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Keywords:

  • band structure;
  • effective masses;
  • ellipsometry;
  • phonon–plasmon coupling;
  • SnO2

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

Generalized infrared spectroscopic ellipsometry (GIRSE) was applied to Sb-doped rutile SnO2(101) films grown by plasma-assisted molecular beam epitaxy (PAMBE). Coupled longitudinal-optical (LO) phonon–plasmon (LPP) modes for the two principal polarization directions of the optically anisotropic SnO2 were identified and their frequencies were determined as a function of electron concentrations obtained by Hall effect measurements. The analysis of these modes yielded very accurate values for the plasma frequencies and finally the anisotropy of the electron effective masses as a function of carrier density. Comparison to Hall effect electron concentrations yielded a non-parabolicity of the conduction band.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

Tin oxide (SnO2) in rutile structure is a wide band gap oxide semiconductor known for decades [1-3]. It is widely used in chemical gas sensors and – highly n-doped – as transparent contact layer for efficient displays, solar cells, and smart windows [4]. More recently, SnO2 has gained renewed attention [5] due to possible applications as active semiconductor material in electronic or optoelectronic devices. Single crystalline SnO2 thin films can be grown by molecular beam epitaxy [6], allowing detailed control of the conductivity by doping with foreign atoms [7]. While p-type doping has proven to be difficult or impossible, [7-10] precisely controlled n-type doping could be achieved by introduction of antimony on tin site (Sbsn) [11]. Previous studies described the growth procedure in detail [6] and investigated the structural and electrical properties of Sb-doped, single-crystalline SnO2 films [11]. However, the optical properties of SnO2:Sb were not characterized so far.

In this article, we focus on the infrared linear optical response of rutile SnO2:Sb as measured by generalized infrared spectroscopic ellipsometry (GIRSE) [12, 13]. The anisotropy of the rutile crystal is reflected in the optical properties of the material and is fully taken into account. Besides phonon contributions we focus on coupled longitudinal-optical (LO) phonon–plasmon (LPP) modes [14] allowing for analysis of the anisotropic effective electron mass.

2 Samples

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

Thin SnO2 films with (101) orientation were grown by plasma-assisted molecular beam epitaxy (PAMBE) on r-plane Al2O3. The film thickness was measured by scanning electron microscopy of the cleaved samples cross section. Besides an unintentionally doped (UID) reference sample of about 500 nm thickness, we investigated a series of six electron doped samples where antimony was used as a dopant. These films were grown on top of UID layers of approx. 500 nm thickness and are also around 500 nm thick. Antimony concentrations were confirmed by secondary ion mass spectroscopy measurements and carrier densities were independently determined by Hall effect experiments at room temperature. More details about growth and electrical characterization can be found elsewhere [6, 11]. Fundamental properties of the samples are summarized in Table 1.

Table 1. Summary of the samples investigated with their antimony concentration determined by secondary ion mass spectrometry [Sb], Hall carrier density nHall, Hall mobility μHall, and the overall film thickness d as determined from SEM cross-section micrographs
sample[Sb] (cm−3)nHall (cm−3)μHall (cm2 V−1 s−1)d (nm)
AUID5.6 × 101795537
B4.5 × 10183.4 × 1018851080
C2.1 × 10191.6 × 1019621005
D3.5 × 10192.6 × 1019611010
E1.4 × 10201.3 × 1020471128
F2.8 × 10202.6 × 1020351154

3 Ellipsometry

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

GIRSE experiments were carried out in the spectral range from 300 cm−1 to 6000 cm−1 using a variable angle commercial IR ellipsometer (J.A. Woollam Co., Inc.). All spectra were recorded at room temperature with a spectral resolution of 4 cm−1. The optical axis [001] in our (101) oriented films is tilted by θ = 34° from the surface normal, allowing the determination of both the ordinary and extraordinary dielectric function tensor components. Spectra were taken in two different sample geometries, one with the optical axis of the rutile crystal oriented in the plane of incidence (Φ = 0°, inline image are parallel to the plane of incidence [6]) and one with the sample rotated by 90° (Φ = 90°, inline image are parallel to the plane of incidence [6]). The sample orientation was known from earlier X-ray diffraction experiments. Alignment in Φ was done manually along the orientation of sample edges. The angle of incidence was varied between 48° and 72° in steps of 6°.

For experimental access to the full optical response of arbitrary sample surface orientations GIRSE is needed. In GIRSE, three ratios of complex reflection coefficients are measured, namely

  • display math(1)

The indices s and p indicate polarization of the electric field vector of the incident light beam perpendicular and parallel to the plane of incidence, respectively.

As our samples are grown on r-plane sapphire, we have measured and analyzed bare substrates using the same experimental conditions to use the results later in a multi layer approach. Note, that only marginal differences to results published earlier for sapphire [15] are found which are attributed to crystal quality differences for different suppliers.

4 Model

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

A model dielectric function accounting for phonon modes is expressed by a factorized ansatz [16]

  • display math(2)

which is a product over all infrared active phonon modes of the material. Here, ϵ is the high frequency limit of the dielectric constant, ωLO (ωTO) the characteristic frequency of the LO (TO) phonon mode, and γLO (γTO) the corresponding broadening parameter of LO (TO) phonon mode. For E || c (||) and E ⊥ c (⊥) l equals 1 and 3, respectively, in the case of rutile crystals. This leads to three (one) LO and TO modes for ⊥ (||), respectively, which are labeled in order of increasing energy as Ek (Ak). Equation (2) accounts for anharmonic effects in polar crystals by allowing different and independent damping parameters for LO and transversal optical (TO) modes [16]. From the Drude model a free carrier contribution to the dielectric function can be taken additionally into account by the form

  • display math(3)

where γp is the broadening parameter of the plasmon. Here, we use the screened plasma frequency

  • display math(4)

As a consequence of this plasmon contribution, the LO phonon modes which are defined as zeros in Eq. (2) shift to new frequencies when adding Eq. (3). These modes are now called LPP modes [17]. Their occurrence can be easily understood when inline image is rewritten in the following form

  • display math(5)

From Eq. (5) it becomes obvious that there are l + 1 LPP modes for l infrared active modes. Thus, for l = 1 we obtain two LPP modes, also known as upper and lower LPP branch. For l = 3 there are four LPP modes with their frequencies below the lowest energy TO mode, between the LO and TO modes, and above the highest energy LO mode. The expected situation for rutile SnO2 is shown in Fig. 1 for all broadening parameters set to zero. It can be seen, that the lowest and highest energy LPP modes follow asymptotically the plasmon frequency, for low and high carrier densities, respectively. The damping parameter of the plasma frequency yields additional information, namely an estimate for the carrier mobility. Within the Drude model

  • display math(6)

where e is the elementary charge. For an entire analysis of the infrared spectra, input parameters are needed for phonon frequencies, especially those which are not within our spectral range. We adopted values from Ref. [18]. For the spectral range observed within our study, there was no detectable difference between using uncoupled phonon mode frequencies for ω < 300 cm−1 or correcting these mode frequencies to assumed LPP mode energies. For comparison, values for the anisotropic effective electron masses were taken from cyclotron resonance results of high quality bulk samples having a room temperature carrier density of 7 × 1015 cm−3 as inline image and inline image [19].

image

Figure 1. Phonon modes and experimentally determined LPP modes as a function of the plasma frequency in rutile SnO2. The black diagonal lines represent the plasmons. Experimentally found LPP mode energies are marked by blue filled circles. For E ⊥ c the two highest LPP modes are observed leading to slightly different screened plasma energy values ωp,⊥.

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High frequency limits of the dielectric constants were taken from our own measurements of the reference sample A and are inline image and inline image. Furthermore, layer thicknesses were fitted within the model as well as small variations around the expected Euler angle Φ. For layer thicknesses, SEM results (see Table 1) have been proven to be very accurate. Euler angles varied around ±3° reflecting good repeatability when mounting samples manually according to known crystallographic directions relative to sample edges.

5 Results and discussion

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

Figure 2 shows the fit results of the ellipsometric parameter ψpp, ψps, and ψsp of the UID reference sample A measured at three different angles of incidence and different orientations. The multilayer sample structure consisting of Al2O3 in inline image orientation and the (101) oriented SnO2 yields a multitude of detectable characteristic phonon frequencies visible mostly as sharp dips and peaks in the spectra. The phonon contributions are modeled according to Eq. (2).

image

Figure 2. Ellipsometric parameters ψpp (upper panels) and ψps, ψps (lower panels) measured (symbols, black) at three different angles of incidence (60°, 66°, 72°) in two different sample orientations (φ = 90° and 0°) for the undoped sample A. Data for higher angles are shifted vertically by 50° and 100°, respectively. Modeled data are represented by continuous lines (red).

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Our model fully represents the complicated optical response of the anisotropic sample on the anisotropic substrate both having tilted optical axes. The excellent agreement between model fit and experimental data yields the applicability of our model allowing to extract the phonon frequencies of infrared active phonon modes from the model (Table 2) which are compared to calculated values taken from Ref. [18] which served us as starting point and to complement infrared modes not detectable within our spectral range.

Table 2. Phonon frequencies and broadening parameters as determined from sample A at room temperature in both sample orientations. All values are in cm−1
  this workRef. [18]
  ωTOγTOωLOγLOωTOωLO
Eu1    223269
2290838742285335
3613977641613745
A2u4992771251456670

GIRSE results for ψpp of the antimony doped layers are shown in Fig. 3 exemplarily, for the angle of incidence of 60°. The results from undoped sample A are shown again for comparison. While for the weakly doped films the phonon features are visible similar to those in sample A, increasing dopant concentration and thus free carrier density leads to characteristic changes in the spectra. With increasing Sb concentration, the steep decrease of ψpp which is located at around 1000 cm−1 for sample A shifts to higher wave numbers and broadens significantly. This feature in ψpp resembles the energy position of the highest LPP mode whose exact value is taken from a best fit model. Please note the Fabry-Pérot fringes at the high energy side of the spectra due to the full SnO2 thickness including both the SnO2:Sb film and the buried UID buffer layer which is modeled by using the results obtained from sample A. The agreement of modeled and measured ψpp is outstanding for both sample orientations and all carrier densities. The LPP mode positions and broadening parameters are summarized in Table 3. Because the carrier density is not dependent on the sample orientation, we can use Eq. (4) to accurately determine the effective electron mass ratio inline image without input from further experimental techniques. Results are presented in Fig. 4. The error bars in Fig. 4 decrease significantly with increasing carrier density and thus increasing plasma frequency, resembling the increasing distance of ωLPP from the highest undisturbed LO phonon allowing more accurate fitting results. The increasing ratio of inline image in Fig. 4 directly reflects an increasing difference in band curvature which is related to the effective mass by

  • display math(7)

E(k) is the conduction band dispersion as a function of momentum k. Because the ratio between inline image is not constant as a function of carrier density, at least one of both effective electron masses is carrier density dependent, too. Our analysis reveals the effective masses at the Fermi energy and thus the Fermi vector kF which translates to a non-parabolic conduction band of SnO2 for at least one direction in reciprocal space. We would like to emphasize that up to this point in analysis no data from additional techniques was required. However, Hall-effect carrier densities of our samples are available. When used in Eq. (4), the effective electron mass of SnO2 is obtained, but with larger uncertainty than the mass ratio as additional measurement uncertainties from Hall analysis come into play. Results are shown in Fig. 5 and indicate a non-parabolicity of the conduction band for both reciprocal space directions to a different extent. For the lowest carrier density of inline image the experimental accuracy was too low to reliably extract the effective masses. Therefore, for the low-carrier-concentration limit we are using the effective mass values from Ref. [19].

image

Figure 3. Ellipsometric parameter ψpp measured (symbols, black) at an angle of incidence of 60° in two different sample orientations. Modeled data are represented by continuous lines (red). The spectra are shifted vertically for clarity.

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Table 3. Highest LPP mode frequencies ωLPP, plasma frequencies ωp, and plasma broadening parameters γp determined from the fit of our model to the ellipsometry results for both sample orientations. All values are in cm−1
sampleφ = 0°, θ = 34°φ = 90°, θ = 34°
ωLPP,⊥ωp,⊥γp,⊥ωLPP,∥ωp,∥γp,∥
B822378234771392320
C143213061159144613181494
D166315581075176816871995
E321231651148366436281530
F40063966949434046952382
image

Figure 4. Effective electron mass ratio as a function of carrier density as determined by Hall effect measurements in antimony doped rutile SnO2. The horizontal line represents literature data taken from Ref. [19]. The non-constant value of inline image is direct proof of non-parabolicity of at least one of the two conduction band directions parallel and perpendicular to the c-axis. A 20% error was estimated for Hall effect carrier densities.

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image

Figure 5. Effective electron masses as a function of carrier density as determined from Hall effect measurements in antimony doped rutile SnO2. Values from Ref. [19] are shown by arrows.

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The broadening parameters γp are used to estimate the mobility using Eq. (6). Table 4 summarizes the mobility as determined from Hall effect measurements and the two different mobility values determined from γp,⊥ and γp,∥ as given in Table 3, respectively. While the absolute values of ellipsometrically obtained mobilities are low compared to those from Hall data, we find agreement in relative mobility decrease for increasing carrier density between samples B and F.

Table 4. Electron mobilities as determined by Hall effect measurements μHall and by Eq. (6) using plasma broadening parameters perpendicular (parallel) to the optical axis μ (μ). All values are in cm2 V−1 s−1
sampleμHallμμ
B857059
C623631
D613423
E472628
F352515

6 Summary

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

We have investigated a series of MBE grown rutile SnO2:Sb samples on r-plane sapphire by GIRSE in two orientations. Detailed analysis of the anisotropic infrared dielectric function yields energy positions of coupled LPP modes, which in turn are used to determine the effective electron mass anisotropy as a function of carrier density. The absolute values of effective electron masses are estimated using carrier densities determined by Hall effect measurements. A non-parabolic conduction band is found for both directions in reciprocal space.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References

We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the framework of Major Research Instrumentation Program No. INST 272/211-1. Part of this work was supported by the National Science Foundation NSF MWN Program under Award No. DMR09-09203.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Samples
  5. 3 Ellipsometry
  6. 4 Model
  7. 5 Results and discussion
  8. 6 Summary
  9. Acknowledgments
  10. References