Optical studies of carrier transport and fundamental absorption in 4H-SiC and Si

Optical Studies

of Carrier Transport and Fundamental Absorption in 4H-SiC and Si

Paulius Grivickas

Laboratory of Material and Semiconductor Physics Department of Microelectronics and Information Technology

Royal Institute of Technology (KTH) Stockholm 2004

Cover illustration: Motion of electron and hole in a bound excitonic state in analogy to the classical mechanics.

Optical Studies

of Carrier Transport and Fundamental Absorption in 4H-SiC and Si

A dissertation submitted to the Royal Institute of Technology (KTH) for the degree of Doctor of Philosophy.

© 2004 Paulius Grivickas

Royal Institute of Technology (KTH)

Department of Microelectronics and Information Technology Laboratory of Material and Semiconductor Physics

Electrum 229, SE-16440 Sweden

ISRN KTH/FTE/FR-2004/1-SE ISSN 0284-545

TRITA FTE

Forskningsrapport 2004:1

Printed in 200 copies by Universitetsservice US AB, Stockholm 2004

To tenacious Queen of my life…

To tender Gold of my soul…

To warbling Lark of my mind…

and…

To sunrise Dew of my heart…

Abstract

The Fourier transient grating (FTG) technique and a novel spectroscopic technique, both based on free carrier absorption (FCA) probing, have been applied to study the carrier diffusivity in 4H-SiC and the fundamental absorption edge in 4H-SiC and Si, respectively.

FTG is a unique technique capable of detecting diffusion coefficient dependence over a broad injection interval ranging from minority carrier diffusion to the ambipolar case. In this work the technique is used for thin epitaxial 4H-SiC layers, increasing the time- and spatial-resolution of the experimental setup by factors of ~100 and ~10, respectively, in comparison to the established Si measurements. It is found that the diffusion coefficient within the detected excitation range in n-type 4H-SiC appears to be lower than the analytical prediction from Hall-mobility data. To explain this, it is suggested that the minority hole mobility is reduced with respect to that of the majority one or that the hole mobility value is in general lower than previously reported. Observed differences between the temperature dependency of the ambipolar diffusion and the Hall-prediction, on the other hand, are attributed to the unknown Hall factor for holes and the additional carrier-carrier scattering mechanism in Hall measurements. Furthermore, at high excitations a substantial decrease in the ambipolar diffusion is observed and additionally confirmed by the holographic transient grating technique. It is shown that at least half of the decrease can be explained by incorporating into the theoretical fitting procedure the calculated band-gap narrowing effect, taken from the literature. Finally, it is demonstrated that numerical data simulation can remove miscalculations in the analytical Fourier data analysis in the presence of Auger recombination.

Measurements with variable excitation wavelength pump-probe are established in this work as a novel spectroscopic technique for detecting the fundamental band edge absorption in indirect band-gap semiconductors. It is shown that the technique provides unique results at high carrier densities in doped or highly excited material. In intrinsic epilayers of 4H-SiC, absorption data are obtained over a wide absorption range, at different temperatures and at various polarizations with respect to the c-axis. Experimental spectra are modeled using the indirect transition theory, subsequently extracting the dominat phonon energies, the approximate excitonic binding energy and the temperature induced band-gap narrowing (BGN) effect in the material. Measurements in highly doped substrates, on the other hand, provide the first experimental indication of the values of doping induced BGN in 4H- SiC. The fundamental absorption edge is also detected in highly doped and excited Si at carrier concentrations exceeding the excitonic Mott transition by several orders of magnitude. In comparison to theoretical predictions representing the current understanding of absorption behavior in dense carrier plasmas, a density dependent excess absorption is revealed at 75 K. Summarizing the main features of the subtracted absorption, it is concluded that an excitonic enhancement effect is present in Si.

Contents

List of appended papers

ii

Acknowledgements

v

1 Introduction

1

2 Indirect semiconductors SiC and Si

2.1 Basic properties 7

2.2 Band structure and vibrational spectra 12

2.3 Band gap narrowing

17

3 Carrier transport

3.1 Mobility and diffusion 25

3.2 Carrier recombination 34

4 Optical absorption

4.1 Excitons 43

4.2 Fundamental absorption 53

4.3 Free carrier absorption 68

5 Measurement methods

5.1 FCA probing 77

5.2 Free carrier gratings 83 5.3 Spectroscopic measurements 90

6 Summary

95

List of appended papers

I Free carrier diffusion measurements in epitaxial 4H-SiC with a Fourier transient grating technique: injection dependence P. Grivickas, J. Linnros and V. Grivickas

Mater. Sci. Forum 338-342, 671(2000) II Free carrier diffusion in 4H-SiC

P. Grivickas, A. Martinez, I. Mikulskas, V. Grivickas, J. Linnros, U.

Lindefelt and R. Tomašiunas

Mater. Sci. Forum 353-356, 353 (2001)

III Carrier lifetime investigation in 4H-SiC grown by CVD and sublimation epitaxy

P. Grivickas, A. Galeckas, J. Linnros, M. Syväjärvi, R. Yakimova, V.

Grivickas and J. A. Tellefsen

Mater. Sci. in Semic. Proc. 4, 191 (2001)

IV Carrier diffusion characterization in epitaxial 4H-SiC P. Grivickas, J. Linnros and V. Grivickas

J. Mater. Res. 16, 524 (2001)

V Temperature dependence of the absorption coefficient in 4H- and 6H- silicon carbide at 355 nm laser pumping wavelength

A. Galeckas, P. Grivickas, V. Grivickas, V. Bikbajevas, and J. Linnros Phys. Stat. Sol. (a) 191, 613 (2002)

VI Characterization of 4H-SiC band-edge absorption properties by free- carrier absorption technique with a variable excitation spectrum P. Grivickas, V. Grivickas, A. Galeckas and J. Linnros

Mater. Sci. Forum 389-393, 617(2002)

VII Single- and two-photon band edge characterization in Si and 4H-SiC by temporally excited free-carrier absorption as a novel technique P. Grivickas, V. Grivickas, A. Galeckas and J. Linnros

in Physics of Semiconductors 2002, edited by A. R. Long and J. H.

Davies, Institute of Physics Conference Series Number 171, N3.1 (Institute of Physics Publishing, Bristol, UK, 2003)

VIII Excitonic absorption above Mott transition in Si P. Grivickas, V. Grivickas and J. Linnros

Phys. Rev. Lett. 91, 246401 (2003)

IX Intrinsic band-edge absorption in 4H-SiC

P. Grivickas, V. Grivickas, J. Linnros and A. Galeckas (in manuscript, intended to Phys. Rev. B)

Published work not appended

Relevance of the carrier transport change in porous silicon at the onset of one excited pair per crystallite

V. Grivickas, J. Linnros, N. Lalic, J. A. Tellefsen and P. Grivickas Thin Solid Films 297, 125 (1997)

Band edge absorption, carrier recombination and transport measurements in 4H-SiC epilayers

V. Grivickas, J. Linnros, P. Grivickas and A. Galeckas Mater. Sci. & Engin. B61-62, 197 (1999)

Determination of the polarization dependency of the free-carrier- absorption in 4H-SiC at high-level photoinjection

V. Grivickas, A. Galeckas, P. Grivickas and J. Linnros Mater. Sci. Forum 338-342, 555(2000)

Thermopower measurements in 4H-SiC and theoretical calculations considering the phonon drag effect

V. Grivickas, M. Stölzer, E. Velmre, A. Udal, P. Grivickas, M. Syväjärvi, R.

Yakimova and V. Bikbajevas

Mater. Sci. Forum 353-356, 491 (2001)

Excess free carrier optical excitation spectroscopy in indirect semiconductors

V. Grivickas, A. Galeckas, P. Grivickas, J. Linnros and V. Bikbajevas Mater. Sci. (ISSN 1392 - 1320 Medziagotyra) 7, 203 (2001)

Impact of phonon drag effect on Seebeck coefficient in p-6H-SIC:

experiment and simulation

V. Bikbajevas, V. Grivickas, M. Stölzer, E. Velmre, A. Udal, P. Grivickas, M.

Syväjärvi and R. Yakimova

Mater. Sci. Forum 433-436, 407 (2003)

Two-photon spectroscopy of 4H-SiC by using laser pulses at band-gap frequencies

V. Grivickas, P. Grivickas, J. Linnros and A. Galeckas Accepted in Mater. Sci. Forum (2004)

Author’s contribution to the appended papers I-IX:

Planning, sample preparation, data acquisition, data analysis, theoretical simulations and writing of the manuscript were done exclusively by the author in all articles appended to the thesis except the following cases:

i) In the article V the author’s contribution consists of the following: sample preparation (~30%), data acquisition (~30%), data analysis (~20%) and manuscript writing (~20%).

ii) In the article VIII all enumerated contributions consist of 100% except ~20%

in the sample preparation and the data acquisition.

Acknowledgements

First of all, I am very grateful to Jan Linnros for being my supervisor and supporting me during the long and interesting PhD journey. He has shared his scientific insights with me openly and has patiently discussed all of my new and sometimes bizarre ideas. Thanks go also to Ulf Lindefelt and Bo Breitholtz, who accepted me as a PhD student within the SiCEP program.

This work would have been almost impossible without my father, Vytautas Grivickas. Only a few of us have the wonderful opportunity to get to know the other side of our parents, where they are valued as professionals. As a scientist I am lucky to have shared my time with one of the greatest researchers I know, while as a son I am proud to have had some battles with my teacher in life.

I also would like to thank Augustinas Galeckas for sharing with me several precious things: homeland, coffee breaks, conference hotel rooms, lists of coauthors and a laboratory. Many researchers say that high quality equipment is not enough to be able to do high quality science. One also needs a person with

“golden arms” to run it, and I have had a good opportunity to see the truth of this phrase.

I am also greatly in debt to Marianne Widing. All departments have secretaries, but only a few have people who help you so much. None of my everyday life problems during these years were allowed to become really big problems, and it has been a pleasure to watch how one person can make life so much brighter for so many of us.

I would like to send special thanks to Antonio Martinez for being my colleague and coworker in this project (I miss your voice already), to Xavier Badel for sharing corridors not only at work, but also in our flat, to Pascal Kleimann for introducing me to the mysterious world of the clean-room, to Patrick Leveque for discussing the strange world of physics, to Uwe Zimmermann for honorably accepting my cookies in exchange of being my computer mentor and to Nenad Lalic and Aliette Mouroux for being my guides into the world of the PhD student. I am especially grateful to Christopher Castleton who accepted the difficult task of turning the language of this thesis into proper English. Many thanks also to the people who have kept me company at the department: Robert, Jan, Marylene, Arturas, Leonardo, Martin, Jens, Hanne, John, Paolo, Edouard, Andrej, Ilya, Maciej…

During these years I met a lot of people who have become good friends:

Rimas (for ever), Ausrius and Alvydas (I believe we share much more than just being Lithuanian physics PhD students in Stockholm…), Giedre and Korkut, Renata and Romanas, Daiva, Kestas 1 and Kestas 2, Paulius… Hi five to my basketball teammates! I am pretty sure that I am forgetting someone, but it doesn’t mean that your picture is not in my heart☺.

Finally, I would like to thank my mother and sister for accepting my exile for so many years, and, even more, I would like to thank Rasa for becoming my new inspiration, my sole mate and my new family. We already have so many things to say to each other and I feel it is just the beginning of the story.

The work summarized in this thesis has been done within the Silicon Carbide Electronic Program (SiCEP) supported by the Swedish Strategic Research Board (SSF). Most of the SiC samples were provided by the SiCEP group in Linköping.

1 Introduction

Since the middle of the last century semiconductors, as a group of materials, have invaded our everyday life at an unimaginable speed. Even though today many people are unfamiliar with words like diode, transistor or semiconductor laser, it would be difficult to find someone unaware of such things as computers, CD players or mobile phones, which provide the foundations of our technological civilization. At the heart of this fancy terminology lies the simple observation that some materials substantially change their electric conductivity due to illumination, temperature variation or incorporation of minute amounts of impurity atoms. Investigation of these properties led to the current understanding that semiconductors represent a group of solids in which covalently bound atoms share common electrons in the completely occupied low energy valence band. If additional energy is provided to the solid, the latter electrically active particles can be excited into the empty conduction band. There, these induced free charges become carriers of electrical current, creating an electrically conductive matter. Macroscopic behavior of the carriers is described by a set of transport equations consisting of the Poisson’s equation, the continuity equation, the drift-diffusion equation plus Fermi-Dirac statistics for carrier concentration. In these equations the most important parameters characterizing the semiconductor are the charge carrier lifetime, mobility and diffusivity.

If one is only concerned with practical material questions for device performance, the task would appear to be that of finding the numerical values of these parameters. However, the true story is far more complicated. There are a vast number of physical mechanisms, which interplay unpredictably under different conditions to determine a particular parameter. In the case of the carrier diffusivity, for example, the very incomplete physical model includes the description of majority and minority carrier diffusivity at various temperatures and injections, which in turn depend on the impurity-carrier, phonon-carrier or electron-hole scattering, which in turn depend on the screening of the impurities by the charge carriers, clustering of impurities or mutual carrier attraction due to the Coulomb interaction and so on. Moreover, the parameters listed in textbooks and manuals usually refer to bulk material properties or to the extracted values under special experimental conditions, while practical device modeling contains a number of constants and variable parameters referring to the sizes and interfaces of the device. The full story should also account for the complexity of modern processing technologies applied to the material, e.g. ion implantation, surface passivation or internal gettering, which produce complex impurity profiles and other structural non-linearities.

An answer to this complicated material challenge from the world of physics was the invention of numerous characterization techniques, capable of

providing detailed answers in specific circumstances. Though only a few techniques can be labeled as standard and not a single technique dominates, all together they provide a rather consistent insight into the properties of semi- conducting materials. Individual characterization strategies depend on the type of specimen, ranging from bulk material to material structure or even the complete device, and the particular group of the physical phenomena pursued.

This work focuses on optical characterization, which is favored both within the scientific community and in industry due to its undestructive nature and fairly simple data interpretation. More specifically, the results of this thesis were attained using optical methods based on the selective probing of excess free carrier concentration.

Pioneering investigations employing free-carrier absorption (FCA) as a tool were done in the fifties in heavily doped Si and Ge [1] and the basic absorption mechanisms were established at that time. However, the real break through in the utilization of FCA came with the discovery of the laser, which provided the means of generating instant excess carrier excitations by short, highly energetic pulses [2]. Non-diverging beams of monochromatic and coherent laser light together with sufficient instrumentation for the detection system provided a powerful pump-probe technique capable of resolving carrier dynamics within a crystal. The principle of perpendicular pump-probe measurements is illustrated in Fig. 1(a). Light from the pump pulse homogeneously excites the lateral side of the sample while a focused probe beam monitors the time behavior of the created excess carrier concentration.

Employment of spatially resolved and calibrated FCA probing in such measurements results in several unique methods to examine material properties.

Probe

Sample

E_G

a) b)

Pump Probe Pump

hω hω

hω hω

probe

pump

Probe

Sample

E_G

a) b)

Pump Probe Pump

hω hω

hω hω

probe

pump

Fig. 1 Principles of (a) the perpendicular pump-probe measurement, (b) the FCA probing mechanism.

The idea of FCA probing originates from the fact that light with photon energy below the material band-gap (_hω_probe<E_G) is strongly absorbed by the excited carriers. The latter are introduced by pump light with photon energy above the material band-gap (_hω_pump > E_G). The whole scenario is shown in Fig.

1(b). In the FCA process a single photon of light is annihilated with the simultaneous transition of an electron or a hole to a higher energy state in the conduction or valence band, respectively, which subsequently relaxes into the initial state. While no additional excess carriers are introduced in the system, the efficiency of the process is proportional to the instantaneous excess carrier concentration present in the material. Therefore, looking at the intensity changes in a weak probe beam passing through the excited volume of the specimen, it is possible to correlate the signal to the absolute carrier concentration.

It is worth mentioning that a great deal of the development of FCA methods was carried out by the two supervisors of this work: V. Grivickas and J.

Linnros [3,4]. They first used FCA probing several years ago to determine the dependence of the bulk lifetime in Si on the injection density. Since the fundamental work of Shockley, Read and Hall (SRH) [5] it was known that recombination via localized impurities changes at different excess carrier densities, and FCA probing was revealed to be a straight forward and extremely accurate tool for studying this phenomenon. It becomes even more advantageous at injected densities greater than 10¹⁷ cm^-3, where nonlinear Auger- recombination becomes the leading recombination mechanism. Scanning measurements into the depth of the sample at that moment provided first insights into the diffusion processes of the free carriers and surface recombination rates.

As a consequence, in 1991 the two invented a novel Fourier Transient Grating (FTG) technique [6], which stands out from other transient grating techniques, like modulated diffraction or thermal deflection, due to its applicability over a very wide excitation range. The technique resulted in detailed studies of injection-dependent mechanisms of ambipolar diffusion and minority carrier drift mobility in Si. Recently it was realized that FCA probing could also serve as an instrument to detect fundamental absorption coefficients, while monitoring the depth distribution of the excited carriers. In 2001 the idea was extended to full spectroscopic measurements [7] when a laser excitation with a tunable wavelength became available in the laboratory. As discussed in the rest of this work, spectroscopy based on FCA detection shows exclusive applicability in the case of complex material structures and reveals its unique capability to access the fundamental absorption edge in the case of highly doped or excited semiconductors.

In this work most of the enumerated advantages of the FCA probing techniques were utilized to investigate a very ambitious, but still relatively mysterious candidate among the indirect band gap semiconductors – silicon carbide (SiC). This binary compound synthesized from the two well-established

representatives of the group IV of elements is famous for its extraordinary mechanical, thermal and electrical properties. Starting its journey more than a hundred years ago as a simple abrasive powder, it now challenges various fields of applied technology ranging from fusion plasma devices and high temperature sensors to light emitting diodes and high-power switches and rectifiers. The turning point in history was marked 15 years ago, with the growth of high quality epitaxial layers on commercial single crystalline SiC wafers. However, despite the continuous achievements during recent years in material quality and understanding of its basic properties, many unresolved questions remain. One of them concerns the diffusion coefficients of free carriers in SiC, which has only preliminary theoretical estimates from drift mobility values. The fundamental absorption edge at all temperatures is only available in a few SiC polytypes, while only a general assessment exists in the commercially attractive 4H-SiC polytype. Even the complete analysis of material lifetime at different temperatures is lacking. The list of the question marks can be continued with the lack of data for band gap narrowing, the influence of anisotropy, unknown exciton binding energy etc. This work attempts to provide at least some of the answers to the mentioned directions.

As a final point, it should be noted that probably the most interesting and weighty result of the thesis, however, was obtained in the most thoroughly investigated of the indirect semiconductors – crystalline Si. The novel fundamental absorption edge measurements were tested on the well-grounded data of this material simultaneously extending them to the boundaries of high doping and excitation. The previous understanding of the absorption constituents in indirect semiconductors suggested that the pronounced bound excitonic component should disappear at high plasma densities due to carrier-carrier screening. Nevertheless, an unexpected excess excitonic-like absorption was found at carrier concentrations far exceeding the critical Mott density for the bound excitons. Though this effect, named excitonic enhancement, was previously observed and established in direct gap semiconductors due to the many-body interactions, it has never been confirmed for any indirect semiconductor, because of the inherent limitations in conventional absorption methods. Current verification of this carrier correlation effect should influence explanation of some carrier transport properties as well as the modeling and simulation of the device performance.

The thesis is organized as follows. Chapter 2 presents the crystal and band gap structure of the two indirect semiconductors Si and SiC, also introducing the concepts of lattice vibration and band gap narrowing. The data are presented in a comparative manner with the main focus on SiC. Chapter 3 briefly overviews relevant carriers transport phenomena and recombination mechanisms, providing the basic relations. Chapter 4 describes in detail all the possible absorption mechanisms starting from the investigated fundamental edge and ending with FCA, which both interferes with fundamental absorption

measurements and provides the probing technique. The concept of the exciton as a correlated electron-hole particle together with the excitonic Mott transition and the excitonic enhancement effect are also elucidated. Chapter 5 presents FCA as a probe tool enumerating the spin-off techniques. Finally, the main conclusions and future outlook are provided in the summary Chapter 6, including a step-by- step commentary on the appended papers.

Note that indexation of equations, figures and references in the work starts from “1” in each subchapter (e.g. 2.1, 3.2), and they are referred in the text in the following manner: Eq. (5) in Ch. 4.2, Fig. (7a) in Ch. 3.1 or Ref. 6 in Ch.

2.2. The list of references is provided after each subchapter. The nine appended articles of the thesis are referred in the text according their indexation with Roman numbers: II, VII. The following two abbreviations are commonly used in the text: lhs – left hand side, rhs – right hand side.

References

[1] H. Y. Fan, Repts. Prog. Phys. 10, 107 (1956).

[2] D. K. Schroder, Semiconductor material and device characterization (Wiley

& Sons, New York, 1990).

[3] J. Linnros and V. Grivickas, Carrier lifetime: Free carrier absorption, Photoluminescence and Photoconductivity, Ch. 5b.2 in Methods in Material Research, edited. by E. N. Kaufman (John Wiley & Sons Inc., 2000).

[4] V. Grivickas et al., Ch. 13.1, 13.4, 13.5, 13.6 in Properties of Crystalline Silicon (EMIS Data reviews Series No. 20, INSPEC), edited by R. Hull (London, 1999).

[5] W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952); R. N. Hall, ibid.

87, 387 (1952).

[6] V. Grivickas and J. Linnros, Appl. Phys. Lett. 59, 72 (1991).

[7] V. Grivickas, A. Galeckas, P. Grivickas, J. Linnros and V. Bikbajevas, Material science (ISSN 1392 - 1320 Medziagotyra) 7, 203 (2001).

2 Indirect Semiconductors SiC and Si

2.1 Basic properties

2.1.1 Crystal structure and polytypes of SiC

Pure Si crystallizes in the diamond structure, with an fcc Bravais lattice and a two-atom basis. The chemical bond is solely covalent, each atom being tetrahedrally coordinated. Its crystal properties are well investigated and documented in textbooks [1]. SiC in this respect is more exotic material and will be discussed in more details.

Both Si and C are elements of group IV of the periodic table. One would expect, therefore, that their atoms would easily create covalent bonds resulting in alloys of the SixC1-x form. However, a large bond-length mismatch appears between the big Si and the much smaller C atoms, which is well illustrated by the corresponding 2.35 Å and 1.55 Å values in their respective tetrahedral diamond structures. As a consequence, a single composition exists between Si and C, where each Si atom has a covalent bonding with four C atoms and vice versa, providing an equal number of the two elements in the total crystal. The expected crystal structure of such SiC material would be zincblende or wurtzite.

Nevertheless, early investigators were left in confusion after finding a variety of stable SiC configurations ranging from cubic (C) to hexagonal (H) and rhombohedral (R) categories. The result was explained by the polytypism phenomenon in which several crystals exist with the same chemical composition but different sequence of close packed atomic layers. Nowadays around 200 polytypes of SiC have been observed, while hundreds more are predicted theoretically.

The stacking sequence of a particular SiC polytype is described using the capital letters A, B and C. Each letter is associated with a single covalently bond Si and C layer, distinguished by its position relative to the previous layer. Such a notation system is explained in Fig. 1 for the four most common polytypes. The simplest AB-AB-… repetition sequence results in the 2H polytype, where 2 stands for the number of layers in a single repetition and H signifies a hexagonal crystal lattice. The ABC-ABC-… array produces the 3C polytype, the single polytype in the cubic crystal system. Historically this polytype has also been called β-SiC, whereas all other crystal types have been collectively referred to as

α-SiC. The ABCB-ABCB-… and ABCACB-ABCACB-… sequences correspond, respectively, to the 4H and 6H polytypes in the hexagonal configuration. From a crystallographical point of view, however, the latter polytypes as well as all the rest are a mixture of cubic and hexagonal bonding and form one-dimensional superlattice structures. For example, 4H-SiC consists of an equal number of hexagonal and cubic bonds, while 1/3 hexagonal and 2/3 cubic bonds gives 6H-SiC. Quasi-cubic or quasi-hexagonal nature with respect to the neighboring stacking plane produces inequivalent lattice sites in the crystal, i.e. an atom in the same hexagonal or cubic site finds itself in a different environment. Therefore, the same donor or acceptor atom placed at an inequivalent site has a different activation energy. Note that in all polytypes the interatomic distance, and thereby the density, is the same, it is only the sequence of stacked layers that differs. Nevertheless, crystals with a shorter repeat distance are generally easier to manufacture in large quantities without inducing too many dislocations and defects. Finally, apart from the cubic 3C polytype, all polytypes are expressed in the hexagonal coordinate system consisting of three a-plane coordinates (oriented by 120 degree angles relative to each other) and a

Fig. 1. The most common SiC polytypes represented by their corresponding stacking sequences (adopted from [2]).

perpendicular principal or c-axis. The latter lies in the stacking direction of the hexagonally close packed layers.

2.1.2 SiC growth and doping

High quality low-doped SiC is usually grown by the low-pressure chemical vapor deposition (CVD) technique, which is illustrated in the upper part of Fig. 2: a highly-doped (~10¹⁹ cm^-3) SiC substrate wafer is placed on a graphite susceptor within a quartz tube. Gaseous precursors containing Si and C are transported into the tube by a H2 carrier gas and cracked inside by heat from an RF coil at typical temperatures of 1400 – 1800 ^oC. Separate atoms or radicals produced by the cracking process nucleate at the atomic steps on the substrate surface, which are intentionally created by polishing the growth surface by the common 8 degrees in respect to the basal a-plane. Two types of the CVD system can be distinguished: i) cold and ii) hot wall, which is more advantageous from the growth rate point of view. In the former case the substrate is simply laid on the susceptor, while in the latter it is enclosed within a tubular susceptor. For growth speed it is also beneficial to build the whole system in a vertical (or chimney) design [4]. Nevertheless, the typical growth rate by CVD technique is

Fig. 2. Principles of the CVD (top) and sublimation (bottom) growth techniques (adopted from [3]).

< 5 µm/h, indicating that only thin < 100 µm epitaxial layers are typically produced.

Epitaxial growth by sublimation has also been studied as a suitable choice for high-growth rates such as ~ 100 µm/h [5]. This approach uses the same principles as the modified basic Lely’s method, which is a standard technique for growing large single crystal boules for SiC substrate wafers. As illustrated by the schematic view of the process shown in the lower part of Fig. 2, SiC source material is heated up to produce the vapor-phases of Si and C species such as Si2C and SiC2. The latter are transported from the source to the nucleating substrate by introducing a temperature gradient. Unfortunately, material purity using this sublimation technique is highly affected by growth conditions such as source material quality, temperature gradient control and contamination from the graphite susceptor. In fact, sublimation epilayers in the low-doped range are produced only by donor and acceptor compensation.

However, in the CVD technique the desired doping levels can be achieved by well-controlled substitution of the doping gases into the main flow, as illustrated in Fig. 2.

These days intrinsic SiC epilayers produced by the CVD technique have unintentional doping levels reduced to ~10¹³ cm^-3. In general, n-type conductivity in SiC originates from N or P donors, which have ionization energies varying between 40 and 140 meV [6], depending on polytype and type of lattice site. The main “shallow” acceptors in the p-type SiC are B or Al, exhibiting ~200 and ~300 meV levels above the valence band, respectively [6].

2.1.3 Comparison of 4H-SiC and Si

At this point it should be noted that of all the possible SiC polytypes, 4H becomes the object of the current work as it is the common polytype of choice for applications due to its superior transport parameter values. These, as well as the following properties are summarized in Table 1 for 4H-SiC and Si: the indirect band-gap EG, critical field at breakdown Ec, thermal conductivity σ, saturated drift velocity vsat, mobilities for electrons and holes at room temperature and low doping µe and µh, relative dielectric constant ε, density ρ and lattice constant a (and c in 4H-SiC). The values provided can be weighted using different figures of merit, but the basic guidelines are clear. 4H-SiC has tremendous potential in power devices due to its higher critical field at breakdown and thermal conductivity. It can be shown that specific on- resistances of unipolar 4H-SiC devices can be as little as 1/300 for an equivalent Si device. Already now, power-blocking voltages in 4H-SiC diodes exceed 5 kV while more than 15 kV is expected theoretically. Moreover, SiC emerges as a superior candidate for use in conditions of high temperature or in hash environments.

At the same time, it is important to remember that α-SiC, due to its hexagonality, has anisotropic crystal structure, i.e. measured parameters can vary substantially in the directions parallel and perpendicular to the c-axis. The degree of anisotropy in 4H-SiC, compared to other polytypes, is not very high.

For the carrier mobilities a ratio of 0.83 was found, while other constants vary even less.

Table 1. Main properties of Si and 4H-SiC

Si 4H-SiC

E_G (eV) 1.12 3.26

Ec (MV/cm) 0.25 2.2

σ (W/cm K) 1.5 3.7

νsat (10⁷ cm/s) 1 2

µe (cm²/Vs) 1350 1000

µh (cm²/Vs) 471 115

εr 11.6 10

ρ (g/cm³) 2.3 3.2

a=/c= (Å) 5.43 3.08/10.0

References

[1] Properties of Crystalline Silicon, edited by R. Hull (INSPEC, London, 1999).

[2] K. Nassau, J. of Gemmology 26, 425 (1999).

[3] http://www.ifm.liu.se/Matephys/new_page/research/sic/.

[4] A. Ellison, Mater. Sci. Forum 338-342, 131 (1999).

[5] M. Syväjärvi, R. Yakimova, H. Jacobsson, M. Linnarsson, A. Henry and E.

Janzen, Proc. of ICSCRM’99 (in press).

[6] W. J. Choyke and G. Pensl, MRS Bulletin 22, 25 (1997).

2.2 Band structure and vibrational spectra

This section provides a basic description of the band structure of the two semiconductors relevant for the optical transitions presented in later chapters.

The well-determined band structure of the crystallographically simpler Si serves as the comparative foundation for the far more complicated and not yet completely elucidated one in the 4H-SiC polytype. In the same manner, the structure of the lattice vibrations in the two materials is summarized, focusing on the phonon branches active in optical transitions.

2.2.1 Band structure

The lhs of Fig. 1 shows a segment of the Si band structure calculated within the k⋅p method [1]. The high symmetry directions in the diagram are indicated by the letters Λ and ∆, while the high symmetry points are labeled by the letters L,Γ and X. The maximum of the valence band in Si appears at the Γ- point, which represents the center of the first Brillouin zone (k = (0,0,0)). The maximum is six-fold degenerate and splits due to spin-orbit coupling into two bands, as indicted schematically on the rhs of Fig. 1. Moreover, the top valence band consists of the two branches, which have different energy dispersion and are known as heavy- and light-hole bands. Their average effective masses are m_hh = 0.476m₀ and m_lh = 0.159m₀, respectively [2], where m₀ is the electron mass in free space, which enter calculations of the total density of states mass mh

through the Luttinger parameters [2], providing mh = 0.527m0. The minimum of the conduction band emerges in the ∆1 direction at a distance k0/kmax = 0.85 from

Fig. 1. Band structure of Si [1].

the Γ-point, where kmax indicates the wave vector size between Γ and X. The minimum is parabolic, but has different effective masses in different directions (effective mass anisotropy). Therefore, the density of states mass me* is expressed as:

0 3

/ 1

2) 0.321 (

* mm m

m_e = _{l t} = _{, (1)}

where ml = 0.916m0 is the longitudinal effective mass in the direction from Γ to k0 and mt = 0.19m0 is the transverse effective mass in the two perpendicular directions [2]. In addition, Eq. (1) has to be multiplied by a factor of g^2/3, where g = 6 represents the number of equivalent conduction band minima due to the

“multi-valley” structure of Si. Note that the density of states mass expressed by Eq. (1) differs from the effective conductivity mass, which enters into carrier transport equations. The latter is given by:

259 0

. / 0

1 / 1 / 1

3 m

m m

m m

t t

l

e =

+

= + _{. (2)}

Energy differences between the various points in the band structure are indicated on the rhs of Fig. 1.

The lhs of Fig. 2 shows the calculated band structure of 4H-SiC within the local density approximation to density functional theory [3]. This approximation is well known to underestimate the absolute fundamental band gap in semiconductors by ~1.1 eV, but the shape of the bands is usually in agreement with other approaches. The complexity of the diagram is evident in comparison

Fig. 2 Band structure of 4H-SiC [3].

to Si, although symmetry of the both materials is rather similar. The complexity arises from the fact that the unit cell of 4H-SiC contains 8 atoms and is almost twice as long in the hexagonal c-axis direction as the primitive unit cell of Si, which contains 2 atoms. As a consequence, the first Brillouin zone of 4H-SiC is only half as long in c-direction resulting from the back folded branches. This process is more explicitly illustrated in Fig. 3 for the phonon branches. The maximum of the valence band again appears at the Γ-point. However, due to the spin-orbit coupling and the hexagonal crystal field it splits into the three subbands, as schematically shown on the rhs of Fig. 2. Their average effective masses are mh⎜⎢ = 1.6; 1.6 and 0.2m0 and mh⊥ = 0.6; 0.6 and 1.5m0 in the parallel and perpendicular directions to the c-axis, respectively [3,4]. The total density of states mass can be obtained through the parameterisation expressions for the wurtzite crystal structure [3]. The Conduction band emerges in the g = 3 equivalent minima at the M point with the following effective masses: m_e⎜⎢ = 0.29m0, m_e⊥ = 0.42m0, me* = 0.37m0 and me = 0.36m0 [3,5]. Again, various energy differences are shown on the rhs of Fig. 2.

E

π/2a π/a k

π/4a π/8a

Si 4H-SiC

0 E

π/2a π/a k

π/4a π/8a

Si 4H-SiC

0

Fig. 3 (lhs) Phonon branch formation in the form of reflections in the k space due to the folding of the extended zones into the first Brillouin

zone. (rhs) Phonon dispersion in the ∆ direction of Si. The symbols represent experimental results from neutron scattering [2].

2.2.2 Vibration spectra

The simplest representation of a solid is the linear-chain model of single interacting atoms. Its equation of motion has a solution of the form

( /2)

sin ka

ω∝ , which is illustrated on the lhs of Fig. 3 by the line extending through the k space of the first Brillouin zone from 0 to π/a, where a represents the lattice constant. In such a model making the assumption of two equivalent atoms per unit cell results in an increase in the unit cell length by a factor of two and a reduction in the edge of the first Brillouin zone by one half to π/2a.

Consequently, the outer parts of the wave vectors in the previous solution are

Fig. 4 (top) Phonon dispersion in 4H-Si with symbols representing experimental results from neutron scattering. (Table) Characteristic energies at the M symmetry point for the 12 phonons with acoustic

nature (left side) and the 12 with optical nature (right side) [6].

reflected into the new zone creating an optical branch (anti-phase motion of the neighboring atoms), while the previous branch is labeled as the acoustic one (in phase motion), as illustrated by the gray line in Fig. 3. This simplified scenario, nevertheless, predicts well the real phonon dispersion in Si, which is shown in the ∆ crystallographic direction on the rhs in Fig. 3 [2]. The branches are additionally split into the transverse (TA and TO) or longitudinal (LA and LO) with respect to the wave propagation. It is phonons in the Si ∆ valley at the k0/kmax = 0.85 value which complete the indirect interband transitions in the presented earlier band structure. From Fig. 3 values of the dominating TA = 18.4 meV and TO = 57.8 meV phonons can be detected.

In the case of 4H-SiC containing 8 atoms per unit cell, the simplified procedure of phonon branch formation from back reflected folds should be repeated three times, as illustrated by the black line in the 0 to π/8a range of Fig.

3. The result gives an approximate guideline for the complicated vibration spectra of the real crystal shown in Fig. 4 [6]. In the spectrum a distinct gap appears between the optical and acoustical branches at the boundaries of the first Brillouin zone due to the different masses of the vibrating Si and C atoms.

Phonon branches participating in the fundamental interband absorption lie at the M symmetry point. In total 24 branches appear ([1L+2T]×2³ = 24, where power index 3 represents the number of reflections in the simplified model): 12 of acoustic nature and 12 of optical. Their corresponding energies are summarized on the left and right sides of the table in Fig. 4 [6]. Note that, due to the selection rules in the anisotropic 4H-SiC, different phonons are active in the parallel and perpendicular direction with respect to the c-axis, as shown by the symbols ⊥ and ⎜⎢ in the table.

References

[1] M. Cardona and F. H. Pollak, Phys. Rev. 142, 530 (1966).

[2] A. Dargys and J. Kundrotas, Handbook on Physical Properties of Ge, Si, GaAs and InP (Science and Encyclopedia Publishers, Vilnius, 1994).

[3] C. Persson and U. Lindefelt, J. Appl. Phys. 82, 5496 (1997).

[4] N. T. Son et al., Phys. Rev. B 61, R10544 (2000).

[5] N. T. Son et al., Appl. Phys. Lett. 66, 1074 (1995).

[6] J. Serrano et al., Mat. Sc. For. 433-436, 257 (2003).

2.3 Band-gap narrowing

2.3.1 Band-gap changes with temperature

There are two basic contributions, both of a similar order, to the temperature dependence of the band-gap, which can be expressed in the following way:

el ph G G

G

dT dE dT

dE dT

dE

⎟ −

⎠

⎜ ⎞

⎝ +⎛

⎟⎠

⎜ ⎞

⎝

=⎛

exp

(1)

The first term represents the change in the crystal volume with temperature resulting from lattice expansion:

T G T P

G

dP dE dV

dP dT

dV dT

dE ⎟

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

=⎛

⎟⎠

⎜ ⎞

⎝

⎛

exp (2)

where 1/V(dV/dT)⏐P = 3αp, where αp is the linear expansion coefficient at constant pressure, and V(dP/dV)⏐T = -βp, where βp is the isothermal bulk modulus or the inverse of the isothermal compressibility [1]. The last term in Eq. (2) represents the pressure coefficient of the band-gap. The parameters can be easily evaluated once the equation of state for the solid is known. The second term in Eq. (1) is the electron-phonon interaction at constant volume. In theoretical studies it is separated into two different contributions, which are referred as the Debye-Waller and self-energy terms. The first originates from the smearing-out effect of the periodic potential, while the second represents the mutual repulsion of intra-band electronic states through increased electron- phonon coupling in second order perturbation theory [2]. Both these contributions are much more complicated to obtain than the previously introduced thermal lattice expansion and, therefore, are the primary target of theoretical calculations. Nevertheless, it has been shown that all electron-phonon interactions can be approximately taken into account by relating them to a single parameter, namely the change in frequency of the phonons [3]. Lattice vibration frequencies change from ωi(q) to ωf(q), where indexes i and f refer to the initial or lower and final or upper states of the electron, respectively, when an electron is excited across the bang-gap. In such a process the band-gap varies with temperature according to:

∑

+

≈

q i

f B

G

G q

T q k E

T

E ( )

) ln ( )

0 ( )

( ω

ω , (3)

if k_BT >_hω(q). Excitation of an electron weakens the atomic binding and thus lowers the elastic restoring force. Hence ω_f(q)<ω_i(q) and so the logarithm in Eq. (3) is negative and the gap is reduced as the temperature increases. Eq. (3) correctly predicts linear dependency at high temperatures, as approximated by the dashed line in Fig. 1. At lower temperatures, the band-gap tends to a constant value, as shown by the solid line.

In practice, comprehensive EG(T) data sets are widely available from various measurement techniques, and one is more concerned with the problem of fitting them with some analytical expression. Technological applications require that such expressions have as few empirical free fitting parameters as possible. The most conventional and recognized in this respect is the three- parameter formula suggested by Varshni [4]:

(

b T

)

aT E

T

E_G( )= _G(0)− ²/ + _{. (4)}

Here a represents the –dEG(T)/dT limit at T → ∞ and b is a temperature parameter whose magnitude was believed to be comparable to the Debye temperature θD. However, despite the common use of this expression, it was concluded [5] that Varshni’s ad hoc model is generally incapable of providing an accurate and, at the same time, physically adequate interpretation of the available EG(T) data sets for most semiconductors. In fact, in some cases even negative values of the b factor can be found. Moreover, at low temperatures in the cryogenic region Eq. (4) predicts a quadratic dependence, whereas most experiments find a more temperature independent behavior [6].

Among the proposed advanced models the most physically reasonable basis was provided in the model of Passler [5]. It originates from noticing that the electron-phonon interaction varies with lattice temperature just as the Bose-

E

T E_G(0)

E

T E_G(0)

Fig. 1. Typical dependence of the band-gap change with temperature.

Einstein occupation factor nB(Ep,T). Hence, performing a summation over all contributions made by phonon modes with the same energy, it is possible to replace Eq. (3) by:

∫

−

= _{G p B p p}

G T E f E n E T dE

E ( ) (0) ( ) ( , ) _{, (5)}

where f(E_p) is a spectral function capable of incorporating thermal lattice expansion as well. Subsequent expansion of the integral in Eq. (5) suggests the following formula, with four fitting parameters:

( )

( )

( )

^⎪⎪_⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ⎟⎠ −

⎜ ⎞

⎝

⎛ + Θ

⎟⎠

⎜ ⎞

⎝

⎛ + Θ

⎟⎠

⎜ ⎞

⎝

⎛ Θ

− + ∆

⎟⎠

⎜ ⎞

⎝

⎛ Θ

∆ + +

∆ ×

− + Θ

∆

− Θ

−

=

2 1 2

3 8 2

4 1 3 2

1 1 3

2 3 1 / exp

3 1 )

0 ( ) (

6

6 4

2 3 2 2

2

2 2

T T

T T

E T T

E_{G G}

α π .(6)

In this expression α represents the entropy of the material

dT T T dE

S ( )

)

( =− in the T → ∞ limit and

B p

k

= E

Θ is the average phonon temperature. The essential difference is made by introducing the fourth fitting parameter, ∆, which represents the degree of phonon dispersion in the material. Passler shows that in comparison to such a framework the Varshni’s formula represents an unrealistic approximation giving the case of extremely large degrees of phonon dispersion,

∆ > 1, whereas the upper limit for semiconductors is estimated to be at ∆ ≈ 0.75.

Though the values of EG(0), α, Θ and ∆ provided by Eq. (6) are very informative for comparison between materials, sometimes it is convenient to use a simpler phonon dispersion related model earlier proposed by the same author (so called “p-representation” model) [7]:

(

^/2

) [

¹

(

^{2 /}

)

¹

]

) 0 ( )

( = _G − Θ_p ^p + Θ_p ^p −

G T E T

E α _{, (7)}

where index p is related to the phonon dispersion byp≈ 1/∆²+1 and Θp is roughly comparable with the average phonon temperature.

2.3.2 Doping and plasma induced band-gap narrowing

The width of the band-gap also monotonically decreases with increasing e-h density due to the exchange and correlation effects. Carriers of opposite

charge are attracted by the Coulomb interaction, while carriers of like charge are repelled. The energies of the attraction and repulsion cancel each other if electrons and holes are randomly distributed. However, the exchange interaction of identical fermions (the Pauli principle) forbids two carriers with parallel spin from occupying the same energy state. This fact increases the average distance between the carriers and reduces the total repulsive energy. Moreover, the repulsion of one type of carriers redistributes the system in such a way that in the vicinity of some particular hole electrons are found with higher probability than other holes and vice versa, increasing the total correlation energy. As a result, the total energy of the interacting particles becomes lower, lowering the effective band-gap.

Calculations of the doping induced band-gap narrowing (BGN) are performed in the following way: in an unperturbed (intrinsic) semiconductor the Hamiltonian H0 describes interactions between the valence electrons and ions of the intrinsic crystal. Solution of the H0 eigenvalue problem results in single- particle energies Ej0(k) for electron states with wave vector k in the j^th energy band. However, in an n-type semiconductor there are also donors, which above the Mott transition [8] produce free carriers in the conduction-band. Thus, additional interactions arise between electrons and electrons, electrons and ionized impurities and, in polar material, electrons and optical phonons. For p- type material the situation is analogous if the appropriate substitutions for holes,

Fig. 2. Doping induced BGN in Si (left) and 4H-SiC (right) [13].

acceptors and the valence-band are made. The interactions can be described using the perturbation Hamiltonian H1 resulting in new single electron energies Ej(k). (The total Hamiltonian is then H = H0 + H1). The energy shift Ej(k) - Ej0(k)

= ^Re

{

_hΣj

[

^k,E0j^(k)/_h

] }

is the real part of the self-energy of the electron state. This self-energy originates from the electronic self-energy operator, which is a kind of dynamic, non-local potential that represents the effect of the electronic exchange and correlation on the motion of each individual electron. The operator is expressed in terms of the unperturbed electron Green’s function Gj0(k,ω) and the dielectric function ε(k’-k,ω). Calculations of the dielectric screening are performed in the simplified single plasmon-pole approximation [9]

or in the more exact random phase approximation (RPA) [10,11], which is capable of giving the exact energy dispersion of the band structure. These models usually assume zero temperature formalism and neglect local fluctuations in the density of impurities. Averaging over statistical fluctuations may be performed, if necessary, at the end of the calculations leading to the band tailing effect [12]. BGN calculated in n-type Si and 4H-SiC within the RPA method is illustrated in Fig. 2 [13]. The solid lines represent fundamental BGN ∆EG, while the dashed ones show BGN detected in the optical type of measurements ∆EGopt = ∆EG + EF, where EF is the Fermi energy of the e-h system. Note that the BGN values presented are the sum from almost equal effects in both conduction and valence bands, though only one type of carriers is present in the semiconductor.

The doping induced BGN is additionally explained on the top of Fig. 4, where the horizontal axis represents the density of states. The constant shift of

Fig. 3. Plasma induced BGN in Si (left) and 4H-SiC (right) [14].

the conduction and valence band edges towards each other with increasing doping (the diagrams from left to right, respectively) stands for the theoretically calculated rigid band-gap shift: EG - ∆EG. In the experiments, however, one detects the optical band-gap, which appears above the Fermi energy: EGopt = EG -

∆EG + EF. The Mott transition in the picture corresponds to the moment when the spreading donor level merges with the conduction band. The band tailing

E N(E)

E_G-∆E_G

E_G E_G^opt

Mott transition

Band tailing Donor

E_F

E N(E)

E_G-∆E_G

E_G E_G^opt

Mott transition

Band tailing Donor

E_F

E_G-∆E_G

E_G E_G^opt

E_Fe

E_Fh

∗

∗

E_Fe

E_Fh

∗

∗ E_G-∆E_G

E_G E_G^opt

E_Fe

E_Fh

∗

∗

E_Fe

E_Fh

∗

∗

Fig. 4 Schematic representation of the doping (top) and plasma (bottom) induced BGN. The top picture also explains the Mott

transition and the band tailing effects.

effect on the other hand represents the band states spreading below the rigid band-gap EG - ∆EG.

Optical excitation of a specimen creates an e-h plasma consisting of equal amounts of electrons and holes. The basic description of the BGN effect in this case is, nevertheless, analogous to that in the doped material, except that the major interactions now appear between electron-electron, hole-hole, electron- hole and, in polar material, electron-optical phonon and hole-optical phonon.

BGN calculated in the e-h plasma of Si and 4H-SiC is illustrated in Fig. 3 [14], where solid and dashed lines have the same meaning as in the case of the doped material. The plasma induced rigid band-gap shift EG - ∆EG is also illustrated on the bottom of Fig. 4. Contrary to the case of the doping induced BGN, however, the Mott transition and the band tailing effect do nor appear in an excited material and the optical band-gap is now determined by the quasi-Fermi levels in the both conduction and valence bands: E_G^opt = E_G - ∆E_G + E_Fe*. + E_Fh*.

References

[1] K. J. Malloy and J. A. VanVechten, J. Vac. Sci. Technol. B 9 (4), 2212 (1991).

[2] M. L. Cohen and D. J. Chadi, in Semiconductor Handbook, edited by M.

Balkanski, Vol. 2, Chap. 4b (North-Holland, Amsterdam, 1980).

[3] V. Heine and J. A. VanVechten, Phys. Rev. B 13, 1622 (1976).

[4] Y.P. Varshni, Physica 34, 149 (1967).

[5] R. Passler, Phys. Rev. B 66, 085201 (2002).

[6] K. P. O’Donnell and X. Chen, Appl. Phys. Lett. 58 (25), 2924 (1991).

[7] R. Passler, Phys. Stat. Sol. (b) 216, 975 (1999).

[8] N. F. Mott, in Metal-Insulator Transitions (Taylor and Francis, London, 1974).

[9] J. C. Inkson, J. Phys. C: Solid State Phys. 9, 1177 (1976).

[10] R. A. Abram, J. Phys. C: Solid State Phys. 17, 6105 (1984).

[11] K. F. Berggren, Phys. Rev. B 24, 1971 (1981).

[12] P. Van Mieghem, Rev. Mod. Phys. 64, 755 (1992).

[13] C. Persson, Phys. Rev. B 60, 16479 (1999).

[14] C. Persson, Solid State El. 44, 471 (2000).

3 Carrier transport

3.1 Diffusion and mobility

The following is a short introduction to the diffusion coefficients for free carriers and their injection dependence. The focus is on the high carriers densities, where the unified motion of correlated electrons and holes results in ambipolar diffusivity. Certain physical phenomena, which effect simplified predictions based upon the continuity equation in this range, will be commented on.

3.1.1 Transport equations and ambipolar diffusion

Carrier (electron is assumed unless otherwise specified) current density in a semiconductor consists of both drift and diffusion components:

n eD nE e

J= µ − ∇ , (1)

where E and n∇ stand for the electric field and the gradient of the carrier concentration with the corresponding proportionality constants µ, the carrier mobility, and D, the carrier diffusion coefficient. Under equilibrium conditions J

= 0 and:

n D n

nE=µ (−∇Φ)= ∇

µ , (2)

where Φ is the electrostatic potential. In the case of non-degeneracy the carrier concentration in a band obeys a classical exponential dependency:

⇒ Φ

−

×

=const exp( e /k T)

n _B (3a)

T k e n

n/ =− ∇Φ/ _B

∇ . (3b)

Substituting n∇ from Eq. (3b) into Eq. (2) one obtains the Einstein relation for the concentration of electrons (n) and holes (p):

p n e i

T

D_i=k^B µ_i, = , _{. (4)}

In general form, Eq. (3) obeys the carrier Fermi-Dirac statistics:

)

2(

/

1 η

NF

n= , (5a)

where

∫

∞

+

= −

0exp( ) 1

)

(η η

x dx F x

m

m , (5b)

is the Fermi integral. N represents the concentration of donors or acceptors and η equals (E_Fn−E_c)/k_BT for electrons and (E_v−E_Fp)/k_BT for holes, where EFn

and EFp are their respective quasi-Fermi levels and Ec and Ev are the edges of the conduction and valence bands. Performing the same procedure one obtains instead of Eq. (4):

i B

i e

T k F

D F µ

η η

) (

) (

2 / 1

2 / 1

−

= , (6a)

where

η η

∂

=∂

−

)

2(

/ 1 2 / 1

F F . (6b)

The Einstein relation as expected in Eq. (4) applies only to one type of carrier, while in a real semiconductor both types are present and related through the mass action law:

0 0

2 n p

n_i = , (7)

where n0 and p0 represent the electron and hole concentrations at thermal equilibrium. In the case of inter-band excitation these concentrations change to n

= n0 + ∆n and p = p0 + ∆p (∆n = ∆p). The carrier distribution of each type, nevertheless, maintains charge neutrality resulting in the continuity equation

n R G e J

t

n+ ∇ = − ∆

∂

∂ 1

, (8)

where G is the generation term and R stands for the recombination rate. When analyzing processes after an excitation (G = 0) Eq. (8) becomes [1]: