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Quantum confinement model for electric transport phenomena in fresh and stored photoluminescent porous silicon films
PII:
Solid-State Electronics Vol. 42, No. 10, pp. 1893±1896, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/98/$ - see front matter S0038-1101(98)00160-9
LETTER QUANTUM CONFINEMENT MODEL FOR ELECTRIC TRANSPORT PHENOMENA IN FRESH AND STORED PHOTOLUMINESCENT POROUS SILICON FILMS V. IANCU1 and M. L. CIUREA2 Physics Department, University ``Politehnica'' of Bucharest, Splaiul Independentei 313, Bucharest 77206, Romania 2 National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Magurele 76900, Romania
1
(Received 25 November 1997; in revised form 5 February 1998) AbstractÐA quantum con®nement model is proposed to explain the electrical transport properties in fresh and stored porous silicon (PS) ®lms. In the present paper, the model is veri®ed for the temperature dependence of the dark current. The studied samples are formed mainly by a network of nanowires, suggesting the separation of the electron Hamiltonian into a longitudinal and a transversal part. The last one can be well described by a two-dimensional (2D) cylindrical in®nite quantum well, whose levels determine the dark current activation energies. The storage oxidation induces modi®cations both in the number and the values of the activation energies, which are in excellent agreement with our model. # 1998 Elsevier Science Ltd. All rights reserved
The discovery of the highly ecient photoluminescence (PL) of the porous silicon (PS) in the visible range at room temperature[1,2] has started a systematic study of this phenomenon, because of the enormous interest in the possibility of obtaining high quality optoelectronic devices with silicon microtechnology. Two kinds of models were proposed to explain the PL. The ®rst one is that of the quantum con®nement of the carriers inside the nanocrystallites, that determines an increase of the band gap[1,2] and, possibly, transforms the indirect band gap of crystalline silicon in a direct one. The second one considers the dierent surface radiative and nonradiative contributions[3±5]. The electrical properties of PS ®lms were also investigated using dierent experimental techniques and dierent theoretical mechanisms[6±10]. We will try to show that the frame of the quantum con®nement model is quite able to explain the main features of the electrical transport properties in PS ®lms, as obtained in our measurements. We have prepared PS ®lms, having approximately 25 6 35 mm thickness, on (100) p-type wafers of crystalline silicon (c-Si), with aluminum electrodes in sandwich con®guration, Al/PS/c-Si/Al, the bottom Al contact being prepared ohmic[11,12]. The transmission microscopy measurements[13] showed that the surface of our ®lms is formed mainly by a network of nanowires, having diameters of the order of 2.5 6 7 nm. Measurements were at ®rst performed on the fresh samples. After that, the
samples were stored and the measurements repeated at dierent time intervals, up to 2 years. In the present paper we will discuss only the dark current measurements. All the other electrical measurements and the corresponding theoretical analysis within the frame of the present model are presented in Ref.[12]. The temperature dependence of the dark current in the 1506 300 K range for the fresh samples shows an Arrhenius behavior (Fig. 1), with only one activation energy, Efdl, whose value lies in the dispersion interval 0.49 6 0.55 eV (i.e. dierent values were obtained for dierent samples, prepared, stored and measured in the same conditions). This value is practically independent of the bias polarity at low voltage (ÿ2 6 + 2 V). The annealing of the samples at 508C for 0.5 h produces only a small reversible decrease of the activation energy, down to 0.43 6 0.51 eV. The storage in ambient produces changes in the properties of PS ®lms, due to natural oxidation. We observed that, after approximately 1.5 years, the electrical parameters become practically constant. In the following, only the results obtained on stabilized samples are discussed. The dark current behavior strongly diers from that of the fresh samples. The I±T curves present two activation energies (Fig. 2), one at low temperatures, Esdl10.50 6 0.60 eV, independent of the bias polarity at low voltage and the other at high temperatures, Esdh11.20 6 1.80 eV, slightly dierent for opposite bias polarities. The dierences are small enough to
1893
1894
V. Iancu and M. L. Ciruea
Fig. 1. The temperature dependence of the dark current in the 150 to 300 K range for fresh samples. Q, Ua= ÿ 1 V (+ on PS); w, Ua=1 V (+ on c-Si).
be neglected in a ®rst approach. The activation energy value changes abruptly at about 280 K. This temperature also is slightly dierent for opposite bias polarities. The annealing produces small revers-
ible increases of both activation energies, up to 0.58 6 0.66 eV, respectively, 1.35 6 1.80 eV. All these results can be explained within the frame of a simpli®ed quantum con®nement model.
Fig. 2. The temperature dependence of the dark current in the 150 to 300 K range obtained on stabilized samples. Q, Ua=2 V (+ on c-Si); q, Ua= ÿ 2 V (+ on PS).
Quantum con®nement model for electric transport phenomena
In our samples, the length of the wires is 1036104 times greater than their diameter. Under these conditions, the electron Hamiltonian can be expressed as the sum of a longitudinal and a transversal part. Such a splitting is not exact, because of the tetrahedral symmetry of the silicon. However, it represents a good ®rst approximation. The solution of the longitudinal equation is a one-dimensional (1D) band structure, presenting an increased (and probably direct) band gap compared to the bulk c-Si gap (see[14]). The simplest description of the transversal part is given by a 2D cylindrical in®nite quantum well. This description also is not exact, due to the c-Si symmetry, but is a good approximation. Under these conditions, the electron energy in a nanowire is 2h2 2 2h2 2 E en,kz x e x n,k z m*d2 m,p m*d2 0,1
2h2 x 2 ÿ x 20,1 esn,kz Em,pÿ1 , m*d2 m,p
1
where esn,kz is the shifted energy corresponding to the longitudinal motion, Em,p the discrete energy levels corresponding to the transversal motion (by de®nition E0,0=0), m* the eective mass of the electron, d the wire diameter and xm,p is the pth zero of the Bessel function Jm(x), m being the orbital (azimuthal) quantum number. Therefore, the quantum con®nement produces both the increase of the band gap and the introduction of a number of discrete energy levels into the gap. This means that the Fermi level is pinned on the ®rst discrete level until this one is fully occupied, then it jumps on the following one, while the longitudinal conduction is supported by the holes formed in the valence band. Then Em,p represent the values for the activation energy of the dark current. We can use the expression of the discrete energy levels to evaluate the crystallite diameter. Using the average value EÄsdl30.55 eV, identi®ed with E0,1, we obtain with m* = m0e a diameter d 3 2.62 nm and with m* = mt* 3 0.66 m0e a diameter d 3 3.22 nm, in good agreement with our data[13]. We have to specify that this is the silicon core diameter, the core of the stored nanowires being surrounded by a thin oxide layer. Under these conditions, if the ®rst activation energy becomes equal with the band gap (r2 eV), the core diameter reduces below 1.3 6 1.6 nm, less than 6 6 7 interatomic distances (i.e. three elementary cells), so the wire is no longer a proper (3D) silicon crystallite. Because of the angular momentum conservation in a longitudinal electric ®eld, the levels with m $ 0 cannot be excited in ordinary dark current measurements, even if E1,0 and E2,0 are smaller than E0,1 (for instance, if E0,130.55 eV, then E1,030.20 eV). To observe them, we would have to measure the dark current at low temperatures, in a strong longi-
1895
tudinal magnetic ®eld, or to study the infrared photoluminescence spectrum, where the desexcitation of the E0,1 level through the E1,0 one should be observed in two lines, one with l 3 3.5 mm and the other with l 3 6.2 mm. In our simpli®ed model, the activation energies are proportional with dÿ2, typical for the eective mass approximation. However, the calculation of the electronic band structure of silicon nanocrystallites, using the technique of the linear combination of atomic orbitals[14], showed that the band gap follows approximately a dÿa dependence, with a R 1.39. If we accept this dependence, the variation of the activation energy values for our diameter range is in good agreement with our data. At the same time, the degeneracy of the levels is proportional with the number of atoms in a given transversal section, that is with d2. Consequently, the decrease of the diameter of the silicon wire (due to the natural oxidation during the storage) will produce a small increase of the activation energy E0,1, but a strong decrease of the degeneracy. At a given temperature, corresponding to the saturation of the ®rst level, the Fermi level will jump on the second one and the value of the activation energy will become E0,2. Their ratio is E0,2 x 20,3 ÿ x 20,1 2:799, E0,1 x 20,2 ÿ x 20,1
2
in excellent agreement with our experimental result, EÄsdh/EÄsdl32.727. If we take into account that the real quantum well is ®nite and not quite cylindrical, the agreement is improved. The potential barrier at the c-Si/ SiO2 separation wall is known to be approximately 5.25 eV[15]. The Si/SiOx barrier in our case seems to be only 2.2 eV, as resulting from the current±voltage characteristics (see Ref.[12]), because the imperfect oxide layer formed by natural oxidation during the storage behaves like a very high resistivity semiconductor, not like a proper insulator. Under these conditions, for a cylindrical symmetry, the theoretical ratio becomes Er0,2/Er0,132.78. Remember that the theoretical ratio for an in®nite square quantum well is 8/3, meaning that the dierence between the theory and experiment can be fully attributed to the lack of symmetry. The small reversible changes produced by annealing are due to the surface desorption±adsorption processes (e.g. H2O), then varying the height of the real (®nite) quantum well. To summarize, we have analyzed the temperature dependence of the dark current in PS ®lms and we have shown that it can be fully explained by the quantum con®nement eects. Our experimental data are well described by a 2D cylindrical in®nite quantum well model, which permits to explain the observed activation energies by means of the discrete energy levels introduced into the longitudinal
1896
V. Iancu and M. L. Ciruea
band gap. We have to remark that these levels do not depend (in the limit of an in®nite quantum well) upon the nature of the crystallite, but only upon its dimensions. This suggests that the present model could work for other nanocrystallites as well. REFERENCES
1. Canham, L. T., Appl. Phys. Lett., 1990, 57, 1046. 2. Lehmann, V. and Gosele, V., Appl. Phys. Lett., 1991, 58, 826. 3. Petrova-Koch, V., Muschik, T., Kux, A., Meyer, B. K. and Koch, F., Appl. Phys. Lett., 1992, 61, 943. 4. Koch, F., Mater. Res. Soc. Symp. Proc., 1993, 298, 319. 5. Hamilton, B., Semicond. Sci. Technol., 1995, 10, 1187. 6. Koshida, N. and Koyama, H., Mater. Res. Soc. Symp. Proc., 1993, 283, 337. 7. Ben-Chorin, M., Moller, F. and Koch, F., Phys. Rev. B, 1994, 49, 2981.
8. Ben-Chorin, M., Moller, F. and Koch, F., Phys. Rev. B, 1995, 51, 2199. 9. Dimitrov, D. B., Phys. Rev. B, 1995, 51, 1562. 10. Peng, C., Hirschmann, K. D. and Fauchet, P. M., J. Appl. Phys., 1996, 80, 295. 11. Ciurea, M. L., Pentia, E., Manea, A., Belu-Marian, A. and Baltog, I., Phys. status solidi (b), 1996, 195, 637. 12. Ciurea, M. L., Baltog, I., Lazar, M., Iancu, V., Lazanu S. and Pentia, E., Thin Solid Films, accepted for publication. 13. Ciurea, M. L., Iancu, V., Nistor, L., Teodorescu, V. and Lazanu, S., to be presented at the Fifth Int. Symp. on Quantum Con®nement, Boston, November 1998. 14. Delerue, C., Allan, G. and Lanoo, M., Phys. Rev. B, 1993, 48, 11024. 15. Sze, S. M., Physics of Semiconductor Devices, 1st edn. John Wiley and Sons, New York, 1969, p. 471.
Solid-State Electronics Vol. 42, No. 10, pp. 1893±1896, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/98/$ - see front matter S0038-1101(98)00160-9
LETTER QUANTUM CONFINEMENT MODEL FOR ELECTRIC TRANSPORT PHENOMENA IN FRESH AND STORED PHOTOLUMINESCENT POROUS SILICON FILMS V. IANCU1 and M. L. CIUREA2 Physics Department, University ``Politehnica'' of Bucharest, Splaiul Independentei 313, Bucharest 77206, Romania 2 National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Magurele 76900, Romania
1
(Received 25 November 1997; in revised form 5 February 1998) AbstractÐA quantum con®nement model is proposed to explain the electrical transport properties in fresh and stored porous silicon (PS) ®lms. In the present paper, the model is veri®ed for the temperature dependence of the dark current. The studied samples are formed mainly by a network of nanowires, suggesting the separation of the electron Hamiltonian into a longitudinal and a transversal part. The last one can be well described by a two-dimensional (2D) cylindrical in®nite quantum well, whose levels determine the dark current activation energies. The storage oxidation induces modi®cations both in the number and the values of the activation energies, which are in excellent agreement with our model. # 1998 Elsevier Science Ltd. All rights reserved
The discovery of the highly ecient photoluminescence (PL) of the porous silicon (PS) in the visible range at room temperature[1,2] has started a systematic study of this phenomenon, because of the enormous interest in the possibility of obtaining high quality optoelectronic devices with silicon microtechnology. Two kinds of models were proposed to explain the PL. The ®rst one is that of the quantum con®nement of the carriers inside the nanocrystallites, that determines an increase of the band gap[1,2] and, possibly, transforms the indirect band gap of crystalline silicon in a direct one. The second one considers the dierent surface radiative and nonradiative contributions[3±5]. The electrical properties of PS ®lms were also investigated using dierent experimental techniques and dierent theoretical mechanisms[6±10]. We will try to show that the frame of the quantum con®nement model is quite able to explain the main features of the electrical transport properties in PS ®lms, as obtained in our measurements. We have prepared PS ®lms, having approximately 25 6 35 mm thickness, on (100) p-type wafers of crystalline silicon (c-Si), with aluminum electrodes in sandwich con®guration, Al/PS/c-Si/Al, the bottom Al contact being prepared ohmic[11,12]. The transmission microscopy measurements[13] showed that the surface of our ®lms is formed mainly by a network of nanowires, having diameters of the order of 2.5 6 7 nm. Measurements were at ®rst performed on the fresh samples. After that, the
samples were stored and the measurements repeated at dierent time intervals, up to 2 years. In the present paper we will discuss only the dark current measurements. All the other electrical measurements and the corresponding theoretical analysis within the frame of the present model are presented in Ref.[12]. The temperature dependence of the dark current in the 1506 300 K range for the fresh samples shows an Arrhenius behavior (Fig. 1), with only one activation energy, Efdl, whose value lies in the dispersion interval 0.49 6 0.55 eV (i.e. dierent values were obtained for dierent samples, prepared, stored and measured in the same conditions). This value is practically independent of the bias polarity at low voltage (ÿ2 6 + 2 V). The annealing of the samples at 508C for 0.5 h produces only a small reversible decrease of the activation energy, down to 0.43 6 0.51 eV. The storage in ambient produces changes in the properties of PS ®lms, due to natural oxidation. We observed that, after approximately 1.5 years, the electrical parameters become practically constant. In the following, only the results obtained on stabilized samples are discussed. The dark current behavior strongly diers from that of the fresh samples. The I±T curves present two activation energies (Fig. 2), one at low temperatures, Esdl10.50 6 0.60 eV, independent of the bias polarity at low voltage and the other at high temperatures, Esdh11.20 6 1.80 eV, slightly dierent for opposite bias polarities. The dierences are small enough to
1893
1894
V. Iancu and M. L. Ciruea
Fig. 1. The temperature dependence of the dark current in the 150 to 300 K range for fresh samples. Q, Ua= ÿ 1 V (+ on PS); w, Ua=1 V (+ on c-Si).
be neglected in a ®rst approach. The activation energy value changes abruptly at about 280 K. This temperature also is slightly dierent for opposite bias polarities. The annealing produces small revers-
ible increases of both activation energies, up to 0.58 6 0.66 eV, respectively, 1.35 6 1.80 eV. All these results can be explained within the frame of a simpli®ed quantum con®nement model.
Fig. 2. The temperature dependence of the dark current in the 150 to 300 K range obtained on stabilized samples. Q, Ua=2 V (+ on c-Si); q, Ua= ÿ 2 V (+ on PS).
Quantum con®nement model for electric transport phenomena
In our samples, the length of the wires is 1036104 times greater than their diameter. Under these conditions, the electron Hamiltonian can be expressed as the sum of a longitudinal and a transversal part. Such a splitting is not exact, because of the tetrahedral symmetry of the silicon. However, it represents a good ®rst approximation. The solution of the longitudinal equation is a one-dimensional (1D) band structure, presenting an increased (and probably direct) band gap compared to the bulk c-Si gap (see[14]). The simplest description of the transversal part is given by a 2D cylindrical in®nite quantum well. This description also is not exact, due to the c-Si symmetry, but is a good approximation. Under these conditions, the electron energy in a nanowire is 2h2 2 2h2 2 E en,kz x e x n,k z m*d2 m,p m*d2 0,1
2h2 x 2 ÿ x 20,1 esn,kz Em,pÿ1 , m*d2 m,p
1
where esn,kz is the shifted energy corresponding to the longitudinal motion, Em,p the discrete energy levels corresponding to the transversal motion (by de®nition E0,0=0), m* the eective mass of the electron, d the wire diameter and xm,p is the pth zero of the Bessel function Jm(x), m being the orbital (azimuthal) quantum number. Therefore, the quantum con®nement produces both the increase of the band gap and the introduction of a number of discrete energy levels into the gap. This means that the Fermi level is pinned on the ®rst discrete level until this one is fully occupied, then it jumps on the following one, while the longitudinal conduction is supported by the holes formed in the valence band. Then Em,p represent the values for the activation energy of the dark current. We can use the expression of the discrete energy levels to evaluate the crystallite diameter. Using the average value EÄsdl30.55 eV, identi®ed with E0,1, we obtain with m* = m0e a diameter d 3 2.62 nm and with m* = mt* 3 0.66 m0e a diameter d 3 3.22 nm, in good agreement with our data[13]. We have to specify that this is the silicon core diameter, the core of the stored nanowires being surrounded by a thin oxide layer. Under these conditions, if the ®rst activation energy becomes equal with the band gap (r2 eV), the core diameter reduces below 1.3 6 1.6 nm, less than 6 6 7 interatomic distances (i.e. three elementary cells), so the wire is no longer a proper (3D) silicon crystallite. Because of the angular momentum conservation in a longitudinal electric ®eld, the levels with m $ 0 cannot be excited in ordinary dark current measurements, even if E1,0 and E2,0 are smaller than E0,1 (for instance, if E0,130.55 eV, then E1,030.20 eV). To observe them, we would have to measure the dark current at low temperatures, in a strong longi-
1895
tudinal magnetic ®eld, or to study the infrared photoluminescence spectrum, where the desexcitation of the E0,1 level through the E1,0 one should be observed in two lines, one with l 3 3.5 mm and the other with l 3 6.2 mm. In our simpli®ed model, the activation energies are proportional with dÿ2, typical for the eective mass approximation. However, the calculation of the electronic band structure of silicon nanocrystallites, using the technique of the linear combination of atomic orbitals[14], showed that the band gap follows approximately a dÿa dependence, with a R 1.39. If we accept this dependence, the variation of the activation energy values for our diameter range is in good agreement with our data. At the same time, the degeneracy of the levels is proportional with the number of atoms in a given transversal section, that is with d2. Consequently, the decrease of the diameter of the silicon wire (due to the natural oxidation during the storage) will produce a small increase of the activation energy E0,1, but a strong decrease of the degeneracy. At a given temperature, corresponding to the saturation of the ®rst level, the Fermi level will jump on the second one and the value of the activation energy will become E0,2. Their ratio is E0,2 x 20,3 ÿ x 20,1 2:799, E0,1 x 20,2 ÿ x 20,1
2
in excellent agreement with our experimental result, EÄsdh/EÄsdl32.727. If we take into account that the real quantum well is ®nite and not quite cylindrical, the agreement is improved. The potential barrier at the c-Si/ SiO2 separation wall is known to be approximately 5.25 eV[15]. The Si/SiOx barrier in our case seems to be only 2.2 eV, as resulting from the current±voltage characteristics (see Ref.[12]), because the imperfect oxide layer formed by natural oxidation during the storage behaves like a very high resistivity semiconductor, not like a proper insulator. Under these conditions, for a cylindrical symmetry, the theoretical ratio becomes Er0,2/Er0,132.78. Remember that the theoretical ratio for an in®nite square quantum well is 8/3, meaning that the dierence between the theory and experiment can be fully attributed to the lack of symmetry. The small reversible changes produced by annealing are due to the surface desorption±adsorption processes (e.g. H2O), then varying the height of the real (®nite) quantum well. To summarize, we have analyzed the temperature dependence of the dark current in PS ®lms and we have shown that it can be fully explained by the quantum con®nement eects. Our experimental data are well described by a 2D cylindrical in®nite quantum well model, which permits to explain the observed activation energies by means of the discrete energy levels introduced into the longitudinal
1896
V. Iancu and M. L. Ciruea
band gap. We have to remark that these levels do not depend (in the limit of an in®nite quantum well) upon the nature of the crystallite, but only upon its dimensions. This suggests that the present model could work for other nanocrystallites as well. REFERENCES
1. Canham, L. T., Appl. Phys. Lett., 1990, 57, 1046. 2. Lehmann, V. and Gosele, V., Appl. Phys. Lett., 1991, 58, 826. 3. Petrova-Koch, V., Muschik, T., Kux, A., Meyer, B. K. and Koch, F., Appl. Phys. Lett., 1992, 61, 943. 4. Koch, F., Mater. Res. Soc. Symp. Proc., 1993, 298, 319. 5. Hamilton, B., Semicond. Sci. Technol., 1995, 10, 1187. 6. Koshida, N. and Koyama, H., Mater. Res. Soc. Symp. Proc., 1993, 283, 337. 7. Ben-Chorin, M., Moller, F. and Koch, F., Phys. Rev. B, 1994, 49, 2981.
8. Ben-Chorin, M., Moller, F. and Koch, F., Phys. Rev. B, 1995, 51, 2199. 9. Dimitrov, D. B., Phys. Rev. B, 1995, 51, 1562. 10. Peng, C., Hirschmann, K. D. and Fauchet, P. M., J. Appl. Phys., 1996, 80, 295. 11. Ciurea, M. L., Pentia, E., Manea, A., Belu-Marian, A. and Baltog, I., Phys. status solidi (b), 1996, 195, 637. 12. Ciurea, M. L., Baltog, I., Lazar, M., Iancu, V., Lazanu S. and Pentia, E., Thin Solid Films, accepted for publication. 13. Ciurea, M. L., Iancu, V., Nistor, L., Teodorescu, V. and Lazanu, S., to be presented at the Fifth Int. Symp. on Quantum Con®nement, Boston, November 1998. 14. Delerue, C., Allan, G. and Lanoo, M., Phys. Rev. B, 1993, 48, 11024. 15. Sze, S. M., Physics of Semiconductor Devices, 1st edn. John Wiley and Sons, New York, 1969, p. 471.