 1D superlattice band structures and simulates well the observed resonance. The third remark is that the quantum-wire generation method by controlled dislocation slipping [2] allows effective generation of 1D quantum-wire superlattices from 2D superlattices. The
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unidimensional character is confirmed by the opening of minigaps at the point K for m t along x and at the points K and M2 for m l along x. That miniband family, being the lowest, makes the observation of negative differential resistance in the y direction theoretically possible, which has already been experimentally observed [7].
References [1] — M. Tsuchiya, J. M. Gaines, R. H. Yan, R. J. Simes, P. O. Holtz, L. A Coldren, and P. M. Petroff, Phys. Rev. Lett. 62, 466 (1989). [2] — F. Voillot, M. Goiran, C. Guasch, J. P. Peyrade, L. Dinh, A. Rocher, and E. Bedel, J. Physique III 1803, (1993). [3] — M. Guzzi and J. L. Staehli, Solid State Phenomena 52, 10–22 (1989). [4] — J. A. Porto and J. S´anchez-Dehesa, Phys. Rev. B51, 14352 (1995). [5] — H. Nokuchi, H. Sakaki, T. Takamatsu, and N. Miura, Phys. Rev. B45, 12148 (1992). [6] — F. Aristone, PhD Thesis, Contribution a` l’´etude des processus de diffusions dans les propri´et`es de transport vertical par minibande, INSA (1994). [7] — B. Goutiers, T. Ferrus, J. A. Porto, J. Sanchez-Dehesa, and L. Ressier, Phys. Low-Dim. Struct. 77, 1–2 (1998).
Study and characterization by magnetophonon resonance of the energy structuring in GaAs/AlAs quantum-wire superlattices T. F ERRUS† , B. G OUTIERS , L. R ESSIER , J. P. P EYRADE Laboratoire de Physique de la Mati`ere Condens´ee UMR 58-30, INSA, 31077 Toulouse Cedex, France
J. G ALIBERT Service National des Champs Puls´es UMR 58-30, INSA, 31077 Toulouse Cedex, France ´ J. A. P ORTO , J. S ANCHEZ - DEHESA Departamento de F´ıs´ıca Te´orica de la Materia Condensada, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain (Received 26 October 1998) We present the characterization of the band structure of GaAs/AlAs quantum-wire 1D superlattices performed by magnetophonon resonance with pulsed magnetic fields up to 35 T. The samples, generated by the ‘atomic saw method’ from original quantum-well 2D superlattices, underwent substantial modifications of their energy bands built up on the X-states of the bulk. We have calculated the band structure by a finite element method and we have studied the various miniband structures built up of the masses m t and m l of GaAs and AlAs at the point X. From an experimental point of view, the main result is that in the 2D case we observe only resonances when the magnetic field B is applied along the growth axis whereas in the 1D case we obtain resonances in all magnetic field configurations. The analysis of the maxima (or minima for B//E) in the resistivity ρx y as a function of B allows us to account, qualitatively and semi-quantitatively, for the band structure theoretically expected. c 1999 Academic Press
Key words: GaAs/AlAs 1D superlattices, quantum wires, miniband structure, magnetophonon resonance.
1. Introduction In practice, the study of the electronic properties of quantum-wire 1D superlattices, except T-wires, relies mainly on two costly modes of sample generation, those using growth vicinal steps [1] and those generated by controlled dislocation slipping from quantum-plane 2D superlattices [2]. Due to the thickness of the steps, the first mode induces an important wire overlapping whereas the second allows the best separation of the wires. We propose in this paper to analyse the restructuring of the energies in k-space at † E-mail: [email protected]
0749–6036/99/010213 + 07
$30.00/0
c 1999 Academic Press
214
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
z
z
x
y
x
A
y
B
Fig. 1. Representation of the space restructuring by the ‘atomic saw method’: A, 2D superlattice; B, 1D quantum-wire superlattice.
x (001) ml mt mt
(010) mt ml mt
z (110)
mt mt
ml (100)
y (110) Fig. 2. Isoenergy surfaces in the bulk sample.
the time of the transition from 2D to 1D induced by the ‘atomic saw method’ [2] and to experimentally check these modifications by magnetophonon resonance on GaAs/AlAs:Si.
2. Samples and miniband characteristics The initial 2D sample is a GaAs/AlAs:Si superlattice with AlAs wells and GaAs barriers both 20 Å wide, doped at 1 × 1016 cm−3 . The direction of the initial growth layers is (001). The final 1D sample is a wire superlattice with mean wire width of 20 Å along (001), a height of 150 Å along (110) and infinite length along (110) (Fig. 1). Both samples are indirect, the lower minibands being built on the bulk X-states because of the thickness of the wells and barriers. The energy shift between wells and barriers is taken to be approximately equal to 260 meV. For the 2D sample we have two families of minibands: those built with the bulk mass m l of the X-states along x and m t along y and z; and the doubly-degenerate minibands, built with m t along x, m l and m t or m t and m l along y and z, respectively (Fig. 2). Similarly, for the 1D sample, we have a first family of minibands generated by m l and m t , respectively, along x and y and a second family with m t along x and m l and m t or m t and m l along y and z respectively. The first Brillouin zone of the last sample is represented in Fig. 3.
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ky M2
K2 M3 K1 kx
G M1
K3 Fig. 3. Representation of the first Brillouin zone and characteristic points K1 , K2 , K3 , M1 , M2 and M3 .
500 B
A
400
E(k) (meV)
E(k) (meV)
500
300 200 100 0
400 300 200 100
0
1
2
3
0
0
1
kδ
2
3
kδ
Fig. 4. 2D band structures calculated with A, m t = 0.19m e− and B, m l = 1.4m e− .
3. Band structure calculation In both cases, the two miniband structures of the 2D sample are trivial and are represented in Fig. 4A and B. As for the 1D sample, we have to write the Hamiltonian in relation to the natural axes x (001), y (110) and z (110). These axes have been chosen so as to ensure the best homogeneity of the wire dimensions and an optimization of the width of the sliding steps. For the first family (m l along x) we have H =−
~2 2 ~2 2 kx • − (k • − k z2 ) + V (x, y), 2m l 2m t y
(1)
and for the second one (m t along x) H =−
~2 2 ~2 2 ~2 2 kx • − kY • − k + V (x, y). 2m t 2m Y 2m z z
(2)
k x • , k y • and kY • being the k operator component in the reference (001) and (110) and k z its scalar value along (110). Taking the wavefunction φ = exp(ik z z)ϕ(x, y), we find the following equalities: −1 1 1 1 1 1 1 1 = + − + , , α = m −1 Y 2 mt ml mt ml mt ml (3) −1 −1 2 = m Y (1 − α ) kY • = k y • + αk z . mz
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Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
100 50 0
E(kx, ky) (meV)
E(kx, ky) (meV)
150
A G
M2
K2
G
200 E(kx, ky) (meV)
E(kx,ky) (meV)
200
150 100 50 0
C G
M3
K1
G
300 250 200 150 100 50 B 0 G 300 250 200 150 100 50 0
K2
M2
G
D G
M3
K1
G
Fig. 5. 1D band structures calculated along the path G–M2 –K2 –G for A, the nondegenerated family and B, the twice degenerated family and along G–M3 –K1 –G for C, the first family and D, the second one.
For the calculations, we have replaced the GaAs and AlAs longitudinal mass by their mean value m l = 1.4m e− according to Guzzi and Staehli [3]; as for the transverse mass, it is taken to be m t = 0.19m e− for the two materials. Concerning the second family, it is adequate to calculate the dispersion equation of the energy for k z = 0 and move the origin of kY for k z 6= 0. The band structures calculated by the spline-function method [4] are represented in Fig. 5A and B for a mean wire section of approximately 20 Å × 150 Å.
4. Magnetic field effect in the 1D case In the absence of a magnetic field, all band structures could be approximated by two half parabolas: E i (k x , k y , k z ) =
~2 2 ~2 2 ~2 2 kx + ky + k + Ctei . 2m xi 2m yi 2m zi z
(4)
We have used i as the index characterizing the miniband number for each miniband family. We assign positive values (respectively, negative values) for the masses if the half parabola is set at the first Brillouin zone centre (respectively, at the first Brillouin zone boundary), the two half parabolas being tangentially connected in the inflexion points. In such conditions, in the presence of a magnetic field, Landau levels and cyclotron frequencies could appear only if, for example, in the case of B along x: 1 1 ~eB 1xi =√ ≤ (5) ni + ~ωci n i + 2 m yi m zi 2 2 where 1xi is the smallest width of the miniband i profile along y or z, and n i the index of the Landau level which is considered. We remark that m yi and m zi are necessarily of the same sign to obtain Landau levels. For the other field configurations, it suffices to cyclically permute x, y and z. The relation of type (5) allows to define
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the transitions between two Landau levels, intra- or inter-minibands, that would be authorized or, on the contrary, would not exist, for a given magnetic field B configuration and for each miniband family.
5. Analysis of the allowed magnetophonon resonance in the 1D case We will use magnetotransport measurement to analyse them, the electric field being applied along y with a low level so that the only the transitions of a single phonon would be considered. The drift mobility is µd ≈ 6000 cm2 V−1 s−1 . The constraints are eBm −1 c τ > 1 and ~ω > kT where m c is the cyclotron mass of the transition under consideration, τ being given by µd . Considering the values of the masses calculated theoretically, we ought to observe resonance only for B higher than 13 T. Thus, the maximum experimental field being 30 T because of field return, we will explore the allowed resonances between 13 and 30 T. The energy of the phonon responsible for the transition is taken at ~ωLO = 36 meV [5, 6]. In the case of m l along x, because of the weak miniband widths and the energy distance, any inter- or intra-band transition is possible in the range of the magnetic field B we consider. On the contrary, for the second family (m t along x) various transitions are allowed. (a) B along x: the calculations give m yi = 0.2m e− ≈ m y , m z ≈ 0.795m e− and m c = (m yi m zi )1/2 ≈ 0.399m e− . The energy is very nearly parabolic along y and z. We also have to deal with classical intra-band resonance. The resonance fields would be 30.8, 24.7 and 20.6 T, and would correspond to transitions induced by a phonon of 36 meV between, respectively, n = 0 and n = 4, n = 0 and n = 5, n = 0 and n = 6. (b) B along y: the values of the cyclotron masses calculated for the first three minibands likely to interact are: m c1 = 0.373m e− , m c2 = 0.357m e− , m c3 = 0.347m e− . As for the distance between the first and the second miniband, the first and the third and finally the second and the third (between two successive miniband minima) are, respectively, 6.5, 22.5 and 16 meV. The only allowed transitions are all inter-miniband transitions, between the Landau level n 1 = 0 of the first miniband and the level n 3 = 2 of the third miniband which corresponds to a 20 T field, between n 2 = 0 and n 3 = 2 with a 29 T field, between n 2 = 0 and n 3 = 3 with a 20 T field and between n 2 = 0 and n 3 = 4 with a 14 T field. (c) B along z: the cyclotron masses are m c1 = 0.179m e− , m c2 = 0.171m e− and m c3 = 0.167m e− . The energy distance between the first and the third miniband and between the second and the third are, respectively, 22.5 and 16 meV. The only allowed transitions are all inter-miniband transitions between n 1 = 0 of the first miniband and n 3 = 1 with a 18.8 T field, between n 2 = 0 and n 3 = 1 with a 28.5 T field and between n 2 = 0 and n 3 = 2 with a 14 T field.
6. Analysis of the allowed magnetophonon resonance in the 2D case Taking into account the miniband widths along x and the distance between minibands, we cannot have magnetophonon resonance for B in the plane (yz), whichever family we consider. For B along x, there are two cases: (a) family with m l along x: in the range of 13–30 T, the resonance fields corresponding to a cyclotron mass equal to 0.19m e− are 29.6, 19.8 and 14.8 T; (b) family with m t along x: the cyclotron mass is equal to 0.516m e− . No intra-band transitions ought to be observed below 30 T which is the upper limit of our experimental field of investigation.
218
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0.50 0.25
0.1
d2I/dB2
d2I/dB2
0.2 0.0 –0.1 –0.2 A 0 5
0.00 –0.25 –0.50
10 15 20 25 30 35 B (T) 1.5
–0.75
B 0
5
10 15 20 25 30 35 B (T)
d2I/dB2
1.0 0.5 0.0
–0.5 –1.0
C 0
5
10 15 20 25 30 35 B (T)
Fig. 6. Magnetophonon oscillations obtained on 1D samples at T = 100 K in the configuration E parallel to y and B parallel to A, x, B, y and C, z.
7. Experimental results Magnetophonon resonance shots have been performed with pulsed magnetic fields upto 35 T at S.N.C.M.P. of Toulouse on the two types of sample (1D and 2D) described previously. The curves I (B) exhibit a monotonic behaviour, so we use the current second derivatives in terms of the magnetic field obtained through a signal treatment by a double derivative amplifier to analyse oscillations. The resistivity maxima or minima for B along y due to the absorption of a phonon correspond to the maxima of the second derivatives and are defined at 0.5 T. Figures 6A, B and C represent some typical characteristics of d 2 I /d B 2 obtained for the three field configurations on the 1D sample at 100 K which is the optimal temperature for the observation of these resonances. The series we observe for this sample are: (a) B along x: 30, 23 and 20 T; (b) B along y: 29, 19 and 12 T; (c) B along z: 28 and 18 T. For the 2D sample, no magnetoresistance oscillations have been observed in the configuration B ⊥ x. On the contrary, when B is parallel to x, resonances are apparent at 22 and 13.5 T. We have to compare these values to the 19.8 and 14.8 T previously calculated, which suggests that the real masses are slightly heavier than those calculated in the 2D case.
8. Conclusion The first remark to make is that there is relatively good agreement between the measured resonance fields and the calculated ones, in all cases (1D or 2D) and for the various field configurations. The second remark is that the modelling in the mean slipping-step approximation for the miniband calculations, although rough, is a good enough approach for the 1D superlattice band structures and simulates well the observed resonance. The third remark is that the quantum-wire generation method by controlled dislocation slipping [2] allows effective generation of 1D quantum-wire superlattices from 2D superlattices. The
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unidimensional character is confirmed by the opening of minigaps at the point K for m t along x and at the points K and M2 for m l along x. That miniband family, being the lowest, makes the observation of negative differential resistance in the y direction theoretically possible, which has already been experimentally observed [7].
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