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Hole transport mechanisms in CuGaSe2
Thin Solid Films 480–481 (2005) 312 – 317 www.elsevier.com/locate/tsf
Hole transport mechanisms in CuGaSe2 Susanne Siebentritt* Hahn-Meitner-Institut, Glienicker Str. 100, 14109 Berlin, Germany Available online 19 December 2004
Abstract Hall and conductivity measurements have been performed on epitaxial and polycrystalline CuGaSe2 films. Temperature dependencies of charge carrier concentration and mobility are analyzed in terms of band transport model and standard scattering mechanisms, and are fitted simultaneously. Defect concentrations are lower than previously reported. Dominant scattering mechanisms in the single crystalline films are nonpolar optical phonon and acoustical phonon scattering, and charged defect scattering. Room temperature mobility in polycrystalline films is considerably lower than in epitaxial films. This can not only be attributed to scattering at extended defects. D 2004 Elsevier B.V. All rights reserved. Keywords: CuGaSe2 film; Nonpolar optical phonon; Epitaxial film
1. Introduction When modeling chalcopyrite solar cells, information on the mobility of charge carriers and its temperature dependence is needed. Likewise enters the mobility into the analysis of many electrical measurements, like admittance spectroscopy. A few studies exist on single crystals, epitaxial and polycrystalline films of CuInSe2 [1–3] and of CuGaSe2 [4–7]. Most analyses rely on the basic paper by Wiley and DiDomenico [8] on the modeling of the basic scattering mechanisms for holes in III–V semiconductors. Some of these equations are based on empirical considerations. Here, we will use equations strictly derived from quantum mechanical considerations and the Boltzmann transport equation [9]. We have previously reported on mobilities in our epitaxial CuGaSe2 films [7], where the scattering mechanisms were derived from the general trends in the temperature dependence of the mobility. It was concluded that transport in the bands is influenced by acoustical and optical phonon scattering, as well as charged impurity scattering. It was also shown that for high defect concentrations and temperatures below 200 K, transport in defect zones becomes important.
* Fax: +49 30 806 23199. E-mail address: [email protected]. 0040-6090/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2004.11.015
But no modeling was performed. On the other hand, we have presented mobility data of polycrystalline films [10]. It was shown that the mobility in polycrystalline films is thermally activated, indicating a barrier at the grain boundaries. At room temperature and above, the observed mobility is lower than what is expected from the activated behavior. This has been interpreted as due to scattering in the bulk of the grains. However, the room temperature mobility is about one order of magnitude lower in the polycrystalline films than in the epitaxial films. The problem that arises here is that if the deviation from the activated behavior of the mobility is indeed due to the mobility in the bulk, it is hard to understand why it should be much lower than in epitaxial films. In this paper, a consistent model is given that can describe the temperature dependence of the mobility in epitaxial and polycrystalline films of CuGaSe2. It is based on the modern description of scattering mechanism [9] and on the latest results available for material constants used in this description. Cu-rich epitaxial and polycrystalline films of CuGaSe2 were grown by metal organic vapor phase epitaxy or by thermal coevaporation, as described elsewhere [11,12]. The data presented here are based on Hall and conductivity measurements in the van der Pauw configuration as described before [10,13]. These measurements yield charge carrier concentration and mobility data. The temperature dependencies of both depend, among others, on the defect
S. Siebentritt / Thin Solid Films 480–481 (2005) 312–317
densities. The charge carrier concentrations have been analyzed before [13], without taking the mobility data into account. In the following, it is shown that by this approach defect concentrations are overestimated. Nevertheless, the general conclusions on self-compensation and concentration dependence of the defect energy drawn in Ref. [13] are supported by this new analysis based on simultaneous fits of the temperature dependence of the charge carrier concentration and the mobility.
2. Theory
313
acoustical phonon scattering and l npo, limited by nonpolar optical phonon scattering, are then given by: 2 pffiffiffiffiffiffi eh- 4 qv2s 2p lac ¼ ð4Þ 3=2 2 3 ðkB T Þ ðm4me Þ5=2 Eac lnpo
pffiffiffiffiffiffi 4 2peh- qðkB HÞ1=2 T ¼ f 5=2 2 H 3ðm4me Þ D
with
f ðT =HÞ ¼ ð 2zÞ5=2 ðexpð 2zÞ 1Þ Zl y3=2 expð 2zyÞdy pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi y þ 1 þ expð 2zÞReð y 1Þ
with
0
Scattering mechanism for holes in the valence band of chalcopyrites can include charged and neutral defect scattering, acoustical, polar and nonpolar optical phonon scattering, and barriers at grain boundaries. The description given here follows the models by Seeger [9]. Neutral defect scattering can be described using the Bohr radius a B of the defect: er e0 h- 2 ð1Þ aB ¼ m4me e2 with e r the low-frequency dielectric constant of the semiconductor, e 0 the vacuum permittivity, [ Planck’s constant divided by 2p, m* the effective mass of the holes, m e the free electron mass, e the unit charge. The mobility limited by neutral defect scattering l n is then given by: e ln ¼ ð2Þ 20haB Nn with N n the density of neutral defects. It has been shown previously that compensation is high in CuGaSe2 and other chalcopyrites (see, e.g., Ref. [13]), therefore scattering at charged defects is important. The mobility, when mainly limited by such charged defect scattering, is described within the Brooks–Herring formalism as: 27=2 ð4per e0 Þ2 ðkB T Þ3=2 pffiffiffiffiffiffiffiffiffiffiffiffi with p3=2 e3 m4me NI f ðbÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 6kB T m4me er e0 b¼ ehp
lBH ¼
and f ð xÞ ¼ ln 1 þ x2
x2 1 þ x2
ð3Þ
with k B Boltzmann’s constant, T temperature, N I density of charged (ionized) defects, and p density of free charge carriers. Acoustical and nonpolar optical phonon scattering is described in the deformation potential formalism, using E ac, the deformation potential constant, describing the energy shift of the free carriers with the gradient of the atomic displacement and D the optical deformation constant, describing the energy shift of the free carriers with the atomic displacement itself. The mobilities l ac, limited by
z ¼ H=2T
and y ¼ E=kB H
ð5Þ
with q the density of the material, v s the velocity of sound, H the characteristic temperature of the optical phonons, and E the energy of the charge carriers within the band. The function f(T/H) can be excellently approximated by ! 1 f ðT =HÞcexp 5:44 pffiffiffiffiffiffiffiffiffiffi 1 ð6Þ T =H Since chalcopyrites are partially ionic crystals, the polar effect of optical phonons needs to be taken into account as well. In Boltzmann approximations, one obtains: 3p1=2 jej sinhðH=2T Þ with 5=2 m4m ax 3=2 2 e 0 ðH=2T Þ K1 ðH=2T Þ h- jejE0 with a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m4me ðkB HÞ3=2 jejm4me kB H 1 el e1 E0 ¼ ð7Þ r 2 4pe0 h with x 0 the characteristic frequency of the optical phonons ([ x 0=k BH), K 1 the modified Bessel function of first order, and e l the high-frequency dielectric constant. In polycrystalline material, high densities of defects are expected at the grain boundaries which are often charged with majority carriers. In this case, the localized charged defects create a space charge region next to the grain boundaries and thus a band bending and a barrier for the majority carriers. The effects on the mobility have been analyzed within the frame of the thermionic emission theory first by Seto [14] and later summarized by Orton and Powell [15]: lopt ¼
lGB ¼ l0 expðU=kB T Þ
ð8Þ
where l 0 is not the bulk mobility but a quantity proportional to the grain size, U is the barrier height. The applicability of this model to our polycrystalline films has been shown before [10]. The total mobility l tot is obtained from Matthiesen’s rule by summing up the inverse mobilities of the individual scattering mechanisms. The parameters in the model are described in the following. The low-frequency dielectric constant e r can be taken from IR reflectivity measurements [16], the highfrequency dielectric constant e l from UV ellipsometry measurements [17], to be 11 and 8, respectively. The effective
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Table 1 Parameters used for the fits to mobilities and charge carriers er
e l m* q
11 8
1
vs 3
5.57 g cm
H
Eac
D
3020 m/s 394 K 2.65 eV 6.8 108 eV/cm
For references and a discussion, see text.
mass m* is unknown. No experimental data exist for CuGaSe2, the numbers used in the literature are estimates using data from other materials. Therefore, m*=1 is used here. The density of the material q has been measured: 5.57 g/ cm3 [18]. The velocity of sound v s is not measured for CuGaSe2. In a simple phonon model, the velocity of sound is proportional to the Debye temperature and the average atomic radius of the material. Taking into account the measured Debye temperatures of CuGaSe2 [19] and CuInSe2 [20], the known volumes of the unit cells [18], and the measured velocity of sound of CuInSe2 [21], a velocity of sound for CuGaSe2 of 3.02 103 m/s is obtained. The characteristic temperature of the optical phonons H can be determined from the measured energy of the LO phonon, i.e., 34 meV [22,23] to be 394 K. The acoustic and optical deformation potential E ac and D are unknown and have been treated as fit parameters in previous studies of the mobility [1,3]. Here, they are also treated as fit parameters but are assumed material properties that should show the same values for all samples. An overview of constants and fit parameters is given in Table 1. The density of neutral defects, N n, and the density of charged defects, N I, are obtained from the density of acceptors and donors, which are fit parameters of the individual samples, and from the position of the Fermi level which is calculated from p(T) data. l 0 and U are also fit parameters for the individual samples, as they depend on the grain size and the details of the grain boundary formation. Parameter variations are performed to obtain simultaneous fits of the temperature dependence of the charge carrier concentration and the mobility. Charge carrier concentration is described by a three defect model with the density of acceptors and donors N A1, N A2, N D, and the acceptor energies E A1, E A2, as the fit parameters which in most cases was reduced to a two defect model with only one type of acceptor as described previously [13].
It is clearly seen that two mechanisms do not play a role in these films: neutral defect scattering and polar optical phonon scattering. The mobility obtained from neutral defect scattering is so high that it does not influence the observed mobility. Mobility due to polar optical phonon scattering is lower than the observed mobility. Therefore, it must be concluded, that polar optical phonon scattering does not play a role in CuGaSe2. This has been observed before for CuInSe2 and Cu(In,Ga)Se2 films [3,24], and therefore polar optical phonon scattering has been excluded. Two explanations are possible. First: the parameters used could be wrong. For example, the effective hole mass could be much lower than 1. With an effective hole mass of 0.4, the mobility due to polar optical phonon scattering would increase to the values observed experimentally. Or the measurements of the dielectric constants are wrong. Second, the ionic character of the bonds in the chalcopyrite could be very low, inhibiting a major influence of the polar effect of the optical phonons. The same effect has been observed in CuInSe2 where the effective mass has been independently measured and the data on the dielectric constants are expected to be more reliable, since more measurements exist. Therefore, it is assumed for the moment, that polar optical phonon scattering does not play a major role, and the summation to obtain the total mobility is done without polar optical phonon scattering. Nonpolar optical phonon scattering and acoustical phonon scattering dominate the mobility around room temperature, whereas at lower temperatures, charged defect scattering becomes effective. In films with higher compensation or higher defect density, i.e., with a higher density of charged defects, charged defect scattering becomes the dominant scattering mechanism at low temperatures (Fig. 2). In samples with even higher defect densities, no description
3. Modeling of experimental mobility data In this section, first, the bulk behavior, i.e., the mobility in epitaxial films, will be discussed, and second, the effect of grain boundaries will be added.
4. Bulk mobilities Fig. 1 shows a typical experimental l(T) curve together with the contribution of the various scattering mechanisms.
Fig. 1. Experimental temperature dependence of the mobility in an epitaxial film together with the theoretical curves for the different scattering mechanisms. Shown are experimental data (l exp), modeled mobility due to scattering at neutral defects (l n), at charged defects (Brooks–Herring formalism l BH), at acoustical phonons (l ac), at nonpolar optical phonons (l npo), at polar optical phonons (l opt), and the total mobility resulting from these scattering mechanisms, excluding polar optical phonon scattering.
S. Siebentritt / Thin Solid Films 480–481 (2005) 312–317
315
Fig. 2. Mobility in an epitaxial film with higher compensation than in Fig. 1.
of the temperature dependence of the mobility at low temperatures by this standard band transport description is possible anymore. Mobilities decrease much faster with decreasing temperature. This is due to hopping of charge carrier in defects as was discussed previously [7,13] and needs another model approach to the transport which is beyond the scope of this contribution. With this model, the mobility of a large number of Curich epitaxial films is described, see Figs. 1 and 2 for typical examples.
5. Defect densities The fits shown in Figs. 1 and 2 are only possible with lower densities of acceptors and donors than previously reported [13]. Assuming the defect concentrations which give the best fit for the temperature dependence of the charge carrier concentration [13], charged defect scattering alone gives much lower mobilities than those observed experimentally, up to a factor of 5 lower than observed. Therefore, both temperature dependencies—of the mobility and of the charge carrier concentration—were fitted simultaneously. This results in individual fits with somewhat lower quality, but only by this approach, a consistent description of the transport in the band is possible. It should be pointed out, that although the defect densities obtained from this combined approach are lower, the general conclusions drawn in Ref. [13] remain unchanged. A main conclusion was the observation of self-compensation of native defects by native defects which manifests itself by an increase of the degree of compensation with increasing acceptor density. With this new analysis, defect densities are lower but the same general trend is observed as seen in Fig. 3. Besides a rather large scattering of the data for low acceptor densities, the self-compensation trend is still clearly obtained. The same is true for the linear dependence of the obtained acceptor ionization energy on the third root
Fig. 3. Degree of compensation as a function of the density of acceptors for a number of polycrystalline and epitaxial samples. The line is a mere guide to the eye.
of the donor density: this trend is also maintained with the combined approach.
6. Influence of grain boundaries As has been shown before, the mobility in polycrystalline CuGaSe2 films follows an activated behavior in an intermediate temperature range between room temperature and approximately 200 K (Fig. 4) [10], as expected according to Eq. (8). At lower temperatures, the deviation from the activated behavior can be attributed to the onset of hopping transport or tunneling. The deviation at higher temperature was attributed to limitations by the bulk of the grains. A question arising here is due to the fact that this mobility is much smaller than the mobility observed at the same temperature in epitaxial films. Therefore, a new fit of the mobility data of the polycrystalline samples has been
Fig. 4. Temperature dependence of the mobility in a polycrystalline sample with activated behavior at intermediate temperatures.
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S. Siebentritt / Thin Solid Films 480–481 (2005) 312–317
made taking all scattering mechanisms into account, not just the barrier due to the space charge at the grain boundaries. The result is shown in Fig. 5 together with the individual contributions of the scattering mechanisms. Since the results of the epitaxial films showed that polar optical phonon scattering does not play a role for the mobility of holes in CuGaSe2, the summation for the polycrystalline films was also done without taking the polar optical phonons into account, since it is unreasonable to assume this mechanism is important in the bulk of the grains but not in the bulk of the epitaxial films. In the temperature region measured, i.e., up to 350 K, neither nonpolar optical nor acoustical phonon scattering is limiting; the mobilities due to these scattering mechanisms are above 100 cm2/Vs and therefore not seen in Fig. 5. As dominant scattering mechanisms appear neutral and charged defect scattering and grain boundary barriers. For neutral defect scattering to play such a dominant role, a high density of neutral defects is needed. This density is higher than the density of neutral defects deduced from the acceptor and donor densities, obtained from the fits to the temperature-dependent charge carrier concentration and the Fermi statistics. Therefore, additional neutral defects with a density around 1020 cm3 have been introduced into the model which are added to the neutral donors and acceptors. Taking these additional neutral defects into account, the low room temperature mobility of the polycrystalline samples can be described. The source of these defects must be the grain boundaries. Assuming grain sizes of 1 Am, as observed in SEM micrographs of our films and a monoatomic layer representing the grain boundary, this would correspond to an area density of neutral defects at the grain boundaries of 1016 cm2, which is unreasonably high. Such a high defect concentration could only be justified by taking into account a certain thickness of the grain boundary, i.e., the defect layer at the grain boundary is about 10 or more monolayers thick, which appears not very likely. Grain boundaries observed in transmission electron microscopy are usually very sharp and do not
extend over 10 monolayers. Another possibility is that the neutral defects are distributed throughout the bulk of the grains. But then, it is hard to explain why the bulk of the polycrystalline material should contain so much more neutral defects than the bulk of epitaxial films. TEM investigations of our polycrystalline films have shown stacking faults within individual grains [25]. Such extended defects have very large scattering cross-sections. On the other hand, stacking faults are also observed in epitaxial films. Thus, they cannot account for the difference in mobility between epitaxial and polycrystalline films. A general consideration of the low mobilities observed in polycrystalline chalcopyrite films reaches a similar conclusion: if mobilities of the order of 10 cm2/Vs are assumed to be due to the standard scattering mechanisms, the scattering times s are extremely low: 1014 s, as deduced from the Drude formula for the mobility, l=es /(m*m e ) with the effective mass equals 1. With a thermal velocity of the charge carriers, this results in scattering lengths of 109 m. This in return results in a scatterer density of 1021 cm3 in accordance with the fit data. Since such high defect densities should be detectable by structural methods, like X-ray diffraction, high resolution electron microscopy, etc., but have never been detected, it must be concluded that the standard scattering mechanisms cannot account for the low mobilities observed in these polycrystalline films. Other mechanisms, like trapping at grain boundaries, must be tested.
7. Conclusion The temperature dependence of the mobility of epitaxial Cu-rich CuGaSe2 films has been explained by a model including acoustical and nonpolar optical phonon scattering and charged defect scattering. The influence of the latter increases with increasing degree of compensation, i.e., increasing density of charged defects. The much lower mobility of polycrystalline samples can be described by the same model including a barrier at the grain boundaries only by assuming an extremely high concentration of neutral defects at or close to the grain boundaries. Therefore, it is concluded that the low mobility in polycrystalline films is not due to standard scattering mechanisms. New explanations must be sought.
Acknowledgement
Fig. 5. Complete fit of the temperature dependence of the mobility of a polycrystalline sample.
This work has been supported by the German Research Ministry BMBF. I would like to thank Angus Rockett, University of Illinois, and Stephan Brehme, HMI, for many helpful discussions.
S. Siebentritt / Thin Solid Films 480–481 (2005) 312–317
References [1] T. Irie, S. Endo, S. Kimura, Jpn. J. Appl. Phys. 18 (1979) 1303. [2] H. Neumann, N. Van Nam, H.-J. Hfbler, G. Kqhn, Solid State Commun. 25 (1978) 899. [3] S.M. Wasim, Sol. Cells 16 (1986) 289. [4] S.M. Wasim, G.S. Porras, Phys. Status Solidi, A Appl. Res. 79 (1983) K65. [5] J.H. Schfn, F.P. Baumgartner, E. Arushanov, H. Riazi-Nejad, C. Kloc, E. Bucher, J. Appl. Phys. 79 (1996) 6961. [6] M. Rusu, P. Gashin, A. Simashkevich, Sol. Energy Mater. Sol. Cells 70 (2001) 175. [7] S. Siebentritt, A. Gerhard, S. Brehme, M. Lux-Steiner, in: R. Birkmire, R. Noufi, D. Lincot, H.-W. Schock (Eds.), II–VI Compound Semiconductor Photovoltaic Materials, San Francisco, April 16–20, 2001, Materials Research Society Symposium Proceedings, vol. 668, 2001, p. H.4.4.1. [8] J.D. Wiley, M. DiDomenico Jr., Phys. Rev., B 2 (1970) 427. [9] K. Seeger, Semiconductor Physics, Springer, Heidelberg, 2002. [10] S. Schuler, S. Nishiwaki, J. Beckmann, N. Rega, S. Brehme, S. Siebentritt, M.C. Lux-Steiner, in: J. Benner (Ed.), Proceedings of the 29th IEEE Photovoltaic Specialist Conference, New Orleans, May 17–22, 2002, p. 504. [11] M.C. Artaud-Gillet, S. Duchemin, R. Odedra, G. Orsal, N. Rega, S. Rushworth, S. Siebentritt, J. Cryst. Growth 248 (2003) 163. [12] S. Schuler, S. Nishwaki, M. Dziedzina, R. Klenk, S. Siebentritt, M.C. Lux-Steiner, in: R. Birkmire, R. Noufi, D. Lincot, H.-W. Schock
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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(Eds.), II–VI Compound Semiconductor Photovoltaic Materials, San Francisco, April 16–20, 2001, Materials Research Society Symposium Proceedings, vol. 668, 2001, p. H5.14.1. S. Schuler, S. Siebentritt, S. Nishiwaki, N. Rega, J. Beckmann, S. Brehme, M.C. Lux-Steiner, Phys. Rev., B 69 (2004) 45210. J.Y.W. Seto, J. Appl. Phys. 46 (1975) 5247. J.W. Orton, M.J. Powell, Rep. Prog. Phys. 43 (1980) 1263. N.N. Syrbu, M. Bogdanash, V.E. Tezlevan, I. Mushcutariu, Physica. B 229 (1997) 199. M.I. Alonso, K. Wakita, J. Pascual, M. Carriga, N. Yamamoto, Phys. Rev., B 63 (2001) 75203. Landolt-Bfrnstein, Numerical Data and Functional Relationships in Science and Technology, Springer, Berlin, 1982. K. Bohmhammel, P. Deus, G. Kqhn, W. Mfller, Phys. Status Solidi, A Appl. Res. 71 (1982) 505. K.J. Bachmann, F.S.L. Hsu, F.A. Thiel, H.M. Kasper, J. Electron. Mater. 6 (1977) 431. B. Fernandez, S.M. Wasim, Phys. Status Solidi, A Appl. Res. 122 (1990) 235. I.V. Bodnar, A.G. Karoza, G.F. Smirnova, Phys. Status Solidi, B Basic Res. 84 (1977) K65. F.J. Ramirez, C. Rinco´n, Solid State Commun. 84 (1992) 551. D.J. Schroeder, J.L. Hernandez, G.D. Berry, A.A. Rockett, J. Appl. Phys. 83 (1998) 1519. S. Nishiwaki, S. Siebentritt, M. Giersig, M.C. Lux-Steiner, J. Appl. Phys. 94 (2003) 6864.
Hole transport mechanisms in CuGaSe2 Susanne Siebentritt* Hahn-Meitner-Institut, Glienicker Str. 100, 14109 Berlin, Germany Available online 19 December 2004
Abstract Hall and conductivity measurements have been performed on epitaxial and polycrystalline CuGaSe2 films. Temperature dependencies of charge carrier concentration and mobility are analyzed in terms of band transport model and standard scattering mechanisms, and are fitted simultaneously. Defect concentrations are lower than previously reported. Dominant scattering mechanisms in the single crystalline films are nonpolar optical phonon and acoustical phonon scattering, and charged defect scattering. Room temperature mobility in polycrystalline films is considerably lower than in epitaxial films. This can not only be attributed to scattering at extended defects. D 2004 Elsevier B.V. All rights reserved. Keywords: CuGaSe2 film; Nonpolar optical phonon; Epitaxial film
1. Introduction When modeling chalcopyrite solar cells, information on the mobility of charge carriers and its temperature dependence is needed. Likewise enters the mobility into the analysis of many electrical measurements, like admittance spectroscopy. A few studies exist on single crystals, epitaxial and polycrystalline films of CuInSe2 [1–3] and of CuGaSe2 [4–7]. Most analyses rely on the basic paper by Wiley and DiDomenico [8] on the modeling of the basic scattering mechanisms for holes in III–V semiconductors. Some of these equations are based on empirical considerations. Here, we will use equations strictly derived from quantum mechanical considerations and the Boltzmann transport equation [9]. We have previously reported on mobilities in our epitaxial CuGaSe2 films [7], where the scattering mechanisms were derived from the general trends in the temperature dependence of the mobility. It was concluded that transport in the bands is influenced by acoustical and optical phonon scattering, as well as charged impurity scattering. It was also shown that for high defect concentrations and temperatures below 200 K, transport in defect zones becomes important.
* Fax: +49 30 806 23199. E-mail address: [email protected]. 0040-6090/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2004.11.015
But no modeling was performed. On the other hand, we have presented mobility data of polycrystalline films [10]. It was shown that the mobility in polycrystalline films is thermally activated, indicating a barrier at the grain boundaries. At room temperature and above, the observed mobility is lower than what is expected from the activated behavior. This has been interpreted as due to scattering in the bulk of the grains. However, the room temperature mobility is about one order of magnitude lower in the polycrystalline films than in the epitaxial films. The problem that arises here is that if the deviation from the activated behavior of the mobility is indeed due to the mobility in the bulk, it is hard to understand why it should be much lower than in epitaxial films. In this paper, a consistent model is given that can describe the temperature dependence of the mobility in epitaxial and polycrystalline films of CuGaSe2. It is based on the modern description of scattering mechanism [9] and on the latest results available for material constants used in this description. Cu-rich epitaxial and polycrystalline films of CuGaSe2 were grown by metal organic vapor phase epitaxy or by thermal coevaporation, as described elsewhere [11,12]. The data presented here are based on Hall and conductivity measurements in the van der Pauw configuration as described before [10,13]. These measurements yield charge carrier concentration and mobility data. The temperature dependencies of both depend, among others, on the defect
S. Siebentritt / Thin Solid Films 480–481 (2005) 312–317
densities. The charge carrier concentrations have been analyzed before [13], without taking the mobility data into account. In the following, it is shown that by this approach defect concentrations are overestimated. Nevertheless, the general conclusions on self-compensation and concentration dependence of the defect energy drawn in Ref. [13] are supported by this new analysis based on simultaneous fits of the temperature dependence of the charge carrier concentration and the mobility.
2. Theory
313
acoustical phonon scattering and l npo, limited by nonpolar optical phonon scattering, are then given by: 2 pffiffiffiffiffiffi eh- 4 qv2s 2p lac ¼ ð4Þ 3=2 2 3 ðkB T Þ ðm4me Þ5=2 Eac lnpo
pffiffiffiffiffiffi 4 2peh- qðkB HÞ1=2 T ¼ f 5=2 2 H 3ðm4me Þ D
with
f ðT =HÞ ¼ ð 2zÞ5=2 ðexpð 2zÞ 1Þ Zl y3=2 expð 2zyÞdy pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi y þ 1 þ expð 2zÞReð y 1Þ
with
0
Scattering mechanism for holes in the valence band of chalcopyrites can include charged and neutral defect scattering, acoustical, polar and nonpolar optical phonon scattering, and barriers at grain boundaries. The description given here follows the models by Seeger [9]. Neutral defect scattering can be described using the Bohr radius a B of the defect: er e0 h- 2 ð1Þ aB ¼ m4me e2 with e r the low-frequency dielectric constant of the semiconductor, e 0 the vacuum permittivity, [ Planck’s constant divided by 2p, m* the effective mass of the holes, m e the free electron mass, e the unit charge. The mobility limited by neutral defect scattering l n is then given by: e ln ¼ ð2Þ 20haB Nn with N n the density of neutral defects. It has been shown previously that compensation is high in CuGaSe2 and other chalcopyrites (see, e.g., Ref. [13]), therefore scattering at charged defects is important. The mobility, when mainly limited by such charged defect scattering, is described within the Brooks–Herring formalism as: 27=2 ð4per e0 Þ2 ðkB T Þ3=2 pffiffiffiffiffiffiffiffiffiffiffiffi with p3=2 e3 m4me NI f ðbÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 6kB T m4me er e0 b¼ ehp
lBH ¼
and f ð xÞ ¼ ln 1 þ x2
x2 1 þ x2
ð3Þ
with k B Boltzmann’s constant, T temperature, N I density of charged (ionized) defects, and p density of free charge carriers. Acoustical and nonpolar optical phonon scattering is described in the deformation potential formalism, using E ac, the deformation potential constant, describing the energy shift of the free carriers with the gradient of the atomic displacement and D the optical deformation constant, describing the energy shift of the free carriers with the atomic displacement itself. The mobilities l ac, limited by
z ¼ H=2T
and y ¼ E=kB H
ð5Þ
with q the density of the material, v s the velocity of sound, H the characteristic temperature of the optical phonons, and E the energy of the charge carriers within the band. The function f(T/H) can be excellently approximated by ! 1 f ðT =HÞcexp 5:44 pffiffiffiffiffiffiffiffiffiffi 1 ð6Þ T =H Since chalcopyrites are partially ionic crystals, the polar effect of optical phonons needs to be taken into account as well. In Boltzmann approximations, one obtains: 3p1=2 jej sinhðH=2T Þ with 5=2 m4m ax 3=2 2 e 0 ðH=2T Þ K1 ðH=2T Þ h- jejE0 with a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m4me ðkB HÞ3=2 jejm4me kB H 1 el e1 E0 ¼ ð7Þ r 2 4pe0 h with x 0 the characteristic frequency of the optical phonons ([ x 0=k BH), K 1 the modified Bessel function of first order, and e l the high-frequency dielectric constant. In polycrystalline material, high densities of defects are expected at the grain boundaries which are often charged with majority carriers. In this case, the localized charged defects create a space charge region next to the grain boundaries and thus a band bending and a barrier for the majority carriers. The effects on the mobility have been analyzed within the frame of the thermionic emission theory first by Seto [14] and later summarized by Orton and Powell [15]: lopt ¼
lGB ¼ l0 expðU=kB T Þ
ð8Þ
where l 0 is not the bulk mobility but a quantity proportional to the grain size, U is the barrier height. The applicability of this model to our polycrystalline films has been shown before [10]. The total mobility l tot is obtained from Matthiesen’s rule by summing up the inverse mobilities of the individual scattering mechanisms. The parameters in the model are described in the following. The low-frequency dielectric constant e r can be taken from IR reflectivity measurements [16], the highfrequency dielectric constant e l from UV ellipsometry measurements [17], to be 11 and 8, respectively. The effective
314
S. Siebentritt / Thin Solid Films 480–481 (2005) 312–317
Table 1 Parameters used for the fits to mobilities and charge carriers er
e l m* q
11 8
1
vs 3
5.57 g cm
H
Eac
D
3020 m/s 394 K 2.65 eV 6.8 108 eV/cm
For references and a discussion, see text.
mass m* is unknown. No experimental data exist for CuGaSe2, the numbers used in the literature are estimates using data from other materials. Therefore, m*=1 is used here. The density of the material q has been measured: 5.57 g/ cm3 [18]. The velocity of sound v s is not measured for CuGaSe2. In a simple phonon model, the velocity of sound is proportional to the Debye temperature and the average atomic radius of the material. Taking into account the measured Debye temperatures of CuGaSe2 [19] and CuInSe2 [20], the known volumes of the unit cells [18], and the measured velocity of sound of CuInSe2 [21], a velocity of sound for CuGaSe2 of 3.02 103 m/s is obtained. The characteristic temperature of the optical phonons H can be determined from the measured energy of the LO phonon, i.e., 34 meV [22,23] to be 394 K. The acoustic and optical deformation potential E ac and D are unknown and have been treated as fit parameters in previous studies of the mobility [1,3]. Here, they are also treated as fit parameters but are assumed material properties that should show the same values for all samples. An overview of constants and fit parameters is given in Table 1. The density of neutral defects, N n, and the density of charged defects, N I, are obtained from the density of acceptors and donors, which are fit parameters of the individual samples, and from the position of the Fermi level which is calculated from p(T) data. l 0 and U are also fit parameters for the individual samples, as they depend on the grain size and the details of the grain boundary formation. Parameter variations are performed to obtain simultaneous fits of the temperature dependence of the charge carrier concentration and the mobility. Charge carrier concentration is described by a three defect model with the density of acceptors and donors N A1, N A2, N D, and the acceptor energies E A1, E A2, as the fit parameters which in most cases was reduced to a two defect model with only one type of acceptor as described previously [13].
It is clearly seen that two mechanisms do not play a role in these films: neutral defect scattering and polar optical phonon scattering. The mobility obtained from neutral defect scattering is so high that it does not influence the observed mobility. Mobility due to polar optical phonon scattering is lower than the observed mobility. Therefore, it must be concluded, that polar optical phonon scattering does not play a role in CuGaSe2. This has been observed before for CuInSe2 and Cu(In,Ga)Se2 films [3,24], and therefore polar optical phonon scattering has been excluded. Two explanations are possible. First: the parameters used could be wrong. For example, the effective hole mass could be much lower than 1. With an effective hole mass of 0.4, the mobility due to polar optical phonon scattering would increase to the values observed experimentally. Or the measurements of the dielectric constants are wrong. Second, the ionic character of the bonds in the chalcopyrite could be very low, inhibiting a major influence of the polar effect of the optical phonons. The same effect has been observed in CuInSe2 where the effective mass has been independently measured and the data on the dielectric constants are expected to be more reliable, since more measurements exist. Therefore, it is assumed for the moment, that polar optical phonon scattering does not play a major role, and the summation to obtain the total mobility is done without polar optical phonon scattering. Nonpolar optical phonon scattering and acoustical phonon scattering dominate the mobility around room temperature, whereas at lower temperatures, charged defect scattering becomes effective. In films with higher compensation or higher defect density, i.e., with a higher density of charged defects, charged defect scattering becomes the dominant scattering mechanism at low temperatures (Fig. 2). In samples with even higher defect densities, no description
3. Modeling of experimental mobility data In this section, first, the bulk behavior, i.e., the mobility in epitaxial films, will be discussed, and second, the effect of grain boundaries will be added.
4. Bulk mobilities Fig. 1 shows a typical experimental l(T) curve together with the contribution of the various scattering mechanisms.
Fig. 1. Experimental temperature dependence of the mobility in an epitaxial film together with the theoretical curves for the different scattering mechanisms. Shown are experimental data (l exp), modeled mobility due to scattering at neutral defects (l n), at charged defects (Brooks–Herring formalism l BH), at acoustical phonons (l ac), at nonpolar optical phonons (l npo), at polar optical phonons (l opt), and the total mobility resulting from these scattering mechanisms, excluding polar optical phonon scattering.
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Fig. 2. Mobility in an epitaxial film with higher compensation than in Fig. 1.
of the temperature dependence of the mobility at low temperatures by this standard band transport description is possible anymore. Mobilities decrease much faster with decreasing temperature. This is due to hopping of charge carrier in defects as was discussed previously [7,13] and needs another model approach to the transport which is beyond the scope of this contribution. With this model, the mobility of a large number of Curich epitaxial films is described, see Figs. 1 and 2 for typical examples.
5. Defect densities The fits shown in Figs. 1 and 2 are only possible with lower densities of acceptors and donors than previously reported [13]. Assuming the defect concentrations which give the best fit for the temperature dependence of the charge carrier concentration [13], charged defect scattering alone gives much lower mobilities than those observed experimentally, up to a factor of 5 lower than observed. Therefore, both temperature dependencies—of the mobility and of the charge carrier concentration—were fitted simultaneously. This results in individual fits with somewhat lower quality, but only by this approach, a consistent description of the transport in the band is possible. It should be pointed out, that although the defect densities obtained from this combined approach are lower, the general conclusions drawn in Ref. [13] remain unchanged. A main conclusion was the observation of self-compensation of native defects by native defects which manifests itself by an increase of the degree of compensation with increasing acceptor density. With this new analysis, defect densities are lower but the same general trend is observed as seen in Fig. 3. Besides a rather large scattering of the data for low acceptor densities, the self-compensation trend is still clearly obtained. The same is true for the linear dependence of the obtained acceptor ionization energy on the third root
Fig. 3. Degree of compensation as a function of the density of acceptors for a number of polycrystalline and epitaxial samples. The line is a mere guide to the eye.
of the donor density: this trend is also maintained with the combined approach.
6. Influence of grain boundaries As has been shown before, the mobility in polycrystalline CuGaSe2 films follows an activated behavior in an intermediate temperature range between room temperature and approximately 200 K (Fig. 4) [10], as expected according to Eq. (8). At lower temperatures, the deviation from the activated behavior can be attributed to the onset of hopping transport or tunneling. The deviation at higher temperature was attributed to limitations by the bulk of the grains. A question arising here is due to the fact that this mobility is much smaller than the mobility observed at the same temperature in epitaxial films. Therefore, a new fit of the mobility data of the polycrystalline samples has been
Fig. 4. Temperature dependence of the mobility in a polycrystalline sample with activated behavior at intermediate temperatures.
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made taking all scattering mechanisms into account, not just the barrier due to the space charge at the grain boundaries. The result is shown in Fig. 5 together with the individual contributions of the scattering mechanisms. Since the results of the epitaxial films showed that polar optical phonon scattering does not play a role for the mobility of holes in CuGaSe2, the summation for the polycrystalline films was also done without taking the polar optical phonons into account, since it is unreasonable to assume this mechanism is important in the bulk of the grains but not in the bulk of the epitaxial films. In the temperature region measured, i.e., up to 350 K, neither nonpolar optical nor acoustical phonon scattering is limiting; the mobilities due to these scattering mechanisms are above 100 cm2/Vs and therefore not seen in Fig. 5. As dominant scattering mechanisms appear neutral and charged defect scattering and grain boundary barriers. For neutral defect scattering to play such a dominant role, a high density of neutral defects is needed. This density is higher than the density of neutral defects deduced from the acceptor and donor densities, obtained from the fits to the temperature-dependent charge carrier concentration and the Fermi statistics. Therefore, additional neutral defects with a density around 1020 cm3 have been introduced into the model which are added to the neutral donors and acceptors. Taking these additional neutral defects into account, the low room temperature mobility of the polycrystalline samples can be described. The source of these defects must be the grain boundaries. Assuming grain sizes of 1 Am, as observed in SEM micrographs of our films and a monoatomic layer representing the grain boundary, this would correspond to an area density of neutral defects at the grain boundaries of 1016 cm2, which is unreasonably high. Such a high defect concentration could only be justified by taking into account a certain thickness of the grain boundary, i.e., the defect layer at the grain boundary is about 10 or more monolayers thick, which appears not very likely. Grain boundaries observed in transmission electron microscopy are usually very sharp and do not
extend over 10 monolayers. Another possibility is that the neutral defects are distributed throughout the bulk of the grains. But then, it is hard to explain why the bulk of the polycrystalline material should contain so much more neutral defects than the bulk of epitaxial films. TEM investigations of our polycrystalline films have shown stacking faults within individual grains [25]. Such extended defects have very large scattering cross-sections. On the other hand, stacking faults are also observed in epitaxial films. Thus, they cannot account for the difference in mobility between epitaxial and polycrystalline films. A general consideration of the low mobilities observed in polycrystalline chalcopyrite films reaches a similar conclusion: if mobilities of the order of 10 cm2/Vs are assumed to be due to the standard scattering mechanisms, the scattering times s are extremely low: 1014 s, as deduced from the Drude formula for the mobility, l=es /(m*m e ) with the effective mass equals 1. With a thermal velocity of the charge carriers, this results in scattering lengths of 109 m. This in return results in a scatterer density of 1021 cm3 in accordance with the fit data. Since such high defect densities should be detectable by structural methods, like X-ray diffraction, high resolution electron microscopy, etc., but have never been detected, it must be concluded that the standard scattering mechanisms cannot account for the low mobilities observed in these polycrystalline films. Other mechanisms, like trapping at grain boundaries, must be tested.
7. Conclusion The temperature dependence of the mobility of epitaxial Cu-rich CuGaSe2 films has been explained by a model including acoustical and nonpolar optical phonon scattering and charged defect scattering. The influence of the latter increases with increasing degree of compensation, i.e., increasing density of charged defects. The much lower mobility of polycrystalline samples can be described by the same model including a barrier at the grain boundaries only by assuming an extremely high concentration of neutral defects at or close to the grain boundaries. Therefore, it is concluded that the low mobility in polycrystalline films is not due to standard scattering mechanisms. New explanations must be sought.
Acknowledgement
Fig. 5. Complete fit of the temperature dependence of the mobility of a polycrystalline sample.
This work has been supported by the German Research Ministry BMBF. I would like to thank Angus Rockett, University of Illinois, and Stephan Brehme, HMI, for many helpful discussions.
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