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Parameters affecting the accuracy of oxide thickness prediction in thin metal–oxide–semiconductor structures
Superlattices and Microstructures 35 (2004) 85–94 www.elsevier.com/locate/superlattices
Parameters affecting the accuracy of oxide thickness prediction in thin metal–oxide–semiconductor structures J.M. Mohaidata,∗, Riyad N. Ahmad-Bitarb a Department of Electrical Engineering, United Arab Emirates University, P.O. Box 17555, Al-Ain,
United Arab Emirates b Physics Department, University of Jordan, Amman, Jordan
Received 5 March 2003; received in revised form 28 January 2004; accepted 21 February 2004 Available online 10 April 2004
Abstract On the basis of the solution of the time dependent Schr¨odinger equation within the framework of the effective mass theory, a complete quantum mechanical electron tunneling through a biased square potential model with abrupt interfaces was deduced. Barriers of 3 eV height and ˚ were investigated. Current density–voltage (J –V ) curves were computed for widths up to 140 A + Al/SiO2 /n Si structure. The computed J –V curves exhibited oscillations at applied voltages above (Fowler–Nordheim tunneling) and below (direct tunneling) 3 V. For oxide thickness estimation, the position of the oscillation extrema from this quantum mechanical model were fitted to a wave ˚ However, interference formula and showed excellent agreement for oxide layer widths less than 50 A. ˚ We show that the electron energy a systematic deviation appeared for layers larger than 50 A. distribution at the injection layer and the electron effective mass on layers other than the oxide layer are important parameters for accurate oxide thickness estimation. © 2004 Elsevier Ltd. All rights reserved. PACS: 73.40.Gk; 73.40.-c; 73.40.Qv Keywords: MOS structure; Tunneling; Quantum interference; Effective mass; Energy distribution
∗ Corresponding author. Tel.: +971-3-7051606; fax: +971-3-7623156.
E-mail address: [email protected] (J.M. Mohaidat). 0749-6036/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2004.02.018
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1. Introduction ˚ with high dielectric constants, in metal– Ultrathin oxide films, less than 100 A, oxide–semiconductor (MOS) structures for application as storage capacitors for dynamic random access memories are required to scale down the device size. The passage of current through the film is dominated by tunneling. Quantum oscillations in the tunneling current were observed by many investigators [1–6]. Recently, pronounced quantum oscillations were detected for electrons injected over the barrier into a MOS structure with a scanning tunneling microscope [7–9]. The presence of oscillations is useful for determining the film thickness [10, 11] or the conduction electron average effective mass. Different theoretical models [4–6, 12, 13] are used to predict the oscillatory behavior of the tunneling current in order to interpret the experimental results. Most of these models are based on the conventional method of WKB (Wentzel–Kramers–Brillouin) approximation. This method is poor in explaining the oscillations and inaccurate in regions where the potential profile varies abruptly [14]. The discrepancies might arise from the framework of the WKB approximation, which assumes a smooth variation of the barrier potential. Recently published wave interference formulas were deduced for the location of the extrema of a biased square barrier with abrupt interfaces [12]. The phase shift between the direct transmission incident electron waves and the transmission part of the reflection wave, which traverses the barrier after being reflected by the two interfaces of the barrier, was calculated. Direct tunneling was assumed in the region where the electron energy is less than the potential energy and, otherwise, Fowler–Nordheim (FN) injection. The optical interference analogy was assumed in deriving these formulas. In this article we present calculated J –V curves based on a numerical solution of the time dependent Schr¨odinger equation for a complete quantum mechanical tunneling through a square potential model with abrupt interfaces. The barrier potential height = ˚ with a biasing voltage up to 20 V were 3 eV and barrier widths L of 140, 100, 70, and 50 A assumed. These values are in accordance with Al/SiO2 /n+ Si (MOS) devices. We show the importance of both the electron energy distribution at the injection layer of the MOS structure and the electron effective mass throughout the structure for the oxide thickness estimation. Given the fundamental nature of the problem (tunneling and reflection through ideal barriers), the use of the time dependent Schr¨odinger equation adds to the originality of this work. The calculated tunneling currents were found to have oscillations. The dependence of the extrema location and their amplitude on the oxide thickness is to be discussed. The validity of the interference formulas [12] will be checked. A numerical approach based on the solution of the time dependent Schr¨odinger equation is employed. The structure is assumed to consist of three regions: the metal, the insulator, and the heavily non-degenerately doped semiconductor. The metal and semiconductor regions are assumed to be of much larger dimensions relative to the barrier dimension. Square potential barriers with abrupt interfaces are assumed. The effects of changing barrier parameters on the tunneling current are studied. The functional dependence of the current on the potential difference across the barrier is studied. Quantum mechanical oscillations appearing in the J –V characteristic curves are discussed and compared against theoretical models [4–6, 12, 13].
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Fig. 1. A schematic energy band diagram of a MOS structure. (a) shows the structure under flat band conditions while (b) shows the structure under an applied bias Va . In this structure the energy separation between the Fermi level and the semiconductor band edge is very small but is exaggerated here for clarity.
Fig. 1 shows a schematic energy band diagram of the metal–insulator–semiconductor ideal structure. φm is the metal work function, χ is the electron affinity, φB is the barrier potential between the metal and the insulator, and L is the barrier width. Fig. 1(a) shows the structure under flat band conditions (zero applied bias). Fig. 1(b) shows the structure under an applied bias of Va . The exact band structure profile was not calculated here. Rigorous computations (self-consistent solution of the Schr¨odinger and Poisson equations) of the band structure will result in a heterojunction at the structure interfaces. This heterojunction is relatively very shallow especially for structures like the one under consideration in this paper and is usually neglected because they could produce only a negligible effect on the overall characteristics [1, 15]. To support this approximation we used for part of the work the 1D self-consistent Schr¨odinger–Poisson solver called SCHRED.1 Using SCHRED we obtained the exact energy band profile and then used it in the solution of the time dependent
1 SCHRED is installed on the Purdue University Semiconductor Simulation Hub (http://nanohub.purdue.edu/) and is accessible through the Web.
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Schr¨odinger equation. SCHRED treats the structure quantum mechanically and accounts for all six valleys of the structure. The numerical solution starts by assuming a one-dimensional (1D) free electron wavepacket with central energy relative to the semiconductor band edge. This wavepacket which is placed in the metal side at t = 0 assumes the following wavefunction form: (z − z 0 )2 ψ(z, t = 0) = A exp(i kz) exp − (1) 4σ 2 √ where A is a normalization constant. k = (1/) 2m ∗ is the wavepacket’s central energy. z 0 is its center location; σ is the packet’s spatial width, which was chosen to equal 55 nm. m ∗ is the electron effective mass which was assumed to be, in the metal side, m ∗ = m o , in the insulator, m ∗ = 0.5m o , and in the semiconductor region (Si), m ∗ = 0.19m o . m o being the free electron mass. The Hamiltonian operator H associated with the metal–insulator–semiconductor system is taken as the Ben-Daniel and Duke Hamiltonian: 2 ∂ 1 ∂ H =− + V (z) (2) 2 ∂z m ∗ ∂z where the potential V (z) is due to the conduction band discontinuity including the effect of the applied electric field, as shown in Fig. 1. The evolution of the wavefunction ψ(z, t) is determined by solving the time dependent Schr¨odinger equation, H ψ(z, t) = i (∂/∂t)ψ(z, t). To proceed with the numerical solution, the Ben-Daniel and Duke Hamiltonian (Eq. (2)) is converted into a difference equation. The resulting Schr¨odinger finite difference equation is then solved numerically using the Crank–Nicholson method [16]. A spatial mesh ˚ a time mesh size of 0.5 fs, and hard-wall boundary conditions at the two size [17] of 0.5 A, end-points 20 µm away from the structure were assumed. Interface boundary conditions included the continuity of the wavefunction and the continuity of (1/m ∗ )(∂ψ/∂z) to insure particle current continuity. An excellent agreement was obtained between the results of this numerical procedure and the experimental results obtained for a double-quantum-well oscillator [18, 19]. The initial electron wavepacket in Eq. (1), at t = 0, is placed in the left region of the structure of Fig. 1. The tunneling wavefunction ψ(z, t) in the semiconductor region is found in time as the wavepacket moves out of the right region, through the insulator, and into the left region. The time integrated current density J at a particular applied voltage and the wavepacket central energy is calculated at a point 2 nm to the left of the barrier (Fig. 1) using the exact quantum mechanical current density expression: ∂ψ ∗ ∗ ∂ψ ψ − ψ . (3) J= 2m ∗ i ∂z ∂z The energy distribution of the free electrons is given by [20] √ B n() = − 1 + exp k Tf B
(4)
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Fig. 2. The current density versus applied voltage J –V characteristics of the MOS structure depicted in Fig. 1 for various layer oxide thicknesses L. Curves (A), (B), (C), and (D) correspond to oxide layer widths L of 50, 70, ˚ respectively. 100, and 140 A,
where B is a constant, f is the electron Fermi level, kB is the Boltzmann constant, and T is the temperature. Therefore, to compare with experimental results, the overall tunneling current density J is weighted by summing over all current density contributions of all electron wavepackets of different energy . Otherwise, the current density J (V ) is found by considering only Eq. (3). 2. Results and discussion Considering an electron wavepacket with central energy , the wavepacket will tunnel a distance L t = L(φB − )/q V through the barrier, then it will be injected over the barrier and ballistically propagate toward the interface. The electron wavepackets have a certain probability of reflecting and of transmitting at the interface. The transmitted and the twicereflected electron waves interfere. For MOS devices whose thickness is of the order of the electron de Broglie wavelength, quantum oscillations have been observed in the tunneling currents [1, 2, 4, 6, 10, 21, 22]. In general, it is found that the position and the amplitude of the oscillations will vary depending on the shape of the potential barrier in the structure. The tunneling J –V characteristics for MOS structures are shown in Fig. 2. Curves (A), (B), (C), and (D) correspond to oxide (SiO2 ) layer thicknesses L of 50, 70, 100,
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˚ respectively. In all cases, the SiO2 barrier potential is taken as φB = 3 eV. and 140 A, From this figure, the following salient features are to be noted. (1) The behavior of the tunneling current with bias exhibits oscillations throughout the applied voltage range. (2) The magnitude of the tunneling current at a particular applied voltage increases as the barrier width decreases. (3) The voltage between any two successive maxima (minima) increases as the barrier width decreases. (4) The amplitude of extrema points increases as the barrier width decreases. (5) The number of extrema points increases as the width increases which is consistent with the expected wave interference in the conduction band of the silicon oxide. Quantitatively, these results are in good agreement with the recently published experimental and the different model calculations [11–13, 22–24]. Mao and co-workers [12] applied a wave interference method on a biased square potential by representing the electron by a wave where interference occurs between the direct transmission parts of the incident electron wave and the transmission parts of the reflection wave, which traverses the barrier after being reflected by the two interfaces of the barrier. They deduced the following relations for the maxima and minima, respectively: − φB 3/2 4L ∗ 1/2 (2m q V ) 1+ N = 0.25 + (5) 3h qV 4L − φB 3/2 N = 0.75 + (2m ∗ q V )1/2 1 + (6) 3h qV where N is the extrema order. Notice that Eqs. (5) and (6) are not valid for the voltage region q V < − φB (direct tunneling region). Fig. 3 shows the Nth maxima voltage location versus N. The barrier is of 3 eV potential height, with widths of 20, 50, 70, and ˚ (curves A, B, C, D, respectively) and an electron reduced mass of 0.5m o . ×’s are 100 A obtained from the wave interference method given by Mao et al. [12] using 0.1 eV electron energy. Closed circles are obtained from this work taking into account the electron energy distribution and summing over all electron energies. The ∗’s that appear superimposed on the closed circles of curve D are for when the exact energy band structure obtained from the self-consistent solution of the Schr¨odinger and Poisson equations through SCHRED is used. The obvious agreement between the self-consistent solution and the method of this paper supports our initial approximation. The agreement between the wave interference method and the solution of the time dependent Schr¨odinger equation looks excellent for small width L = 20 and ˚ (curves (A) and (B), respectively) and systematic deviation becomes obvious at 50 A larger L. The two results agree well in terms of the number of the extrema as well as the location for small L. If Eqs. (5) and (6) are used to find the barrier width from the location ˚ was obtained for of the extrema of the quantum mechanical calculation, 49.7 ± 0.3 A ˚ the barrier of 50 A width. This agreement reflects the complementarity between the wave interference method and the solution of the time dependent Schr¨odinger equation method. Fig. 3 also shows that there is a systematic positive deviation in the positions of the extrema, which are calculated using the time dependent Schr¨odinger equation relating to the interference method. This deviation grows as the width of the barrier increases and as the extrema number increases. This could be attributed to two factors.
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Fig. 3. Voltage location of the tunneling current density oscillations, N th maximum versus N th maximum count. ˚ respectively. The barrier Curves (A), (B), (C), and (D) correspond to barrier widths of 20, 50, 70, and 100 A, potential is kept at 3 eV. For all the curves ◦’s refer to this work and ×’s are from the wave interference model of [12]. ∗’s on curve (D) are obtained from the self-consistent solution of the Schr¨odinger and Poisson equations using SCHRED.
The first is that electrons in the conduction band have an energy distribution given by Eq. (4). The effect of this distribution compared to a single energy value is twofold. The first result is that there is a difference in the maxima locations for the current density due to single-energy electrons and current density when the electron distribution is taken into ˚ The solid line corresponds account. This difference is shown in Fig. 4 for L = 70 A. to the tunneling current when the electron distribution is taken into account while the dotted curve corresponds to the tunneling current when an electron energy of 0.1 eV is assumed. It is noted that the effect of using single-energy electrons is an extrema position shift toward lower voltages of about 0.1 V. Changing the electron energy from 0.1 to other energies showed the same effect, however with a difference in the tunneling current amplitude. The second result is that at a particular applied voltage, and considering electron distribution, some electrons will experience direct tunneling and others will experience Fowler–Nordheim injection depending on their energy location, resulting in oscillation extrema at applied voltage less than the difference between the barrier potential and the electron energy q V < − φB which are not observable by the wave interference method. ˚ oxide layer Fig. 5 shows on a linear–linear scale the tunneling current density for a 70 A
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Fig. 4. The current density versus applied voltage (J –V ) characteristics of the MOS structure under study versus ˚ width. The solid line shows the applied voltage on a linear–linear scale for a barrier of 3 eV potential and 50 A current density when summation of the electron energy distribution (Eq. (4)) is taken into account. The dotted line shows when single-energy electrons of 0.1 eV are assumed.
width and 3 eV potential for up to 3 V applied voltage. The two maxima locations at 0.85 V and 1.77 V, which are due to high-energy electrons, cannot be determined by the wave interference method. Hence, N (the maxima count) will be different for the wave interference method and the solution of the Schr¨odinger equation and there will be some ambiguity in the use of Eqs. (5) and (6). Therefore, and based on the above two results, the electron energy distribution is to be considered for accurate and detailed analysis. Larger oxide layers showed a larger number of maxima locations for q V < 3 eV. The assumption of a single-energy electron in Mao’s work contributes to the divergence between his work ˚ and this work at barrier widths larger than 50 A. The second factor in the disagreement between our work and Mao’s work for barriers ˚ lies in the fact that the effective mass of the tunneling electron was taken as larger than 50 A variable as the electron wavepacket moves through the MOS device in the time dependent Schr¨odinger equation. However, in the wave interference model the mass of the electron was considered in the oxide layer only. It is necessary to include the electron effective mass for both materials at the interface where interference takes place for accurate calculations. In a previous work [25], the electron effective mass on both sides of the interference
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Fig. 5. The current density versus applied voltage (J –V ) characteristics of a MOS structure with the oxide layer ˚ and a potential of 3 eV. Oscillation maxima locations at applied voltage V = 0.85 V and width L = 70 A V = 1.8 V cannot be observed by the wave interference method of [12].
interface was proved to be important for explaining why oscillations were not observable in metal–insulator–metal structures. In that work it was shown that as the semiconductor effective mass on the SiO2 /Si interface is increased, the oscillations start to disappear, resulting in a decrease in the number of maxima counted. Hence, at a particular effective mass in the semiconductor region the number of maxima counted will be different and, therefore, the use of the wave interference method (Eqs. (5) and (6)) will result in the wrong prediction of the oxide thickness. Finally, it is worth noting that other theoretical models [10, 26] that use Airy functions method also make use of single-energy electrons and the electron effective mass in the oxide layer only. Therefore, it is expected that there will be some discrepancy between the Airy function method and the time dependent Schr¨odinger equation method for larger barriers for the same reasons as mentioned above. 3. Summary This work shows the results of the solution of the time dependent Schr¨odinger equation for an electron tunneling through a MOS device. The J –V characteristics of
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the tunneling electrons exhibit a fast increasing current with oscillations that are barrier width dependent. It is found that as the width decreases the number of extrema decreases and the extrema amplitudes increase. These results are found to be in good agreement with the wave interference model [12]. Finally, the agreement between the quantum mechanical calculation and the wave interference model suggests equivalence between the ˚ The discrepancy between the two methods for barrier widths two methods for L ≤ 50 A. ˚ could be attributed to two factors. The first is that in Mao’s work, only larger than 50 A the mass of the electron in the oxide layer was used to model the interference, while the electron mass in the semiconductor layer was not considered. Second, in Mao’s work, the electron energy distribution was neglected and single-energy electrons were used instead. Acknowledgements The first author would like to acknowledge with appreciation the United Arab Emirates University Research Counsel for financial support and for providing computing facilities. Appreciation is also extended to Purdue University Support for allowing the use of their nanotechnology computation Hub. The second author wishes to express his appreciation for the support of the University of Jordan and the Jordanian Higher Counsel of Research. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
H.J. Hebert, E.A. Irene, J. Appl. Phys. 82 (1997) 291. J.C. Poler, K.K. McKay, E.A. Irene, J. Vac. Sci. Technol. B 12 (1994) 88. F.-C. Chiu, J.-J. Wang, J.Y.-M. Lee, S.C. Wu, J. Appl. Phys. 81 (1997) 6911. E. Miranda, A. Faig´on, F. Campabadal, Solid State Electron. 41 (1997) 67. A. Schenk, G. Heiser, J. Appl. Phys. 81 (1997) 7900. J. Maserjian, N. Zamani, J. Appl. Phys. 53 (1982) 559. R. Ludeke, H.J. Wen, A. Schenk, Appl. Phys. Lett. 73 (1998) 1221. H.J. Wen, R. Ludeke, J. Vac. Sci. Technol. A 16 (1998) 1735. R. Ludeke, H.J. Wen, Appl. Phys. Lett. 71 (1997) 3123. S. Zafar, K.A. Conrad, Q. Liu, E.A. Irene, G. Hames, R. Kuehn, J.J. Wortman, Appl. Phys. Lett. 67 (1995) 1031. K.J. Hebert, S. Zafar, E.A. Irene, R. Kuehn, T.E. McCarthy, E.K. Demirlioglu, Appl. Phys. Lett. 68 (1996) 266. L. Mao, C. Tan, M. Xu, Solid State Electron. 44 (2000) 1501. Y. Ando, T. Itoh, J. Appl. Phys. 61 (1987) 1497. A. Messiah, Quantum Mechanics, North-Holland, Amsterdam, 1961. K. Sakakibara, N. Ajika, K. Eikyu, K. Ishikawa, H. Miyoshi, IEEE Trans. Electron Devices 44 (1997) 1002. A. Goldberg, H. Schey, Am. J. Phys. 35 (1967) 177. H. Guo, K. Diff, G. Neofotistos, J. Gunton, Appl. Phys. Lett. 53 (1988) 131. K. Leo, J. Shah, E. Gbel, T. Damen, S. Schmitt, W. Schafer, K. Khler, Phys. Rev. Lett. 66 (1991) 201. J. Mohaidat, K. Shum, R. Alfano, Phys. Rev. B 45 (1992) 3822. R. Pierret, Semiconductor Fundamentals, Addison-Wesley Publishing Co., Reading, Massachusetts, 1989, (Chapter 2). T. Tuurnala, T. Majamaa, Phys. Scripta T69 (1997) 310. G. Lewicki, J. Maserjian, J. Appl. Phys. 46 (1975) 3032. O. Briere, K. Brrla, A. Hamaui, G. Ghibaudo, Solid State Electron. 41 (1997) 987. K.J. Hebert, E.A. Irene, J. Appl. Phys. 82 (1997) 291. J. Mohaidat, J. Appl. Phys. 90 (2001) 871. L. Mao, C. Tan, M. Xu, J. Appl. Phys. 88 (2000) 6560.
Parameters affecting the accuracy of oxide thickness prediction in thin metal–oxide–semiconductor structures J.M. Mohaidata,∗, Riyad N. Ahmad-Bitarb a Department of Electrical Engineering, United Arab Emirates University, P.O. Box 17555, Al-Ain,
United Arab Emirates b Physics Department, University of Jordan, Amman, Jordan
Received 5 March 2003; received in revised form 28 January 2004; accepted 21 February 2004 Available online 10 April 2004
Abstract On the basis of the solution of the time dependent Schr¨odinger equation within the framework of the effective mass theory, a complete quantum mechanical electron tunneling through a biased square potential model with abrupt interfaces was deduced. Barriers of 3 eV height and ˚ were investigated. Current density–voltage (J –V ) curves were computed for widths up to 140 A + Al/SiO2 /n Si structure. The computed J –V curves exhibited oscillations at applied voltages above (Fowler–Nordheim tunneling) and below (direct tunneling) 3 V. For oxide thickness estimation, the position of the oscillation extrema from this quantum mechanical model were fitted to a wave ˚ However, interference formula and showed excellent agreement for oxide layer widths less than 50 A. ˚ We show that the electron energy a systematic deviation appeared for layers larger than 50 A. distribution at the injection layer and the electron effective mass on layers other than the oxide layer are important parameters for accurate oxide thickness estimation. © 2004 Elsevier Ltd. All rights reserved. PACS: 73.40.Gk; 73.40.-c; 73.40.Qv Keywords: MOS structure; Tunneling; Quantum interference; Effective mass; Energy distribution
∗ Corresponding author. Tel.: +971-3-7051606; fax: +971-3-7623156.
E-mail address: [email protected] (J.M. Mohaidat). 0749-6036/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2004.02.018
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1. Introduction ˚ with high dielectric constants, in metal– Ultrathin oxide films, less than 100 A, oxide–semiconductor (MOS) structures for application as storage capacitors for dynamic random access memories are required to scale down the device size. The passage of current through the film is dominated by tunneling. Quantum oscillations in the tunneling current were observed by many investigators [1–6]. Recently, pronounced quantum oscillations were detected for electrons injected over the barrier into a MOS structure with a scanning tunneling microscope [7–9]. The presence of oscillations is useful for determining the film thickness [10, 11] or the conduction electron average effective mass. Different theoretical models [4–6, 12, 13] are used to predict the oscillatory behavior of the tunneling current in order to interpret the experimental results. Most of these models are based on the conventional method of WKB (Wentzel–Kramers–Brillouin) approximation. This method is poor in explaining the oscillations and inaccurate in regions where the potential profile varies abruptly [14]. The discrepancies might arise from the framework of the WKB approximation, which assumes a smooth variation of the barrier potential. Recently published wave interference formulas were deduced for the location of the extrema of a biased square barrier with abrupt interfaces [12]. The phase shift between the direct transmission incident electron waves and the transmission part of the reflection wave, which traverses the barrier after being reflected by the two interfaces of the barrier, was calculated. Direct tunneling was assumed in the region where the electron energy is less than the potential energy and, otherwise, Fowler–Nordheim (FN) injection. The optical interference analogy was assumed in deriving these formulas. In this article we present calculated J –V curves based on a numerical solution of the time dependent Schr¨odinger equation for a complete quantum mechanical tunneling through a square potential model with abrupt interfaces. The barrier potential height = ˚ with a biasing voltage up to 20 V were 3 eV and barrier widths L of 140, 100, 70, and 50 A assumed. These values are in accordance with Al/SiO2 /n+ Si (MOS) devices. We show the importance of both the electron energy distribution at the injection layer of the MOS structure and the electron effective mass throughout the structure for the oxide thickness estimation. Given the fundamental nature of the problem (tunneling and reflection through ideal barriers), the use of the time dependent Schr¨odinger equation adds to the originality of this work. The calculated tunneling currents were found to have oscillations. The dependence of the extrema location and their amplitude on the oxide thickness is to be discussed. The validity of the interference formulas [12] will be checked. A numerical approach based on the solution of the time dependent Schr¨odinger equation is employed. The structure is assumed to consist of three regions: the metal, the insulator, and the heavily non-degenerately doped semiconductor. The metal and semiconductor regions are assumed to be of much larger dimensions relative to the barrier dimension. Square potential barriers with abrupt interfaces are assumed. The effects of changing barrier parameters on the tunneling current are studied. The functional dependence of the current on the potential difference across the barrier is studied. Quantum mechanical oscillations appearing in the J –V characteristic curves are discussed and compared against theoretical models [4–6, 12, 13].
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Fig. 1. A schematic energy band diagram of a MOS structure. (a) shows the structure under flat band conditions while (b) shows the structure under an applied bias Va . In this structure the energy separation between the Fermi level and the semiconductor band edge is very small but is exaggerated here for clarity.
Fig. 1 shows a schematic energy band diagram of the metal–insulator–semiconductor ideal structure. φm is the metal work function, χ is the electron affinity, φB is the barrier potential between the metal and the insulator, and L is the barrier width. Fig. 1(a) shows the structure under flat band conditions (zero applied bias). Fig. 1(b) shows the structure under an applied bias of Va . The exact band structure profile was not calculated here. Rigorous computations (self-consistent solution of the Schr¨odinger and Poisson equations) of the band structure will result in a heterojunction at the structure interfaces. This heterojunction is relatively very shallow especially for structures like the one under consideration in this paper and is usually neglected because they could produce only a negligible effect on the overall characteristics [1, 15]. To support this approximation we used for part of the work the 1D self-consistent Schr¨odinger–Poisson solver called SCHRED.1 Using SCHRED we obtained the exact energy band profile and then used it in the solution of the time dependent
1 SCHRED is installed on the Purdue University Semiconductor Simulation Hub (http://nanohub.purdue.edu/) and is accessible through the Web.
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Schr¨odinger equation. SCHRED treats the structure quantum mechanically and accounts for all six valleys of the structure. The numerical solution starts by assuming a one-dimensional (1D) free electron wavepacket with central energy relative to the semiconductor band edge. This wavepacket which is placed in the metal side at t = 0 assumes the following wavefunction form: (z − z 0 )2 ψ(z, t = 0) = A exp(i kz) exp − (1) 4σ 2 √ where A is a normalization constant. k = (1/) 2m ∗ is the wavepacket’s central energy. z 0 is its center location; σ is the packet’s spatial width, which was chosen to equal 55 nm. m ∗ is the electron effective mass which was assumed to be, in the metal side, m ∗ = m o , in the insulator, m ∗ = 0.5m o , and in the semiconductor region (Si), m ∗ = 0.19m o . m o being the free electron mass. The Hamiltonian operator H associated with the metal–insulator–semiconductor system is taken as the Ben-Daniel and Duke Hamiltonian: 2 ∂ 1 ∂ H =− + V (z) (2) 2 ∂z m ∗ ∂z where the potential V (z) is due to the conduction band discontinuity including the effect of the applied electric field, as shown in Fig. 1. The evolution of the wavefunction ψ(z, t) is determined by solving the time dependent Schr¨odinger equation, H ψ(z, t) = i (∂/∂t)ψ(z, t). To proceed with the numerical solution, the Ben-Daniel and Duke Hamiltonian (Eq. (2)) is converted into a difference equation. The resulting Schr¨odinger finite difference equation is then solved numerically using the Crank–Nicholson method [16]. A spatial mesh ˚ a time mesh size of 0.5 fs, and hard-wall boundary conditions at the two size [17] of 0.5 A, end-points 20 µm away from the structure were assumed. Interface boundary conditions included the continuity of the wavefunction and the continuity of (1/m ∗ )(∂ψ/∂z) to insure particle current continuity. An excellent agreement was obtained between the results of this numerical procedure and the experimental results obtained for a double-quantum-well oscillator [18, 19]. The initial electron wavepacket in Eq. (1), at t = 0, is placed in the left region of the structure of Fig. 1. The tunneling wavefunction ψ(z, t) in the semiconductor region is found in time as the wavepacket moves out of the right region, through the insulator, and into the left region. The time integrated current density J at a particular applied voltage and the wavepacket central energy is calculated at a point 2 nm to the left of the barrier (Fig. 1) using the exact quantum mechanical current density expression: ∂ψ ∗ ∗ ∂ψ ψ − ψ . (3) J= 2m ∗ i ∂z ∂z The energy distribution of the free electrons is given by [20] √ B n() = − 1 + exp k Tf B
(4)
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Fig. 2. The current density versus applied voltage J –V characteristics of the MOS structure depicted in Fig. 1 for various layer oxide thicknesses L. Curves (A), (B), (C), and (D) correspond to oxide layer widths L of 50, 70, ˚ respectively. 100, and 140 A,
where B is a constant, f is the electron Fermi level, kB is the Boltzmann constant, and T is the temperature. Therefore, to compare with experimental results, the overall tunneling current density J is weighted by summing over all current density contributions of all electron wavepackets of different energy . Otherwise, the current density J (V ) is found by considering only Eq. (3). 2. Results and discussion Considering an electron wavepacket with central energy , the wavepacket will tunnel a distance L t = L(φB − )/q V through the barrier, then it will be injected over the barrier and ballistically propagate toward the interface. The electron wavepackets have a certain probability of reflecting and of transmitting at the interface. The transmitted and the twicereflected electron waves interfere. For MOS devices whose thickness is of the order of the electron de Broglie wavelength, quantum oscillations have been observed in the tunneling currents [1, 2, 4, 6, 10, 21, 22]. In general, it is found that the position and the amplitude of the oscillations will vary depending on the shape of the potential barrier in the structure. The tunneling J –V characteristics for MOS structures are shown in Fig. 2. Curves (A), (B), (C), and (D) correspond to oxide (SiO2 ) layer thicknesses L of 50, 70, 100,
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˚ respectively. In all cases, the SiO2 barrier potential is taken as φB = 3 eV. and 140 A, From this figure, the following salient features are to be noted. (1) The behavior of the tunneling current with bias exhibits oscillations throughout the applied voltage range. (2) The magnitude of the tunneling current at a particular applied voltage increases as the barrier width decreases. (3) The voltage between any two successive maxima (minima) increases as the barrier width decreases. (4) The amplitude of extrema points increases as the barrier width decreases. (5) The number of extrema points increases as the width increases which is consistent with the expected wave interference in the conduction band of the silicon oxide. Quantitatively, these results are in good agreement with the recently published experimental and the different model calculations [11–13, 22–24]. Mao and co-workers [12] applied a wave interference method on a biased square potential by representing the electron by a wave where interference occurs between the direct transmission parts of the incident electron wave and the transmission parts of the reflection wave, which traverses the barrier after being reflected by the two interfaces of the barrier. They deduced the following relations for the maxima and minima, respectively: − φB 3/2 4L ∗ 1/2 (2m q V ) 1+ N = 0.25 + (5) 3h qV 4L − φB 3/2 N = 0.75 + (2m ∗ q V )1/2 1 + (6) 3h qV where N is the extrema order. Notice that Eqs. (5) and (6) are not valid for the voltage region q V < − φB (direct tunneling region). Fig. 3 shows the Nth maxima voltage location versus N. The barrier is of 3 eV potential height, with widths of 20, 50, 70, and ˚ (curves A, B, C, D, respectively) and an electron reduced mass of 0.5m o . ×’s are 100 A obtained from the wave interference method given by Mao et al. [12] using 0.1 eV electron energy. Closed circles are obtained from this work taking into account the electron energy distribution and summing over all electron energies. The ∗’s that appear superimposed on the closed circles of curve D are for when the exact energy band structure obtained from the self-consistent solution of the Schr¨odinger and Poisson equations through SCHRED is used. The obvious agreement between the self-consistent solution and the method of this paper supports our initial approximation. The agreement between the wave interference method and the solution of the time dependent Schr¨odinger equation looks excellent for small width L = 20 and ˚ (curves (A) and (B), respectively) and systematic deviation becomes obvious at 50 A larger L. The two results agree well in terms of the number of the extrema as well as the location for small L. If Eqs. (5) and (6) are used to find the barrier width from the location ˚ was obtained for of the extrema of the quantum mechanical calculation, 49.7 ± 0.3 A ˚ the barrier of 50 A width. This agreement reflects the complementarity between the wave interference method and the solution of the time dependent Schr¨odinger equation method. Fig. 3 also shows that there is a systematic positive deviation in the positions of the extrema, which are calculated using the time dependent Schr¨odinger equation relating to the interference method. This deviation grows as the width of the barrier increases and as the extrema number increases. This could be attributed to two factors.
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Fig. 3. Voltage location of the tunneling current density oscillations, N th maximum versus N th maximum count. ˚ respectively. The barrier Curves (A), (B), (C), and (D) correspond to barrier widths of 20, 50, 70, and 100 A, potential is kept at 3 eV. For all the curves ◦’s refer to this work and ×’s are from the wave interference model of [12]. ∗’s on curve (D) are obtained from the self-consistent solution of the Schr¨odinger and Poisson equations using SCHRED.
The first is that electrons in the conduction band have an energy distribution given by Eq. (4). The effect of this distribution compared to a single energy value is twofold. The first result is that there is a difference in the maxima locations for the current density due to single-energy electrons and current density when the electron distribution is taken into ˚ The solid line corresponds account. This difference is shown in Fig. 4 for L = 70 A. to the tunneling current when the electron distribution is taken into account while the dotted curve corresponds to the tunneling current when an electron energy of 0.1 eV is assumed. It is noted that the effect of using single-energy electrons is an extrema position shift toward lower voltages of about 0.1 V. Changing the electron energy from 0.1 to other energies showed the same effect, however with a difference in the tunneling current amplitude. The second result is that at a particular applied voltage, and considering electron distribution, some electrons will experience direct tunneling and others will experience Fowler–Nordheim injection depending on their energy location, resulting in oscillation extrema at applied voltage less than the difference between the barrier potential and the electron energy q V < − φB which are not observable by the wave interference method. ˚ oxide layer Fig. 5 shows on a linear–linear scale the tunneling current density for a 70 A
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Fig. 4. The current density versus applied voltage (J –V ) characteristics of the MOS structure under study versus ˚ width. The solid line shows the applied voltage on a linear–linear scale for a barrier of 3 eV potential and 50 A current density when summation of the electron energy distribution (Eq. (4)) is taken into account. The dotted line shows when single-energy electrons of 0.1 eV are assumed.
width and 3 eV potential for up to 3 V applied voltage. The two maxima locations at 0.85 V and 1.77 V, which are due to high-energy electrons, cannot be determined by the wave interference method. Hence, N (the maxima count) will be different for the wave interference method and the solution of the Schr¨odinger equation and there will be some ambiguity in the use of Eqs. (5) and (6). Therefore, and based on the above two results, the electron energy distribution is to be considered for accurate and detailed analysis. Larger oxide layers showed a larger number of maxima locations for q V < 3 eV. The assumption of a single-energy electron in Mao’s work contributes to the divergence between his work ˚ and this work at barrier widths larger than 50 A. The second factor in the disagreement between our work and Mao’s work for barriers ˚ lies in the fact that the effective mass of the tunneling electron was taken as larger than 50 A variable as the electron wavepacket moves through the MOS device in the time dependent Schr¨odinger equation. However, in the wave interference model the mass of the electron was considered in the oxide layer only. It is necessary to include the electron effective mass for both materials at the interface where interference takes place for accurate calculations. In a previous work [25], the electron effective mass on both sides of the interference
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Fig. 5. The current density versus applied voltage (J –V ) characteristics of a MOS structure with the oxide layer ˚ and a potential of 3 eV. Oscillation maxima locations at applied voltage V = 0.85 V and width L = 70 A V = 1.8 V cannot be observed by the wave interference method of [12].
interface was proved to be important for explaining why oscillations were not observable in metal–insulator–metal structures. In that work it was shown that as the semiconductor effective mass on the SiO2 /Si interface is increased, the oscillations start to disappear, resulting in a decrease in the number of maxima counted. Hence, at a particular effective mass in the semiconductor region the number of maxima counted will be different and, therefore, the use of the wave interference method (Eqs. (5) and (6)) will result in the wrong prediction of the oxide thickness. Finally, it is worth noting that other theoretical models [10, 26] that use Airy functions method also make use of single-energy electrons and the electron effective mass in the oxide layer only. Therefore, it is expected that there will be some discrepancy between the Airy function method and the time dependent Schr¨odinger equation method for larger barriers for the same reasons as mentioned above. 3. Summary This work shows the results of the solution of the time dependent Schr¨odinger equation for an electron tunneling through a MOS device. The J –V characteristics of
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the tunneling electrons exhibit a fast increasing current with oscillations that are barrier width dependent. It is found that as the width decreases the number of extrema decreases and the extrema amplitudes increase. These results are found to be in good agreement with the wave interference model [12]. Finally, the agreement between the quantum mechanical calculation and the wave interference model suggests equivalence between the ˚ The discrepancy between the two methods for barrier widths two methods for L ≤ 50 A. ˚ could be attributed to two factors. The first is that in Mao’s work, only larger than 50 A the mass of the electron in the oxide layer was used to model the interference, while the electron mass in the semiconductor layer was not considered. Second, in Mao’s work, the electron energy distribution was neglected and single-energy electrons were used instead. Acknowledgements The first author would like to acknowledge with appreciation the United Arab Emirates University Research Counsel for financial support and for providing computing facilities. Appreciation is also extended to Purdue University Support for allowing the use of their nanotechnology computation Hub. The second author wishes to express his appreciation for the support of the University of Jordan and the Jordanian Higher Counsel of Research. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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