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Experimental evidence of resonant tunneling via localized DQW states in an asymmetric triple barrier structure
Superlattices and Microstructures 33 (2003) 227–234 www.elsevier.com/locate/jnlabr/yspmi
Experimental evidence of resonant tunneling via localized DQW states in an asymmetric triple barrier structure Rober Vel´asquez∗ Departamento de F´ısica, Universidade Federal de Vic¸osa, Av. P.H. Rolfs s/n, CEP 36571-000, Vic¸osa, MG, Brazil Received 8 November 2002; received in revised form 6 August 2003; accepted 24 September 2003
Abstract In this work we report on field-induced features appearing in the tunneling current traces of a biased asymmetric triple barrier resonant tunneling device in the presence of an in-plane magnetic field. A theoretical model that satisfactorily explains the origin of these features is discussed. The reported data evidences the localized nature of the quantum states in thin layer asymmetric doublequantum-well structures. © 2003 Elsevier Ltd. All rights reserved.
1. Introduction Magnetotunneling spectroscopy using an in-plane magnetic field ( ⊥ ) has been a powerful tool for probing tunneling transitions involving two-dimensional (2D) states [1–10]. It holds on the fact that an in-plane magnetic field induces a relative shift between the dispersion curves of the emitter and collector states in a tunneling system. In this system, the requirement of conservation of the total energy and the in-plane momentum for a resonant transition restricts the tunneling to occur only between states of specific k-values. In this framework, here we report on low-temperature magnetotunneling spectroscopy data, of a strongly coupled Al0.4 Ga0.6 As/GaAs asymmetric triplebarrier (TB) structure, in order to obtain some information about the localization strength degree of the double-quantum-well (DQW) states in the active region of this system. ∗ Fax: +55-031-899-2483.
E-mail address: [email protected] (R. Vel´asquez). 0749-6036/$ - see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0749-6036(03)00090-9
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Since the resonant tunneling channels are restricted to specific emitter states, then, the main features may reflect basically differences in the effective mass and the occupation probability at finite temperatures of these states. Our main data consist of tunneling current traces, under in-plane magnetic fields up to 10 T, at some fixed bias voltages Vb , here so-called I (B) traces. 2. Theory In general, the main effect of an in-plane magnetic field = (B, 0, 0) on the dispersion curves of a 2D system, with the quantum confinement direction along the z-axis, is given by 2 2 k x2 e2 B 2 z n2 2 eB n|z|n + + − , E( ) = E n + k y 2m ∗ 2m ∗ 2m ∗ = (k x , k y ) being the in-plane wavevector, m ∗ the electron effective mass, E n the edge energy of the nth subband, and z n and zn = n|z|n the mean spread of the wavefunction χn (z) and the expectation value of the z coordinate in the nth subband, when the magnetic field is absent. Therefore, the in-plane magnetic field raises the subbands edges by e2 B 2 z n2 /2m ∗ and shifts each paraboloided E(k x , k y ) dispersion curve’s vertex by k y = eBzn / along the k y axis, remaining unaffected along the k x axis. In order to take tunneling into account, between 2D states in a TB system, we can consider that it is the DQW’s dispersion curves that shift relative to the allowed emitter’s dispersion curve by k y = eBz/, z being the average separation between the quasibound states of the emitter layer and the DQW (i.e., the average separation between their probability density maximum). Semiclassically, this effect can be understood in terms of the momentum eBz acquired by the tunneling electron because of the action of the Lorentz force. Thus, in the presence of an in-plane magnetic field, one electron occupying an emitter state characterized by the quantum numbers (E, k x , k y ) can tunnel resonantly into a DQW state with quantum numbers (E, k x , k y − k y ), where E is the energy with respect to some common origin. Thereby, for a given bias voltage Vb , the crossing points in the intersection curve between the emitter’s and DQW’s E(k x , k y ) paraboloids become resonant tunneling channels as both canonical momentum and total energy are conserved. Further, for better understanding, one can neglect the k x direction, which is not affected by the magnetic field. It follows that at the crossing points between the E(k y ) dispersion parabolas: 2 2 k 2y 2 eB z = (E) + − , k B y 2m ∗ 2m ∗
as schematically shown in Fig. 1 for k x = 0. There, e2 B 2 (z n2 − z 02 ), 2m ∗ accounts for the subband separations E = E n − E 0 at finite B, E 0 being the edge energy of the occupied emitter subband, and e2 B 2 z n2 /2m ∗ accounting for the diamagnetic shift (E) B = (E) B=0 +
R. Vel´asquez / Superlattices and Microstructures 33 (2003) 227–234
229
E
DQW
EF 2DEG
(∆E)B ∆k y − kF
0
ky kF
ky
Fig. 1. Field-induced relative shift between dispersion curves of the emitter (2DEG) and one DQW’s subband, for k x = 0. As discussed in the text, resonant tunneling is allowed to states with wavevectors at the crossing points between both E(k y ) parabolic dispersion curves.
to higher energies of each subband, which increases with increasing extension of their wavefunctions. 3. Experiment The sample investigated was grown by molecular beam epitaxy on (100)-oriented Si-doped n+ -GaAs substrate. Its layer structure in order of growth from the n+ -GaAs ˚ of GaAs substrate is: (i) 1 µm of GaAs doped to n = 1.7 × 1018 cm−3 ; (ii) 600 A ˚ of GaAs doped to n = 2.0 × 1016 cm−3 ; doped to n = 2.0 × 1017 cm−3 ; (iii) 600 A ˚ of undoped ˚ of undoped GaAs; (v) 52 A ˚ of undoped Al0.4 Ga0.6 As; (vi) 99 A (iv) 25 A ˚ of undoped Al0.4 Ga0.6 As; (viii) 58 A ˚ of undoped GaAs; (ix) 52 A ˚ GaAs; (vii) 16 A ˚ of undoped GaAs; (xi) 600 A ˚ of GaAs doped to of undoped Al0.4 Ga0.6 As; (x) 25 A ˚ of GaAs doped to n = 2.0×1017 cm−3 and (xiii) 0.95 µm n = 2.0×1016 cm−3 ; (xii) 500 A of GaAs doped to n = 1.7 × 1018 cm−3 . The device was processed into 100 µm diameter mesa using standard photolithography techniques and chemical etching. Further, an AuGeNi alloy Ohmic contact was processed on its top layer. Due to the step-like doping of the contact layers and the presence of thick enough undoped spacer layers, when biased, the accumulation of charges leads to the formation of a two-dimensional electron gas (2DEG) adjacent to the emitter barrier [11]. In effect, the 2D character of these states was reflected as strong oscillations in the tunnel current of this device, when a magnetic field perpendicular to the 2DEG layer was applied. Therefore, we are dealing with quantum transitions involving purely 2D states.
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2DEG
2DEG
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Vb (V) Fig. 2. I (V ) characteristic of the investigated TB device at 4.2 K. The forward bias peak at ∼0.159 V is magnified × 25. The left (right) inset shows the conduction band profile under resonance for the forward (reverse) bias, where the arrow indicates electron tunneling into the ground DQW subband.
The reported data were measured at ∼4.2 K and under magnetic fields up to 10 T. The electron tunneling from the top contact layer to the substrate (substrate biased positive) corresponds to the forward bias, and that from the substrate to the top contact layer (top contact biased positive) corresponds to the reverse bias. 4. Results and discussion The current–voltage I (V ) characteristic of the investigated TB structure is shown in Fig. 2. It shows resonance peaks at ∼0.159 V, −0.460 V and −0.723 V, respectively. In order to identify the involved DQW states let us consider the discussion reported in [12–14], about the coupling behavior of single quantum well (SQW) states in a DQW system. It reports that even for an asymmetric DQW, a narrow enough interwell barrier leads the SQW states to become coupled DQW states; however, in this case, mainly localized in one of the wells. It differs with a symmetric DQW, where the coupling of the SQW states leads to completely delocalized DQW states, the so-called symmetric and antisymmetric states. In these systems, the anticrossing of the energy levels produces an energy gap SAS , which is responsible for a great variety of tunneling related phenomena, e.g., see [7, 14–17]. In consequence, in the asymmetric DQW region of our TB structure, ˚ the available tunneling channels may be DQW states with thin interwell barrier of ∼16 A, mainly localized in one of the wells. According to the above arguments, the forward bias peak at ∼0.159 V corresponds to resonant tunneling into the ground DQW subband, with wavefunction mainly localized
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at the wider well (see the left inset in Fig. 2). Symmetrically, the first reverse bias peak at ∼−0.460 V corresponds to resonant tunneling into a similar ground DQW subband (see right inset in Fig. 2). However, the second reverse bias peak at ∼−0.723 V is attributed to resonant tunneling into the first excited DQW subband, with wavefunction mainly localized at the narrower well. Therefore, it may explain the high current intensity and bias voltage for the first reverse bias peak in comparison to the first forward bias peak. In particular, the high current intensity for the reverse bias peak may be due to the increase on wavefunction overlap when the DQW subband’s wavefunction is mainly localized in the quantum well adjacent to the 2DEG layer. Another set of experimental data, to be discussed here, is shown in Fig. 3. They consist of the I (B) traces of the investigated TB structure, measured under in-plane magnetic fields B up to 10 T. We note, as a common feature, the presence of one or more negative magnetoresistance regions, indicating with arrows in the figure. In these regions, the current increases up to reach a maximum value, at a magnetic field denoted Bν , followed by a steady decrease to its background value, with the cut off field shifting to higher fields as Vb goes towards a resonance (behavior better resolved in Fig. 3(a)). The lowfield negative magnetoresistance shifts to lower fields when the device is biased towards a resonance, and disappears completely under resonance. Conversely, the second negative magnetoresistance (which appears as a shoulder in the high field region of these traces) shifts to higher fields when the device is biased towards a resonance. Following the theory discussed in Section 2, most of the features observed in the reported I (B) traces can be attributed to the establishment of field-induced resonance for states with wavevector at the crossing region between the emitter’s and DQW’s dispersion curves. For this purpose, in Fig. 4 we depict the field-induced E(k y ) dispersion curves crossing profiles leading to these features, assuming the emitter subband vertex at k y = 0 (see Fig. 1). Firstly, since k y is the emitter wavevector at the crossing point, then, the situation depicted in Fig. 4(a) will correspond to the onset of a negative magnetoresistance region, i.e., k y ∼ k F at low temperatures and k y > k y . There, the current starts to increase basically due to the increase on the occupation probability f (E) of the adjacent low-energy emitter states by sweeping B to higher values. Although, it could also account for the expected progressive suppression of non-parabolicity of these low-energy emitter states, which reduces the electron effective mass. To our knowledge, the decrease on the electron effective mass in the 2DEG should increase the tunneling rate. Next, at k y = k y , situation depicted in Fig. 4(b), the field-induced tunneling current will reach its maximum value, with the field leading to this configuration, Bν , shifting to lower values when Vb goes towards a resonance, as could be inferred when E → 0 in Fig. 1. Further, at high fields, i.e., when k y < k y , see Fig. 4(c), a steady decrease on the tunnel current is expected. In addition, the observed shift to higher fields of the cut off field is attributed to the raise on the Fermi level at the 2DEG layer at increasing bias voltages. Further, if any other DQW’s dispersion curve is also lead to cross with the emitter’s dispersion (see Fig. 4(d)), then other(s) additional negative magnetoresistance feature(s) will appear in the I (B) traces of the device. Indeed, it explains the appearance of the shoulder at high fields on the reported I (B) traces, which has been attributed to the resonant tunneling into the first excited DQW subband.
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(a)
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B (T) Fig. 3. I (B) traces at 4.2 K of the TB structure for a few set of bias voltages: (a) up to the forward bias peak at ∼0.159 V (from bottom to top: 0.110, 0.120, 0.130, 0.135, 0.140, 0.145, 0.150 and 0.159 V); (b) below to the first reverse bias peak at ∼−0.460 V (from the bottom to top: −0.200, −0.230, −0.260 and −0.290 V); (c) below the second reverse bias peak at ∼0.723 V (from bottom to top: −0.560, −0.600, −0.620, −0.640, −0.660 and −0.680 V, respectively). For all traces the arrows mark the current maximum at a given Bν .
Now, in order to obtain some additional information about the localization strength degree of the DQW states, let us analyse the bias dependence of the Bν fields from the I (B) traces of our asymmetric TB structure, plotted in Fig. 5. The Bν values were obtained from the d2 I /dB 2 plots of the corresponding I (B) traces. In the figure, each Bν (Vb ) curve corresponds to field-induced resonant tunneling into one specific DQW subband E n . Firstly, the fact that the Bν for the bias at the onset of tunneling into the ground DQW subband (E 1 ) is lower for the forward bias than for the reverse bias
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DQW
EF 2 DEG
(a)
(b)
(c)
(d)
Fig. 4. Graphical representation of the dispersion curves E(k y ) crossing profiles corresponding to: (a) onset of a negative magnetoresistance region; (b) field-inducing tunneling current maximum at B = Bν ; (c) tunnel current cut off; (d) second current maximum (shoulder) in the I (B) traces.
8
B (T)
6
E3 E2 E2
4
2 E1
E2
E1
0 − 0,6
−0,4
−0,2
0,0
0,2
Vb (V) Fig. 5. Bν vs Vb plots corresponding to the field-induced current maxima observed in the low-temperature I (B) traces of the TB device. There, E 1 , E 2 and E 3 indicate tunneling into the ground, first excited and second excited DQW subband, respectively.
clearly validates our theoretical model. Indeed, a higher field Bν should be necessary to establish the current maximum crossing profile condition for higher E, as could be inferred by inspection of Figs. 4(a) and (b). It also explains the observed shift to lower fields of the low-field region Bν ’s values when the device is biased towards the first resonance. Next, at k y = k y , which corresponds to one field-induced current maxima in the I (B) traces, our model yields (E) B = (e2 Bν2 /2m ∗ )z2 , so the observed differences on the Bν ’s values for tunneling into the same DQW state may reflect basically differences on the effective distance tunneled z. In effect, as stated before, in a coupled asymmetric DQW system their associated wavefunctions are mainly localized in one of the wells. Other interesting phenomenon is related to the observed increase of the gap between Bν (Vb ) curves at the same bias region, when the device is biased towards a resonance.
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It may be attributed basically to the increase on subband separation under a higher voltage drop across the DQW region. Finally, another interesting consequence of our model—as can be inferred by inspection of Figs. 4(b) and (d)—is related to the fact that we could be able to map out the bias dependence of the energy levels of a 2D system, in the active region 2 z2 − B 2 z2 ] of a resonant tunneling device, by computing E n − E m = (e2 /2m ∗ )[Bν,n n ν,m m for each biases Vb . 5. Summary We performed a qualitative analysis of the main features appearing in the lowtemperature I (V ) characteristic and I (B) traces of a strongly coupled asymmetric TB structure. The I (V ) characteristic clearly evidences a high degree of localization of the DQW states. A theoretical model was discussed in order to explain the origin of the fieldinduced negative magnetoresistances in the tunnel current involving quantum transitions between 2D states. As expected, for an asymmetric TB structure, the analysis of the I (B) traces also suggests some degree of localization of the DQW states. Acknowledgements The author acknowledges partial financial support by CNPq. The sample growth and the experimental part of this work were performed at the Physics Department—Universidade Federal de Minas Gerais (UFMG), Brazil. References [1] J. Smoliner, W. Demmerle, G. Berthold, E. Gornik, G. Weimann, Phys. Rev. Lett. 63 (1989) 2116. [2] M.L. Leadbeater, L. Eaves, P.E. Simmonds, G.A. Toombs, F.W. Sheard, P.A. Claxton, G. Hill, M.A. Pate, Solid State Electron. 31 (1988) 707. [3] G. Platero, L. Brey, C. Tejedor, Phys. Rev. B 40 (1989) 8548. [4] J.P. Eisenstein, T.J. Gramila, L.N. Pfeiffer, K.W. West, Phys. Rev. B 44 (1991) 6511. [5] J.A. Simmons, S.K. Lyo, J.F. Klem, M.E. Sherwin, J.R. Wendth, Phys. Rev. B 47 (1993) 15741. [6] L. Zheng, A.H. MacDonald, Phys. Rev. B 47 (1993) 10619. [7] A. Kurobe, I.M. Castleton, E.H. Linfield, M.P. Grimshaw, K.M. Brown, D.A. Ritchie, M. Pepper, G.A.C. Jones, Phys. Rev. B 50 (1994) 4889. [8] R.K. Hayden, D.K. Maude, L. Eaves, E.C. Valadares, M. Henini, F.W. Sheard, O.H. Hughes, J.C. Portal, L. Cury, Phys. Rev. Lett. 66 (1991) 1749. [9] S. Williams, H. Callebaut, Q. Hu, Appl. Phys. Lett. 79 (2001) 4444. [10] G. Rainer, J. Smoliner, E. Gornik, G. B¨ohm, G. Weimann, Phys. Rev. B 51 (1995) 17642. [11] B.R.A. Neves, E.S. Alves, J.F. Sampaio, A.G. de Oliveira, E.A. Meneses, Brazilian J. Phys. 24 (1994) 203. [12] P. Bonnel, P. Lefebvre, B. Gil, H. Mathieu, C. Deparis, J. Massies, G. Neu, Y. Chen, Phys. Rev. B 42 (1990) 3435. [13] R. Vel´asquez, R.V. Sampaio, Solid State Electron. 43 (1999) 2155. [14] G.S. Boebinger, H.W. Jiang, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 64 (1990) 1793. [15] J.P. Eisenstein, G.S. Boebinger, L.N. Pfeiffer, S. He, Phys. Rev. Lett. 68 (1992) 1383. [16] J.A. Simmons, S.K. Lyo, N.E. Harff, J.F. Klem, Phys. Rev. Lett. 73 (1994) 2256. [17] R. Vel´asquez, Solid State Electron. 44 (2000) 697.
Experimental evidence of resonant tunneling via localized DQW states in an asymmetric triple barrier structure Rober Vel´asquez∗ Departamento de F´ısica, Universidade Federal de Vic¸osa, Av. P.H. Rolfs s/n, CEP 36571-000, Vic¸osa, MG, Brazil Received 8 November 2002; received in revised form 6 August 2003; accepted 24 September 2003
Abstract In this work we report on field-induced features appearing in the tunneling current traces of a biased asymmetric triple barrier resonant tunneling device in the presence of an in-plane magnetic field. A theoretical model that satisfactorily explains the origin of these features is discussed. The reported data evidences the localized nature of the quantum states in thin layer asymmetric doublequantum-well structures. © 2003 Elsevier Ltd. All rights reserved.
1. Introduction Magnetotunneling spectroscopy using an in-plane magnetic field ( ⊥ ) has been a powerful tool for probing tunneling transitions involving two-dimensional (2D) states [1–10]. It holds on the fact that an in-plane magnetic field induces a relative shift between the dispersion curves of the emitter and collector states in a tunneling system. In this system, the requirement of conservation of the total energy and the in-plane momentum for a resonant transition restricts the tunneling to occur only between states of specific k-values. In this framework, here we report on low-temperature magnetotunneling spectroscopy data, of a strongly coupled Al0.4 Ga0.6 As/GaAs asymmetric triplebarrier (TB) structure, in order to obtain some information about the localization strength degree of the double-quantum-well (DQW) states in the active region of this system. ∗ Fax: +55-031-899-2483.
E-mail address: [email protected] (R. Vel´asquez). 0749-6036/$ - see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0749-6036(03)00090-9
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Since the resonant tunneling channels are restricted to specific emitter states, then, the main features may reflect basically differences in the effective mass and the occupation probability at finite temperatures of these states. Our main data consist of tunneling current traces, under in-plane magnetic fields up to 10 T, at some fixed bias voltages Vb , here so-called I (B) traces. 2. Theory In general, the main effect of an in-plane magnetic field = (B, 0, 0) on the dispersion curves of a 2D system, with the quantum confinement direction along the z-axis, is given by 2 2 k x2 e2 B 2 z n2 2 eB n|z|n + + − , E( ) = E n + k y 2m ∗ 2m ∗ 2m ∗ = (k x , k y ) being the in-plane wavevector, m ∗ the electron effective mass, E n the edge energy of the nth subband, and z n and zn = n|z|n the mean spread of the wavefunction χn (z) and the expectation value of the z coordinate in the nth subband, when the magnetic field is absent. Therefore, the in-plane magnetic field raises the subbands edges by e2 B 2 z n2 /2m ∗ and shifts each paraboloided E(k x , k y ) dispersion curve’s vertex by k y = eBzn / along the k y axis, remaining unaffected along the k x axis. In order to take tunneling into account, between 2D states in a TB system, we can consider that it is the DQW’s dispersion curves that shift relative to the allowed emitter’s dispersion curve by k y = eBz/, z being the average separation between the quasibound states of the emitter layer and the DQW (i.e., the average separation between their probability density maximum). Semiclassically, this effect can be understood in terms of the momentum eBz acquired by the tunneling electron because of the action of the Lorentz force. Thus, in the presence of an in-plane magnetic field, one electron occupying an emitter state characterized by the quantum numbers (E, k x , k y ) can tunnel resonantly into a DQW state with quantum numbers (E, k x , k y − k y ), where E is the energy with respect to some common origin. Thereby, for a given bias voltage Vb , the crossing points in the intersection curve between the emitter’s and DQW’s E(k x , k y ) paraboloids become resonant tunneling channels as both canonical momentum and total energy are conserved. Further, for better understanding, one can neglect the k x direction, which is not affected by the magnetic field. It follows that at the crossing points between the E(k y ) dispersion parabolas: 2 2 k 2y 2 eB z = (E) + − , k B y 2m ∗ 2m ∗
as schematically shown in Fig. 1 for k x = 0. There, e2 B 2 (z n2 − z 02 ), 2m ∗ accounts for the subband separations E = E n − E 0 at finite B, E 0 being the edge energy of the occupied emitter subband, and e2 B 2 z n2 /2m ∗ accounting for the diamagnetic shift (E) B = (E) B=0 +
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E
DQW
EF 2DEG
(∆E)B ∆k y − kF
0
ky kF
ky
Fig. 1. Field-induced relative shift between dispersion curves of the emitter (2DEG) and one DQW’s subband, for k x = 0. As discussed in the text, resonant tunneling is allowed to states with wavevectors at the crossing points between both E(k y ) parabolic dispersion curves.
to higher energies of each subband, which increases with increasing extension of their wavefunctions. 3. Experiment The sample investigated was grown by molecular beam epitaxy on (100)-oriented Si-doped n+ -GaAs substrate. Its layer structure in order of growth from the n+ -GaAs ˚ of GaAs substrate is: (i) 1 µm of GaAs doped to n = 1.7 × 1018 cm−3 ; (ii) 600 A ˚ of GaAs doped to n = 2.0 × 1016 cm−3 ; doped to n = 2.0 × 1017 cm−3 ; (iii) 600 A ˚ of undoped ˚ of undoped GaAs; (v) 52 A ˚ of undoped Al0.4 Ga0.6 As; (vi) 99 A (iv) 25 A ˚ of undoped Al0.4 Ga0.6 As; (viii) 58 A ˚ of undoped GaAs; (ix) 52 A ˚ GaAs; (vii) 16 A ˚ of undoped GaAs; (xi) 600 A ˚ of GaAs doped to of undoped Al0.4 Ga0.6 As; (x) 25 A ˚ of GaAs doped to n = 2.0×1017 cm−3 and (xiii) 0.95 µm n = 2.0×1016 cm−3 ; (xii) 500 A of GaAs doped to n = 1.7 × 1018 cm−3 . The device was processed into 100 µm diameter mesa using standard photolithography techniques and chemical etching. Further, an AuGeNi alloy Ohmic contact was processed on its top layer. Due to the step-like doping of the contact layers and the presence of thick enough undoped spacer layers, when biased, the accumulation of charges leads to the formation of a two-dimensional electron gas (2DEG) adjacent to the emitter barrier [11]. In effect, the 2D character of these states was reflected as strong oscillations in the tunnel current of this device, when a magnetic field perpendicular to the 2DEG layer was applied. Therefore, we are dealing with quantum transitions involving purely 2D states.
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I(mA)
50
X 25 0
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Vb (V) Fig. 2. I (V ) characteristic of the investigated TB device at 4.2 K. The forward bias peak at ∼0.159 V is magnified × 25. The left (right) inset shows the conduction band profile under resonance for the forward (reverse) bias, where the arrow indicates electron tunneling into the ground DQW subband.
The reported data were measured at ∼4.2 K and under magnetic fields up to 10 T. The electron tunneling from the top contact layer to the substrate (substrate biased positive) corresponds to the forward bias, and that from the substrate to the top contact layer (top contact biased positive) corresponds to the reverse bias. 4. Results and discussion The current–voltage I (V ) characteristic of the investigated TB structure is shown in Fig. 2. It shows resonance peaks at ∼0.159 V, −0.460 V and −0.723 V, respectively. In order to identify the involved DQW states let us consider the discussion reported in [12–14], about the coupling behavior of single quantum well (SQW) states in a DQW system. It reports that even for an asymmetric DQW, a narrow enough interwell barrier leads the SQW states to become coupled DQW states; however, in this case, mainly localized in one of the wells. It differs with a symmetric DQW, where the coupling of the SQW states leads to completely delocalized DQW states, the so-called symmetric and antisymmetric states. In these systems, the anticrossing of the energy levels produces an energy gap SAS , which is responsible for a great variety of tunneling related phenomena, e.g., see [7, 14–17]. In consequence, in the asymmetric DQW region of our TB structure, ˚ the available tunneling channels may be DQW states with thin interwell barrier of ∼16 A, mainly localized in one of the wells. According to the above arguments, the forward bias peak at ∼0.159 V corresponds to resonant tunneling into the ground DQW subband, with wavefunction mainly localized
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at the wider well (see the left inset in Fig. 2). Symmetrically, the first reverse bias peak at ∼−0.460 V corresponds to resonant tunneling into a similar ground DQW subband (see right inset in Fig. 2). However, the second reverse bias peak at ∼−0.723 V is attributed to resonant tunneling into the first excited DQW subband, with wavefunction mainly localized at the narrower well. Therefore, it may explain the high current intensity and bias voltage for the first reverse bias peak in comparison to the first forward bias peak. In particular, the high current intensity for the reverse bias peak may be due to the increase on wavefunction overlap when the DQW subband’s wavefunction is mainly localized in the quantum well adjacent to the 2DEG layer. Another set of experimental data, to be discussed here, is shown in Fig. 3. They consist of the I (B) traces of the investigated TB structure, measured under in-plane magnetic fields B up to 10 T. We note, as a common feature, the presence of one or more negative magnetoresistance regions, indicating with arrows in the figure. In these regions, the current increases up to reach a maximum value, at a magnetic field denoted Bν , followed by a steady decrease to its background value, with the cut off field shifting to higher fields as Vb goes towards a resonance (behavior better resolved in Fig. 3(a)). The lowfield negative magnetoresistance shifts to lower fields when the device is biased towards a resonance, and disappears completely under resonance. Conversely, the second negative magnetoresistance (which appears as a shoulder in the high field region of these traces) shifts to higher fields when the device is biased towards a resonance. Following the theory discussed in Section 2, most of the features observed in the reported I (B) traces can be attributed to the establishment of field-induced resonance for states with wavevector at the crossing region between the emitter’s and DQW’s dispersion curves. For this purpose, in Fig. 4 we depict the field-induced E(k y ) dispersion curves crossing profiles leading to these features, assuming the emitter subband vertex at k y = 0 (see Fig. 1). Firstly, since k y is the emitter wavevector at the crossing point, then, the situation depicted in Fig. 4(a) will correspond to the onset of a negative magnetoresistance region, i.e., k y ∼ k F at low temperatures and k y > k y . There, the current starts to increase basically due to the increase on the occupation probability f (E) of the adjacent low-energy emitter states by sweeping B to higher values. Although, it could also account for the expected progressive suppression of non-parabolicity of these low-energy emitter states, which reduces the electron effective mass. To our knowledge, the decrease on the electron effective mass in the 2DEG should increase the tunneling rate. Next, at k y = k y , situation depicted in Fig. 4(b), the field-induced tunneling current will reach its maximum value, with the field leading to this configuration, Bν , shifting to lower values when Vb goes towards a resonance, as could be inferred when E → 0 in Fig. 1. Further, at high fields, i.e., when k y < k y , see Fig. 4(c), a steady decrease on the tunnel current is expected. In addition, the observed shift to higher fields of the cut off field is attributed to the raise on the Fermi level at the 2DEG layer at increasing bias voltages. Further, if any other DQW’s dispersion curve is also lead to cross with the emitter’s dispersion (see Fig. 4(d)), then other(s) additional negative magnetoresistance feature(s) will appear in the I (B) traces of the device. Indeed, it explains the appearance of the shoulder at high fields on the reported I (B) traces, which has been attributed to the resonant tunneling into the first excited DQW subband.
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(a)
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B (T) Fig. 3. I (B) traces at 4.2 K of the TB structure for a few set of bias voltages: (a) up to the forward bias peak at ∼0.159 V (from bottom to top: 0.110, 0.120, 0.130, 0.135, 0.140, 0.145, 0.150 and 0.159 V); (b) below to the first reverse bias peak at ∼−0.460 V (from the bottom to top: −0.200, −0.230, −0.260 and −0.290 V); (c) below the second reverse bias peak at ∼0.723 V (from bottom to top: −0.560, −0.600, −0.620, −0.640, −0.660 and −0.680 V, respectively). For all traces the arrows mark the current maximum at a given Bν .
Now, in order to obtain some additional information about the localization strength degree of the DQW states, let us analyse the bias dependence of the Bν fields from the I (B) traces of our asymmetric TB structure, plotted in Fig. 5. The Bν values were obtained from the d2 I /dB 2 plots of the corresponding I (B) traces. In the figure, each Bν (Vb ) curve corresponds to field-induced resonant tunneling into one specific DQW subband E n . Firstly, the fact that the Bν for the bias at the onset of tunneling into the ground DQW subband (E 1 ) is lower for the forward bias than for the reverse bias
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EF 2 DEG
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Fig. 4. Graphical representation of the dispersion curves E(k y ) crossing profiles corresponding to: (a) onset of a negative magnetoresistance region; (b) field-inducing tunneling current maximum at B = Bν ; (c) tunnel current cut off; (d) second current maximum (shoulder) in the I (B) traces.
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clearly validates our theoretical model. Indeed, a higher field Bν should be necessary to establish the current maximum crossing profile condition for higher E, as could be inferred by inspection of Figs. 4(a) and (b). It also explains the observed shift to lower fields of the low-field region Bν ’s values when the device is biased towards the first resonance. Next, at k y = k y , which corresponds to one field-induced current maxima in the I (B) traces, our model yields (E) B = (e2 Bν2 /2m ∗ )z2 , so the observed differences on the Bν ’s values for tunneling into the same DQW state may reflect basically differences on the effective distance tunneled z. In effect, as stated before, in a coupled asymmetric DQW system their associated wavefunctions are mainly localized in one of the wells. Other interesting phenomenon is related to the observed increase of the gap between Bν (Vb ) curves at the same bias region, when the device is biased towards a resonance.
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It may be attributed basically to the increase on subband separation under a higher voltage drop across the DQW region. Finally, another interesting consequence of our model—as can be inferred by inspection of Figs. 4(b) and (d)—is related to the fact that we could be able to map out the bias dependence of the energy levels of a 2D system, in the active region 2 z2 − B 2 z2 ] of a resonant tunneling device, by computing E n − E m = (e2 /2m ∗ )[Bν,n n ν,m m for each biases Vb . 5. Summary We performed a qualitative analysis of the main features appearing in the lowtemperature I (V ) characteristic and I (B) traces of a strongly coupled asymmetric TB structure. The I (V ) characteristic clearly evidences a high degree of localization of the DQW states. A theoretical model was discussed in order to explain the origin of the fieldinduced negative magnetoresistances in the tunnel current involving quantum transitions between 2D states. As expected, for an asymmetric TB structure, the analysis of the I (B) traces also suggests some degree of localization of the DQW states. Acknowledgements The author acknowledges partial financial support by CNPq. The sample growth and the experimental part of this work were performed at the Physics Department—Universidade Federal de Minas Gerais (UFMG), Brazil. References [1] J. Smoliner, W. Demmerle, G. Berthold, E. Gornik, G. Weimann, Phys. Rev. Lett. 63 (1989) 2116. [2] M.L. Leadbeater, L. Eaves, P.E. Simmonds, G.A. Toombs, F.W. Sheard, P.A. Claxton, G. Hill, M.A. Pate, Solid State Electron. 31 (1988) 707. [3] G. Platero, L. Brey, C. Tejedor, Phys. Rev. B 40 (1989) 8548. [4] J.P. Eisenstein, T.J. Gramila, L.N. Pfeiffer, K.W. West, Phys. Rev. B 44 (1991) 6511. [5] J.A. Simmons, S.K. Lyo, J.F. Klem, M.E. Sherwin, J.R. Wendth, Phys. Rev. B 47 (1993) 15741. [6] L. Zheng, A.H. MacDonald, Phys. Rev. B 47 (1993) 10619. [7] A. Kurobe, I.M. Castleton, E.H. Linfield, M.P. Grimshaw, K.M. Brown, D.A. Ritchie, M. Pepper, G.A.C. Jones, Phys. Rev. B 50 (1994) 4889. [8] R.K. Hayden, D.K. Maude, L. Eaves, E.C. Valadares, M. Henini, F.W. Sheard, O.H. Hughes, J.C. Portal, L. Cury, Phys. Rev. Lett. 66 (1991) 1749. [9] S. Williams, H. Callebaut, Q. Hu, Appl. Phys. Lett. 79 (2001) 4444. [10] G. Rainer, J. Smoliner, E. Gornik, G. B¨ohm, G. Weimann, Phys. Rev. B 51 (1995) 17642. [11] B.R.A. Neves, E.S. Alves, J.F. Sampaio, A.G. de Oliveira, E.A. Meneses, Brazilian J. Phys. 24 (1994) 203. [12] P. Bonnel, P. Lefebvre, B. Gil, H. Mathieu, C. Deparis, J. Massies, G. Neu, Y. Chen, Phys. Rev. B 42 (1990) 3435. [13] R. Vel´asquez, R.V. Sampaio, Solid State Electron. 43 (1999) 2155. [14] G.S. Boebinger, H.W. Jiang, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 64 (1990) 1793. [15] J.P. Eisenstein, G.S. Boebinger, L.N. Pfeiffer, S. He, Phys. Rev. Lett. 68 (1992) 1383. [16] J.A. Simmons, S.K. Lyo, N.E. Harff, J.F. Klem, Phys. Rev. Lett. 73 (1994) 2256. [17] R. Vel´asquez, Solid State Electron. 44 (2000) 697.