Charge Transport in Single-crystalline CVD Diamond

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(197) List of Papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I J. Isberg, M. Gabrysch, A. Tajani, and D. J. Twitchen, Transient current electric field profiling of single crystal CVD diamond, Semiconductor Science and Technology 21 (8), 1193–1195 (2006). II J. Isberg, M. Gabrysch, A. Tajani, and D. J. Twitchen, High-field Electrical Transport in Single Crystal CVD Diamond Diodes, Advances in Science and Technology 48, 73–76 (2006). III M. Gabrysch, E. Marklund, J. Hajdu, D. J. Twitchen, J. Rudati, A. M. Lindenberg, C. Caleman, R. W. Falcone, T. Tschentscher, K. Moffat, P. H. Bucksbaum, J. Als-Nielsen, A. J. Nelson, D. P. Siddons, P. J. Emma, P. Krejcik, H. Schlarb, J. Arthur, S. Brennan, J. Hastings, and J. Isberg, Formation of secondary electron cascades in single-crystalline plasma-deposited diamond upon exposure to femtosecond x-ray pulses, Journal of Applied Physics 103 (6), 064909 (2008). IV M. Gabrysch, S. Majdi, A. Hallén, M. Linnarsson, A. Schöner, D. J. Twitchen, and J. Isberg, Compensation in boron-doped CVD diamond, Physica Status Solidi (a) 205 (9), 2190-2194 (2008), presented at the Diamond Workshop 2008, SBDD XIII, Hasselt (Belgium). V J. Isberg, S. Majdi, M. Gabrysch, I. Friel, and R. S. Balmer, A lateral time-of-flight system for charge transport studies, Diamond & Related Materials 18, 1163–1166 (2009). VI J. Isberg, M. Gabrysch, S. Majdi, and D. J. Twitchen, Negative differential electron mobility and single valley transport in diamond, submitted to Nature Materials, April 2010. VII M. Gabrysch, S. Majdi, D. J. Twitchen, and J. Isberg, Electron and hole drift velocity in CVD diamond, submitted to Physical Review B, April 2010. Reprints were made with permission from the publishers..

(198) The author has contributed to the following papers which are not included in the thesis.. VIII S. Majdi, M. Gabrysch, R. S. Balmer, D. J. Twitchen, and J. Isberg, Characterization by Internal Photoemission Spectroscopy of Single-Crystal CVD Diamond Schottky Barrier Diodes, accepted for publication in Journal of Electronic Materials, DOI: 10.1007/s11664-010-1255-8, April 2010. IX C. Caleman, C. Ortiz, E. Marklund, F. Bultmark, M. Gabrysch, F. G. Parak, J. Hajdu, M. Klintenberg, and N. Tîmneanu, Radiation damage in biological material: electronic properties and electron impact ionization in urea, Europhysics Letters 85, 18005 (2009)..

(199) Contents. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Common allotropes of carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Natural diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Synthetic diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Diamond properties and applications . . . . . . . . . . . . . . . . . . . . . 1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Diamond as a semiconductor material . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Semiconductor materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Electrical properties of semiconductors . . . . . . . . . . . . . . . . . . . . 2.2.1 Energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Intrinsic carrier concentration . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Advantages of the semiconductor diamond . . . . . . . . . . . . . . . . 2.4 CVD diamond synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Doping diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Future diamond devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Principles of the time-of-flight technique . . . . . . . . . . . . . . . . . . . . . . 3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mobility measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Low injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 High injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Data acquisition and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Free charge carrier transport in diamond . . . . . . . . . . . . . . . . . . . . . . 4.1 Drift-diffusion equations from the BTE . . . . . . . . . . . . . . . . . . . . 4.1.1 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . 4.1.2 Equilibrium distribution function for a Fermi gas . . . . . . . 4.1.3 Uniform electric field with RTA . . . . . . . . . . . . . . . . . . . . . . . 4.2 Charge transport in 1-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Fundamental transport equations . . . . . . . . . . . . . . . . . . . . 4.2.2 Carrier generation by laser, low injection . . . . . . . . . . . . . . . 4.2.3 Carrier transit with homogenous space charge density and trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Carrier diffusion during transit . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Carrier extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Full transit signal and fast processes . . . . . . . . . . . . . . . . . .. 11 11 12 12 13 14 17 17 18 18 19 20 22 24 25 26 27 29 29 30 30 33 34 37 37 37 38 39 42 42 44 45 47 47 48.

(200) 4.3 Electrical field profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Carrier transit simulations in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . 5 Electron cascades in diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Diamond as a detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Pair-creation from ionising radiation . . . . . . . . . . . . . . . . . . . . . . 5.3 Impact ionisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary of results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Investigation of δ-doped structures . . . . . . . . . . . . . . . . . . . . . . . 9.2 High-voltage low-loss converters . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Time-resolved study of electron cascades . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 51 55 55 56 58 61 65 71 73 73 74 74 77 79 83.

(201) Nomenclature and abbreviations. | |X. magnitude of X. e. electron (as index). [X ]. concentration of element X. electrical potential (external) force. ⟨...⟩. crystallographic direction. φ F. ∼. approximately. F. Fermi-Dirac distribution. α. ionisation rate. f. distribution function. αn/p. electron/hole ionisation rate. f0. equilibrium distr. function. a p /b p. hole ionisation coefficients. g. generation rate. β. correction factor for SCL case. ga. spin degeneracy factor. C. capacitance. ħ. reduced Planck’s constant. c. speed of light. h. Planck’s constant. ∂. partial derivative. h. hole (as index). D. diffusion constant. I. current. d. differential. i. intrinsic (as index). d. sample thickness. j. current density. ε0. vacuum permittivity. k. Boltzmann’s constant. εr. relative permittivity. λ. wavelength. pc E ap E. average pair-creation energy. μ. (drift) mobility. electric field. m∗. effective mass. applied electric field. ∇. del operator. sc E. space charge electric field. NA. acceptor doping concentration. E. energy. NC. effective DOS in conduc. band. EA. acceptor ionisation energy. ND. donor doping concentration. EC. E at bottom of conduc. band. NV. effective DOS in valence band. EF. Fermi level. n. electron density. E F. quasi-Fermi level. ni. intrinsic carrier concentration. EV. E at the top of valence band. p. momentum. Eg. bandgap energy. p. hole density. E kin. kinetic energy. Q. total charge. E pot. potential energy. q. elementary charge. E th. thermal energy. ρc. carrier concentration. E tot. total energy. ρ sc. space charge concentration. eˆx. unity-vector in x-direction. r. recombination rate. 9.

(202) 10. σ. variance. V. voltage. τ. time-of-flight. vd. drift velocity. τf. relaxation time. v sat. saturation (drift) velocity. T. (absolute) temperature. v th. thermal velocity. U. (bias) voltage. Z. atomic number. 1-D. one-dimensional. NDM. neg. differential mobility. AC. alternating current. RT. room temperature. BTE. Boltzmann transport eq.. RTA. relaxation time. C. carbon / diamond. CVD. chemical vapour deposition. SC. single-crystalline. DC. direct current. SCL. space charge limited. DOS. density-of-states. SIMS. Secondary Ion Mass. FEM. finite element method. FWHM. full width at half maximum. TCT. transient-current technique. GUI. graphical user interface. ToF. time-of-flight. HPHT. high-pressure high-temp.. UV. ultraviolet. IR. infrared. VUV. vacuum ultraviolet. MC. Monte Carlo. XFEL. X-ray free-electron laser. approximation. Spectrometry.

(203) 1. Introduction. Diamonds have been known as gemstones for several thousand years and were recognised by various early cultures for their religious or industrial uses [1]. The word “diamond” has its origin in the Ancient Greek word “adámas / ἀδάμας” meaning invincible. Besides the property of being the hardest known natural material, diamond is mainly appreciated as a gemstone because of its optical properties: the high refractive index and large colour dispersion result in a unique brilliance.. 1.1 Common allotropes of carbon Carbon is the lightest Group IV element in the periodic table, having a half-filled valence shell with the electronic configuration s 2 p 2 . In the case of diamond these s- and p-states hybridise and form the extremely strong tetrahedral sp 3-bonds. Together with the special three-dimensional arrangement of the atoms in the lattice, the so-called diamond structure, they make diamond so exceptionally hard and also lead to other amazing intrinsic properties such as a high refractive index, extremely high thermal conductivity, and a high melting point. Besides diamond, pure carbon can also be formed as graphite, amorphous carbon, graphene, fullerenes (e.g. buckyballs, nanotubes, nanowires) or lonsdaleite (hexagonal crystal lattice) just to mention some of the best known. Graphite is the most common form of pure carbon on Earth and in contrast to diamond, each carbon shares one electron with two of its neighbours, and two electrons with the third neighbour. The atoms all bond in planes (two-dimensional hexagonal lattice) which are stacked on top of each other resulting in quite weak forces between different planes. Graphite is therefore a rather soft material and differs also considerably in other physical properties from diamond. At normal temperatures and pressures, graphite is thermodynamically favoured, as can be seen from the phase diagram for carbon. The fact that diamond exists at all is due to the very large activation barrier for conversion between the two. In the absence of an easy interconversion mechanism, a phase transition would require almost as much energy as destroying the entire lattice and rebuilding it. Because the barrier is too high, once. 11.

(204) formed, diamond cannot reconvert to graphite. That is why diamond is said to be metastable: it is kinetically stable, but not thermodynamically stable [2].. 1.2 Natural diamond Diamond can form naturally at depths greater than 150 km in the upper mantle of the Earth [3]. Under conditions of extreme pressure and temperature it is the most stable form of carbon. Over a period of millions of years carbonaceous deposits slowly crystallise into single crystal diamond gemstones. Together with magma the crystals are then brought up to the Earth’s crust by kimberlite or lamproite volcanic eruptions. Since this transfer happens within just a few hours, the conversion to graphite does not occur and it is possible to mine diamond in the volcanic pipes containing material that was transported toward the surface by volcanic action, but was not ejected before the volcanic activity ceased. Natural diamond generally contains a significant degree of impurities such as nitrogen or boron. In order to classify the purity of diamond the following scheme is used [1]: – Type Ia diamond contains nitrogen in fairly substantial amounts (in the order of 0.1%). The majority of natural diamonds are of this type. – Type Ib diamond also contains nitrogen but in dispersed substitutional form. Almost all synthetic diamonds are of this type. – Type IIa diamond is effectively free of nitrogen. Diamonds of this type have enhanced optical and thermal properties but are rare in nature. – Type IIb diamond is a very pure type which has p-type semiconducting properties. It is extremely rare in nature and has (uncompensated) boron acceptor impurities which result in a light blue colour.. 1.3 Synthetic diamond Diamond synthesis can be achieved through several routes. The two main methods are high-pressure high-temperature synthesis (HPHT) and chemical vapour deposition (CVD). HPHT diamond Ever since the discovery that diamond was pure carbon in 1797, many attempts were made to convert inexpensive graphite into gemstones. However, it was not before the 1950s that this could be achieved in a reproducible and verifiable way [4–6]. Both Allmänna Svenska Elektriska Aktiebolaget (ASEA) in Sweden and General Electric in the United States developed a technique to raise the pressure in the reaction chamber to more than 12.

(205) 8 GPa and the temperature to above 2000 ℃ in order to reproduce the conditions under which natural diamond forms inside the Earth. (Without the use of suitable catalysts conditions must be even more extreme.) This catalytic high-pressure high-temperature method is the most widely used technique for diamond synthesis today, because of its relatively low cost. The amount of annually produced HPHT diamond exceeds the one of mined natural diamond by approximately a factor of four. Today, HPHT diamond is commonly used in many industrial applications such as cutting, drilling, thermal management etc. However, it invariably contains many crystal defects and impurities. The yellow colour indicates a relatively high nitrogen concentration. CVD diamond The second method, using chemical vapour deposition, was first applied in the 1980s, and basically creates a carbon plasma on top of a substrate onto which the carbon atoms deposit to form diamond [6]. In contrast to HPHT diamond, it is possible to grow CVD diamond under conditions of high purity resulting in fewer defects and impurities. In the beginning, the deposition conditions have invariably resulted in polycrystalline material. Only since the last couple of years it is also possible to grow thick free-standing plates of high-purity single-crystalline CVD (SCCVD) diamond by homoepitaxy [7]. The CVD process is discussed in more detail in Sec. 2.4.. 1.4 Diamond properties and applications Diamond is an outstanding material, as Table 1.1 shows. It is the hardest known material, has the lowest coefficient of thermal expansion, is chemically inert and wear-resistant, offers low friction, has the highest thermal conductivity, is electrically insulating and optically transparent from the ultra-violet (UV) to the far infra-red (IR) region (> 70% transmission for λ > 1μm but 60% at ∼ 5μm). Because of these properties, diamond is already used in many diverse applications besides its appreciation as a gemstone. The most common are: – Mechanical applications: abrasive and wear-resistant coatings for cutting tools such as drills, saws, knives, glass cutting and wire dies – Optical applications: lenses, windows for high power lasers and diffractive optical elements – Thermal applications: heat sinks for power transistors and semiconductor laser arrays – Detector applications: “solar blind” photodetectors, radiation-hard and/or chemically inert detectors. 13.

(206) Table 1.1: Some of the outstanding properties of diamond, taken from [8]. Property. Description. Extreme mechanical hardness. ∼ 90 GPa. Highest bulk modulus. 1.2 × 1012 N/m2 8.3 × 10−13 m2 /N. Lowest compressibility Lowest thermal expansion coefficient at RT. a). 0.8 × 10−6 K−1. Highest thermal conductivity at RT. 24 W/(cm K). Good electrical insulator. R 1016 Ω cm. Semiconductor if doped. R 10–106 Ω cm. Wide bandgap. 5.47 eV at RT. Broad optical transparency. from deep UV to far IR. Biologically compatible. nontoxic and tissue equivalent. Low electron affinity. even negative in some cases. Very low coefficient of friction. ∼ 0.001 in water. High resistance to wear and chemical corrosion a) RT stands for room temperature.. Given its many unique properties, it is clearly possible to imagine numerous other potential applications for diamond as an engineering material. However, progress in implementing many such ideas has been constricted by the comparative shortage of (natural) diamond. Most of the electrical applications of diamond are just at their infancy because only for the last couple of years have electronic grade SC-CVD diamond become available for device design and development [7].. 1.5 Outline of the thesis This doctoral thesis deals with several aspects of the electronic properties of single-crystalline CVD diamond, with a special focus on charge transport. It is organised as follows: Chapter 2 is a recap of diamond as a semiconductor material including the CVD process, diamond doping and devices. Chapter 3 focuses on mobility measurements (both with low and high injection) by using the time-of-flight (ToF) technique. Chapter 4 studies charge transport both analytically and using finite element simulation tools. In the same context the electric field profiling method is explained. Chapter 5 is about electron cascades in diamond initiated by ionising radiation such as X-rays or α-particles. Besides the study of CVD diamond as a detector material and model compound, focus is also put on the investigation of important mechanisms such as pair-creation and 14.

(207) impact ionisation that can lead to avalanche breakdown in devices. The last chapters include a summary of the papers the author has contributed to (Chapter 6), a summary of results and their discussion (Chapter 7), the conclusion (Chapter 8) and suggestions for future work (Chapter 9).. 15.

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(209) 2. Diamond as a semiconductor material. In this chapter some of the basic properties of semiconductor materials in general and those of diamond in particular are discussed. The overview of semiconductor materials and some of their electrical characteristics such as energy bands, intrinsic carrier concentration and mobility is followed by a compilation of the advantages of diamond over other semiconductor materials. A description of the single-crystalline CVD diamond growth technique is given before briefly discussing doping diamond and the importance of having low compensation. The chapter is completed by an overview of possible future diamond devices.. 2.1 Semiconductor materials Solid-state materials can be grouped into three classes: conductors, semiconductors, and insulators [9]. Conductors have high conductivities (i.e. low resistances), typically on the order of 104 –106 S/cm; and insulators have very low conductivities, ranging from 10−18 to 10−8 S/cm. The conductivity of semiconductors lies in between those of conductors and insulators and is in general sensitive to temperature, illumination, magnetic field, and minimal amounts of impurity atoms (∼ 1–100 ppm). It is this sensitivity in conductivity which makes semiconductors so important for electronics. Group II. III. IV. V. VI. B. C. N. O. Al. Si. P. S. Zn. Ga. Ge. As. Se. Cd. In. Sn. Sb. Te. Hg. Tl. Pb. Bi. Po. Metals Non-metals. Figure 2.1: Semiconductor related elements of the periodic table. Metals are shown in orange and nonmetals in yellow. Elements with intermediate colours show intermediate properties.. 17.

(210) Figure 2.1 shows the part of the periodic table which is related to semiconductors. The element semiconductors can be found in Group IV and are composed of just a single element such as diamond (C), silicon (Si) or germanium (Ge). If two or more elements from the periodic table are combined, one talks about a compound semiconductor. For example, if the Group III element gallium (Ga) and the Group V element arsenic (As) are combined, the binary III–V compound gallium arsenide (GaAs) is formed. Furthermore, there are also II–VI compounds such as zinc oxide (ZnO) and IV–IV compounds such as silicon carbide (SiC). Ternary and quaternary compounds are formed of three and four elements, respectively, but involve relatively complex processes to prepare them in single-crystalline form. A list of common semiconductors is given below: element semiconductors:. C, Si, Ge. IV–IV semiconductors:. SiGe, SiC. III–V semiconductors:. GaAs, InSb, GaN, AlN, GaP, AlAs, InP, .... II–VI semiconductors:. ZnO, CdS, CdSe, CdTe, .... With the advent of semiconductor electronics in the 1950s, germanium was the major semiconductor material, but silicon has virtually supplanted it since the 1960s because of its superior properties at room temperature (RT) and lower costs. Nowadays, silicon is one of the most studied elements in the periodic table. Device-grade silicon costs much less than any other semiconductor material and silicon technology is by far the most advanced among all semiconductor technologies [9]. However, besides these advantages, there are also major drawbacks of silicon-based electronics. This is the motivation for establishing alternative semiconductor materials such as gallium arsenide, silicon carbide, gallium nitride or – diamond, the object of interest of this thesis.. 2.2 Electrical properties of semiconductors 2.2.1 Energy bands The electrical properties of solid state materials can be analysed by looking at their band structure [9, 10]. Energy bands form as a quantum mechanical consequence (Pauli’s exclusion principle) when isolated atoms are brought together forming a crystal. One distinguishes between the valence band, which consists of the electrons forming the chemical bonds, and the conduction band, which consists of electrons of higher energies which can move freely across the crystal. Electronic conduction can only take place within these bands, i.e. at least one band should be partially populated by electrons.. 18.

(211) In the case of conductors/metals, electrons are free to move with only a small electrical field applied since there exist many unoccupied energy states close to the occupied ones at all temperatures. This is because the conduction band is either partially filled or overlaps the valence band. The case for semiconductors and insulators is different. At zero absolute temperature, electrons can only occupy the lowest energy states. This means that all states in the lower band are occupied and all states in the upper band are unoccupied and thus no conduction can take place. The bottom of the conduction band is called E C , and the top of the valence band is called E V . The bandgap energy E g = E C − E V between these two levels is the width of the forbidden energy gap. It is the amount of energy required to break a bond in the crystal, i.e. to free an electron to the conduction band and to leave a hole in the valence band. In an insulator, the valence electrons form strong bonds between neighbouring atoms and consequently these bonds are difficult to break. Thus, the bandgap is large and there are no free electrons to participate in current conduction at or near room temperature. For a semiconductor the situation is similar but the energy gap is much smaller, on the order of a few electronvolts. This leads to poor conduction at low temperatures, but when the thermal energy E = kT constitutes a larger fraction of the bandgap energy, an appreciable number of electrons are thermally excited from the valence to the conduction band. This excitation does not necessarily have to be thermal. For example, it can also happen by illumination, ionising radiation or strong electric fields. These cases will be discussed in Chapters 3, 4 and 5. The conduction at lower temperatures can be improved by doping – the process of intensionally introducing impurity levels into the bandgap. If these levels are shallow, i.e. close to E C or E V , they can be activated thermally even at low temperatures. Sec. 2.5 will deal with the doping of diamond.. 2.2.2 Intrinsic carrier concentration An intrinsic semiconductor is one that contains relatively small amounts of impurities compared to the thermally generated electrons and holes [9]. In order to obtain the electron concentration (the number of electrons per unit volume) in an intrinsic semiconductor, one has to integrate the electron density n(E ) in an incremental energy range dE from the bottom to the top of the conduction band. The density n(E ) is the product of the Fermi-Dirac distribution F (E ) and the density of allowed energy states per energy range per unit volume N (E ). The Fermi-Dirac distribution e −(E −E F )/kT for (E − E F ) > 3kT 1 F (E ) = 1 + e (E −E F )/kT 1 − e −(E −E F )/kT for (E − E F ) < 3kT 19.

(212) gives the probability at the absolute temperature T that an electron occupies an electronic state with energy E . The Fermi level E F is by definition the energy at which this probability is one-half and k is the Boltzmann constant. The integral returns the electron density in the conduction band n = NC e −(E C −E F )/kT. (2.1). where NC is the effective density-of-states (DOS) in the conduction band. With m e∗ being the density-of-states effective electron mass and h Planck’s constant, it reads in the case of diamond: 2π m e∗ kT 3/2 . (2.2) NC = 2 h2 The hole density p in the valence band can be obtained in a similar way: p = NV e −(E F −E V )/kT. (2.3). where NV is the effective DOS in the valence band. For diamond it reads: 2π m h∗ kT 3/2 (2.4) NV = 2 h2 with m h∗ being the density-of-states effective hole mass. For an intrinsic semiconductor, the number of electrons per unit volume in the conduction band equals the number of holes per unit volume in the valence band, and the intrinsic carrier concentration n i can be calculated from the mass action law which reads np = n i 2 . By using Eqs. (2.1) and (2.3) it follows that (2.5) n i 2 = np = NC NV e −E g /kT . The intrinsic carrier concentration in diamond is shown in Fig. 2.2 as a function of temperature. It can be clearly seen that, because of the wide bandgap, the intrinsic charge carrier concentration is very low and becomes only significant for temperatures exceeding approx. 1000 ℃.. 2.2.3 Mobility The electrons in the conduction band and the resulting holes in the valence band can be considered as free particles that follow the classical equation of motion but have an effective mass that differs from the free electron mass in order to incorporate the fact that they move in a crystal lattice [9]. For finite temperatures, the charge carriers move rapidly and completely randomly in all directions because of thermal excitation. When applying an they will experience a force −q E and become accelerated in electric field E that direction until they collide with lattice atoms, impurity atoms or other 20.

(213) 1018 1017 1016. -3. ni (cm ). 1015 1014 1013 1012 1011 1010 109 108 1000. 1500. 2000. 2500. T (°C) Figure 2.2: Intrinsic carrier concentration in diamond as a function of temperature.. scattering centres. The velocity of the charge carriers will then consist of the thermal velocity v th and the additional drift velocity vd. For small electric fields the drift velocity is much lower than the thermal : velocity and v d is proportional to E v d = μE. (2.6). where the proportionality constant μ is called mobility. Since electrons and holes have different properties, such as effective masses, one distinguishes between electron mobility μe and hole mobility μh . When increasing the electric field, the drift velocity | v d | starts to saturate. For diamond, silicon and germanium, it increases monotonically at RT and converges towards v sat , the so-called saturation (drift) velocity. This behaviour is well described by the empirical model [9, 11]: vd =. μE 1+. | μ|E v sat. ,. (2.7). plotted in Fig 2.3. For diamond, the electron saturation velocity at room temperature is ∼ 2 × 107 cm/s – twice as high as for silicon. Mobility and saturation velocity are important parameters for carrier transport because they describe how strongly the motion of the charge carrier is influenced by an applied electric field. At high impurity or defect concentration mobility is limited by (impurity) scattering [12]. Therefore, a high mobility is also an indicator of low impurity or defect concentration. 21.

(214) Drift velocity vd (cm/s). 20x106. 15x106. 10x106. chosen parameters: vsat = 2 x 107 cm/s. 5x106. μ = 4500 cm2/Vs 0 0. 10. 20. 30. 40. 50. Electric field E (kV/cm). |. The gray straight line Figure 2.3: Drift velocity | v d | versus applied electric field |E is a fit to weak electric fields and therefore reflects Eq. (2.6).. 2.3 Advantages of the semiconductor diamond After this recap of general semiconductor properties, the advantages of SCCVD diamond as a semiconductor material will now be discussed. A selection of electronic and thermal properties [7, 13, 14] of some common semiconductor materials is shown in Fig. 2.4 and Tab. 2.1. Having a closer look shows clearly that diamond has many superior intrinsic properties compared to silicon, the state-of-the-art semiconductor.. GaAs Ge. InP. Diamond. 2. Electron + Hole mobility at 300K (cm /Vs). 10000. GaN. Si. AlAs. 1000. SiC AlSb. ZnSe. GaP. ZnO. CdS InN. ZnS. 100 AlP. AlN. 10 0. 1. 2. 3. 4. 5. 6. 7. Bandgap (eV). Figure 2.4: Added electron and hole mobility versus bandgap for different semiconductors. The area of the circles is proportional to the thermal conductivity.. 22.

(215) Table 2.1: Electronic and thermal properties [7, 13, 14] of some common semiconductor materials in comparison to diamond (C). The data refer to measurements at room temperature. Ge. Si. GaAs. 4H-SiC. GaN. C. Bandgap. 0.7. 1.1. 1.4. 3.2. 3.4. 5.5. eV. Breakdown field. 0.1. 0.3. 0.4. 3. 5. 20. MV/cm. Electron mobility. 3900. 1450. 8500. 900. 2000. 4500. cm2 /Vs. Hole mobility. 1900. 480. 400. 120. 200. 3800. cm2 /Vs. Thermal conductivity. 0.58. 1.5. 0.55. 3.7. 1.3. 24. W/cmK. The mobilities of diamond are very high and the electron mobility at RT is only exceeded by a few materials, e.g. GaAs or InP. High carrier mobilities are desirable for fast-response and high-frequency electronic devices. GaAs is mainly used for high-speed electronics, mobile phones and satellite communication [9], but it is considered highly toxic and carcinogenic. It has also the drawback of a low hole mobility, which makes it unsuitable for certain applications. High thermal conductivity is very suitable for power electronics where devices suffer from a high generation of heat [15]. For traditional power devices heat sinks have to be included in the structure to prevent the device from malfunction. For diamond devices this is not necessary because diamond has the highest thermal conductivity and it is already used as a heat sink for power transistors and semiconductor laser arrays [16]. As a wide bandgap semiconductor, diamond offers the benefit of thermal stability and a high breakdown field – the maximal field strength a material can withstand intrinsically without breaking down. Breakdown is discussed in more detail in Section 5.3. Diamond offers many advantages to other wide bandgap materials such as silicon carbide (SiC) or gallium nitride (GaN). Although diamond epitaxy is still in its infancy, growing epitaxial layers of diamond in a CVD process is in many ways simpler than growing other wide bandgap semiconductor materials. This is due to the simpler structure of diamond, consisting of carbon atoms only. Polyatomic materials such as SiC or GaN require careful control of the stoichiometry and they exist in many hundred different crystalline structures. Single-crystalline configurations of carbon, on the other hand, exist only in a few forms. Epitaxy of SiC is riddled with a particular problem, the formation of tubular channels, called micropipes, during growth [17]. This problem does not exist for diamond. Another advantage associated with growing diamond in a CVD process is that the raw materials are cheap and naturally abundant gases: methane and hydrogen, while epitaxy of SiC and GaN involves extremely toxic substances. 23.

(216) Of all wide bandgap semiconductor materials, diamond clearly has by far the most intriguing and extreme properties: mechanical, optical, thermal, as well as electrical. Diamond exhibits the highest breakdown field strength and thermal conductivity of any material and has the highest carrier mobilities of any wide bandgap semiconductor. Therefore, it enables the development of electronic devices with superior performance regarding power efficiency, power density, high frequency properties, power loss and cooling. See Sec. 2.7 for future application areas for diamond electronic devices.. 2.4 CVD diamond synthesis This section focuses on the process to grow diamond on a substrate. The schematic setup of the CVD synthesis is shown in Fig. 2.5. The substrate that shall be overgrown by diamond is placed in the reaction chamber. Typically for this type of synthesis, the temperature on the substrate is around 800 ℃ and the pressure is kept below 10 kPa [6]. Varying amounts of gases, mostly hydrogen but sometimes also argon (∼ 10%), and a carbon source such as methane (∼ 5%), are fed into the chamber and energised by microwave power. The chemical bounds are broken down and a plasma of highly reactive atoms is formed. The result is both diamond and graphite growth on the substrate. Atomic hydrogen in the plasma removesany graphite phase formed and eventually only diamond stays on the substrate. In contrast to HPHT diamond, it is possible to grow CVD diamond under conditions of high purity, resulting in fewer defects and impurities. Other advantages are the possibilities to grow diamond over larger areas, to deposit it on a substrate, and to control the properties of the diamond produced. This allows the addition of many of diamond’s important qualities to other materials: Coated valve rings or cutting tools benefit from diamond’s hardness and wear resistance. On extensive heat-producing components a diamond coating can function as a heat sink [16], as mentioned before. In the beginning, deposition conditions have invariably resulted in polycrystalline material. Only during the last couple of years has it become possible to grow thick free-standing plates of high-purity SC-CVD diamond by homoepitaxy. This was first achieved by DeBeers Industrial Diamonds1 in England [7]. Besides requiring a very careful control of the growth process, for single-crystalline diamond growth, the substrate itself has to be a single crystal diamond. Most commonly a specially prepared type Ib HPHT sample of high quality is taken2 . At the end of the process, the SC-CVD layer is separated from the substrate by a laser cutting technique and polished in order to obtain a free-standing plate. 1. Today known as Element Six Ltd. They provided the samples investigated in Papers I-VIII. Other substrates, such as silicon, are also possible, but due to the different lattice spacing only polycrystalline growth is possible then.. 2. 24.

(217) Microwave source Microwave cavity Quartz bell jar. Magnetron. Substrate H2 Gas in CH4 (Ag). Plasma. To vacuum pump Water cooling Figure 2.5: Schematic setup of the CVD synthesis of diamond.. 2.5 Doping diamond As mentioned previously, diamond has a very low concentration of intrinsic charge carriers at temperatures below 1000 ℃. If a significant room temperature conductivity is desired, it is necessary to dope diamond. When a semiconductor is doped with impurity atoms, it becomes extrinsic and impurity energy levels are introduced [9]. Unfortunately, there are no shallow dopants known for diamond. As shown in Fig. 2.6 both for p-type dopant boron (B) and for n-type dopants phosphorus (P) and nitrogen (N) the dopant levels are rather deep and result in low thermal excitation of free charge carriers at room temperature 1 (E th = 40 eV).. EC. conduction band P 0.52eV N 1.7eV. 5.47 eV B 0.37eV. EV. valence band. Figure 2.6: Activation energies for some common dopants in diamond.. 25.

(218) Type IIb natural diamond is p-type but extremely rare in nature. Since doping diamond through diffusion is not achievable, the first intentional doping of (natural) diamond was done by ion implantation at end of the 1960s [18]. However, with this technique a considerably amount of damage is done to the crystal lattice which cannot be reversed by annealing. With the advent of the diamond CVD process in the 1980s it became possible to add the dopants to the gas phase during growth. Only a few years later several groups demonstrated p-type boron doping of the diamond films from the gas phase [19–21]. This type of doping can, for instance, be achieved by a diborane B2 H6 addition to the H2 /CH4 /Ar source gas mixture [22]. This leads to a very homogenous distribution in the bulk and allows rather sharp interfaces by quickly changing the gas-phase boron concentration during the growth process (see e.g. Paper IV). n-type diamond does not exist in nature at all and it is even more difficult to form it artificially. However, it is also desirable to have efficient n-type diamond doping for electronic applications such as cold cathode electron emitters, UV photodetection and UV light emission diodes. The first success to form n-type diamond thin films by phosphorus doping with PH3 , CH4 , and H2 gas mixtures during the growth was reported in 1997 [23]. Four years later a UV light emission diamond diode with a pn-junction was demonstrated [24]. There are different approaches of how to circumvent the problem of low thermal excitation of free charge carriers at room temperature. One idea is to use two different layers. One layer is very highly doped and thus has a low mobility but a high number of free charge carriers which diffuse into a second (intrinsic) layer. This combines a high number of carriers with the excellent mobility in the intrinsic layer leading to desirable properties. The concept of the so-called pulse- or δ-doped structure is described in more detail in Sec. 9.1.. 2.6 Compensation In order to achieve successful diamond devices the ability to grow doped diamond films with low concentrations of defects and residual impurities is of very high importance. The presence of deep-level impurities in doped semiconductors causes compensation effects, i.e. electrons (holes), which in the absence of the impurity would have been emitted to the conduction (valence) band, are instead trapped, leading to a reduced free carrier concentration. For this reason, the compensation ratio, i.e. the ratio between the dopant and the compensating defect concentration, should be kept to a minimum. In diamond, known dopants are only partially thermally activated at room temperature. Because of this, very low compensation ratios are 26.

(219) necessary to achieve reasonable carrier densities. As an example, consider boron-doped diamond with a concentration of [B] = 1018 cm−3 . Without compensation, a room-temperature hole density of 2 × 1015 cm−3 is expected. For a compensation ratio ND /NA = 1% the hole concentration drops by almost a decade. Deep impurity states can also be undesirable in devices for other reasons. For example, a high concentration of impurities reduces carrier mobility, mainly by ionised-impurity scattering [12]. In a p-type semiconductor, normal band-conduction occurs in the valence band through the transport of holes, which result from the ionisation of acceptors at an energy E A above the valence-band edge. The hole concentration p in the valence band of a non-degenerate p-type semiconductor is the solution to the equation NV −E A /kT p(p + ND ) − n i 2 = e ≡ NV. 2 NA − ND − p − n i /p ga. (2.8). assuming that both acceptors and compensating donors are present with concentrations NA and ND , respectively, and with NA > ND , see e.g. [25]. In the above equation, g a is the spin degeneracy factor for the valence band and E A the acceptor ionisation energy. Because of diamond’s large bandgap (E g = 5.47eV) one can neglect the intrinsic carrier concentration and obtains for the hole concentration N + ND 2 N + ND V , (2.9) + NV (NA − ND ) − V p= 2 2 which is a function of temperature. By measuring the hole concentration at different temperatures and fitting the data to the equation above, the ionisation energy of the dopant and the impurity concentrations can be obtained. This was done in Paper IV.. 2.7 Future diamond devices This section gives an overview of possible future application areas for diamond electronic devices summarised in Tab. 2.2. Present-day power electronics, mainly based on silicon devices, exhibit a number of serious problems and limitations. Many of these limitations are inherent in the material itself and the only way to make significant progress is to switch to other semiconductor materials. Some of the most significant problems are: high losses, the difficulty to meet high switching frequencies, and limitations on voltages that can be attained [9]. Diamond power devices have the potential to increase efficiencies and reduce total system. 27.

(220) costs due to diamond’s ability to operate successfully at higher voltages than silicon-based devices or even other wide bandgap materials such as SiC or GaN. Diamond power devices also play an important role for a faster introduction of hybrid vehicles. With present technology, separate cooling systems for both the combustion engine and the electro-motor’s converter system are necessary. Using just one (combined) cooling system would result in significantly smaller and lighter motor blocks. The combination of the two systems is possible with diamond power devices because they can operate at such high temperatures due to diamond’s wide bandgap. In this context it is worth noting that more efficient converters also means higher total efficiency of the vehicle and reduced emissions. Table 2.2: Future application areas for diamond electronics. Area/device. related properties. Power electronics. high breakdown field, high thermal conductivity. HV diodes, Power MESFETS RF and Microwave HF MESFETS. high saturation velocity, high breakdown field high thermal conductivity. High Temperature Electronics. wide bandgap, chemical inertness. Photoconductive Switches. high breakdown field, high carrier mobilities. for Pulsed Power applications Radiation and UV detectors. 28. radiation hardness, solar blindness.

(221) 3. Principles of the time-of-flight technique. The time-of-flight (ToF) method1 is of great importance for the study of sample properties such as low-field drift mobilities and charge trapping [26]. This chapter deals with the basic principles by first describing the experimental setup and then explaining how drift mobility and saturation velocity can be measured for the low and high injection regime.. 3.1 Experimental setup The idea behind how to study low-field drift mobility by ToF measurements is quite simple. A sample with semitransparent contacts is illuminated with photons of high enough energy in order to create electron-hole pairs [7, 27, 28] hc . (3.1) λ< Eg In the case of diamond, the bandgap E g is 5.47 eV and hence the wavelength has to be below 226 nm, i.e., we need UV or X-ray photons. Alternatively, other sources of ionising radiation such as α-particles [29], β-particles [30] or pulsed electron beams [31] can be utilised for electron-hole pair creation. When applying an electric field across the sample, the created charge carriers experience a force and will drift. In order to measure the velocity of the charge carriers, a pulsed source with a pulse length much shorter than the time-of-flight is needed. The source should preferably provide a trigger signal to the oscilloscope where the induced current is measured. A second constraint for measuring the velocity is that the distance travelled by the charge carriers is known which means that all charge should be created at the close proximity of the illuminated contact. This is fulfilled for just above bandgap radiation that has a penetration depth of only ∼ 5 μm in diamond. Compared to the standard sample thickness of around ∼ 500 μm this can mostly be neglected.. 1. An alternative name in the literature is “transient-current technique” (TCT).. 29.

(222) The semitransparent contacts allow both illumination and the application of a rather homogenous field across the thickness of the sample. Titanium (Ti) is commonly used as injecting contact metal since it can form a carbide phase at the interface2 during contact annealing at around 450 ℃ [34]. The Ti layer has to be covered by a less oxidising metal such as gold (Au) or aluminum (Al). Figures 3.1 and 3.2 show a patterned diamond sample with a magnification of the mesh contact and the setup for the ToF measurements presented in Papers I,V,VI and VII.. 3.2 Mobility measurements When measuring the low-field drift mobilities of the charge carriers by creating electron-hole pairs of total charge Q within a sample of capacitance C and determining their drift velocity under an applied bias voltage of known magnitude U , three cases are distinguished: – low injection regime: Q C ·U and therefore the applied voltage results | = |U |/d (if charge trapping can be in an homogenous electric field |E neglected and d is the sample thickness). – high injection regime: Q

(223) C · U implies that the electric field is inhomogeneously distributed in the sample which has to be corrected for by a factor β (see below) that can be obtained from simulations. – intermediate regime: Q ≈ C · U and precise knowledge of the created charge is necessary in order to compare to simulations. This makes studies rather complicated and that is why this case is mostly avoided in experiments.. 3.2.1 Low injection The mobilities of the carriers can be determined for low injection in the following way. A bias voltage U synchronised with the illumination source is applied for a short time (∼1 ms) across the sample in order to keep charging effects to a minimum. If the illuminated contact is negatively charged, the created holes get immediately annihilated at this contact but the created electrons travel through the whole thickness d of the sample. In the absence of trapping, the applied electric field is given by | = |E. |U | . d. (3.2). The measured current is constant when the electron cloud travels at constant drift velocity v d through the sample. With the cloud arriving at the 2. The surface termination of the diamond also has an influence on the electrical transport properties [32, 33].. 30.

(224) Figure 3.1: Patterned diamond sample with a semitransparent 40 μm Ti/Al mesh.. 1064. Nd-YAG Laser 5ns. 532+. 2Z nm. 4Z 1064. 266+ 532+ 1064. Halogen or Xe lamp. (4+1)Z. 10 Hz trigger attenuator. Power Supply & Q-switch trigger. 21 3n m. interference filters. SP150 Monochromator. shutter. trig. LN-cryostat. Pulse generator. 10m koax Sample. Temperature controller. Computer. trig data acqusition. Heater. Oscilloscope TDS 640 5 Gs/s. Figure 3.2: Setup for the ToF measurements presented in Papers I,V,VI and VII. 16 14. -11.3V -30.1V -89.5V. current (μA). 12 10 8 6 4 2 0 -2 0. 20. 40. 60. 80. 100. 120. 140. 160. time (ns). Figure 3.3: Typical ToF curves for the low injection regime. The contribution from immediately annihilated charge carriers cannot be seen due to limited bandwidth.. 31.

(225) electron time-of-flight (ns). 100. 80. sample thickness: d = 490μm fit coefficients: slope = a = 1095.7 offset = b = 9.00. 60. 40. electron mobility: μe = d2 / a = 2190 cm2/Vs. 20. 0 0.00. 0.02. 0.04. 0.06. 0.08. 1 / |bias voltage| (V-1) Figure 3.4: Typical plot of electron ToF τe versus the inverse applied voltage |U |−1 . The mobility can be extracted from the linear best fit as discussed in the text.. positively charged contact, the current drops (see Fig. 3.3). The full width at half maximum (FWHM) time interval with respect to the current plateau is the time-of-flight τe of the electrons. It follows from Eqs. (2.7) and (3.2) | μ |E. 1 + vesat d d d =d τe = = + | | v sat | vd| μe |E μe |E =. d d2 + μe |U | v sat. (3.3). Applying a set of negative bias voltages and plotting τe versus the inverse applied voltage |U |−1 should theoretically lead to a straight line with the slope a only depending on the sample thickness and the electron mobility and the offset b only depending on the sample thickness and the saturation velocity. A typical plot of a medium quality sample is shown in Fig. 3.4. A completely analogous consideration for holes leads to the hole mobility and one can write for both cases τe/h = a e/h |U |−1 + b e/h. (3.4). with the following identifications made d2 , a e/h d e/h v sat = . b e/h. μe/h =. 32. (3.5) (3.6).

(226) 0.74 0.73 0.72 0.71 0.70. E. 0.69 0.68 0.67 0.66 0.65 0.64 0.63 0.62 200. 300. 400. 500. 600. 700. 800. 900. 1000. thickness (μm). Figure 3.5: Correction factor β obtained from simulations in case of high injection.. 3.2.2 High injection In the case of high injection, the carrier transport is space charge limited (SCL) and the electric field has an inhomogeneous distribution because of the non-negligible charge in the carrier cloud [35, 36]. A reservoir of carriers is created near the illuminated electrode [7], which thermalises quickly. A first current peak results from the screening of the electrical field in this electron-hole plasma and the second from the arrival of the first charge carriers at the opposite electrode. The charge carriers do not travel at a constant drift velocity like in the low injection case. Instead, they experience an increasing electric field during the space charge limited transit through the sample. To compensate for that effect, a correction factor β is introduced, which is a slowly varying function of the sample thickness as shown in Fig. 3.5. This factor can be obtained from simulations. Just as before, the polarity of the bias determines whether electron or hole drift is observed. In this case, the electric field in Eq. (3.2) has to be divided by the correction factor β. Thus, the mobility and the saturation velocity can be extracted from the linear fit with slope a e/h and offset b e/h by the identifications βd 2 , a e/h βd e/h v sat = . b e/h. μe/h =. (3.7) (3.8). 33.

(227) 3.3 Data acquisition and evaluation Mobility measurements by the time-of-flight technique involve the recording of several current transits with different bias voltages applied, as pointed out in the previous section. The ToF setups for both vertical and lateral are shown in Fig. 3.6. The author of this thesis was involved in developing a fully automated measurement software of which a screenshot is shown in Fig. 3.7. The program was written in TESTPOINT and controls the temperature, the bias voltage and also, if desired, a magnetic field that can be applied perpendicular to the electrical field. The current signals are sampled by a digital oscilloscope and stored on the harddrive of the PC. The communication with the equipment is via the GPIB interface (also known as IEEE-488). For the evaluation of the recorded data a graphical user interface (GUI) was written in MATLAB. The software allows to extract the time-of-flight from the current transit curves not only by marking the plateau and the related FWHM manually but also by fitting the data to Eq. (4.37) – to be derived in the next chapter. Error bars can be obtained in this way, which can be taken into account (by weighting) when extracting the low-drift mobility from ToF versus 1/U plots. The GUI is shown in Fig. 3.8.. vertical ToF. lateral ToF. Figure 3.6: The time-of-flight setup with the cryostat used for vertical measurements (on the left) and the holder used for lateral measurements (on the right).. 34.

(228) Figure 3.7: Screenshot of the program controlling the measurement hardware via the GPIB bus. The software was written in TESTPOINT.. Figure 3.8: Screenshot of the evaluation GUI. The software was written in MATLAB.. 35.

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(230) 4. Free charge carrier transport in diamond. This chapter deals with carrier transport in more detail. The aim is to show how the different stages between carrier generation by a short laser pulse and collection at the opposite contact influence the current transient. Or the other way round, to show what conclusions we can draw from the observed current transient. In this context, the electrical field profiling method presented in Paper I will be explained.. 4.1 Drift-diffusion equations from the BTE In this section, the fundamental drift-diffusion equations will be derived by solving the Boltzmann transport equation (BTE) in the case of a Fermi gas with an applied electric field using the relaxation time approximation. For a more detailed discussion see e.g. [37]. Our objectives in this section are to understand the origin of the drift-diffusion equation and the assumptions that limit its validity1 .. 4.1.1 The Boltzmann transport equation The Boltzmann transport equation2 is a particle continuity equation describing particle flow in the six-dimensional (6-D) phase space. ∂f ∂ f df · ∇p f = . (4.1) = + v · ∇x f + F dt ∂t ∂t coll , t ) is a function of time and the The distribution function f ( x, p six-dimensional (6-D) phase space which is normalised in such a way that it describes the probability of finding a carrier with crystal momentum , at location p x , at time t . The carrier’s velocity is connected to the momentum by (4.2) v = ∇p E kin 1. To establish the limitations of the drift-diffusion equation is especially important since it serves as the cornerstone of semiconductor device analysis. 2 The BTE is an approximation because it is a single particle description of a many particle system of carriers. But correlations between carriers are not treated, even though carrier interact through their electric field.. 37.

(231) is the external force which is experienced by the particles described and F by f . The term on the right hand side (RHS) is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. This term is also called the collision integral and is in general a higher dimensional integral by which f is connected non-linearly. The explicit form of this integral depends on the type and interactions of the particles we are investigating and has to be determined from the microscopic theory (e.g. quantum mechanics). That is why the BTE is an integro-differential equation and cannot be solved by standard means of classical mechanics.. 4.1.2 Equilibrium distribution function for a Fermi gas In equilibrium nothing changes with time and the BTE reads · ∇p f = 0 . v · ∇x f + F. (4.3). The solution is the equilibrium distribution function commonly denoted by f 0 which in the case of a Fermi gas is the previously mentioned Fermi-Dirac distribution F (E ) ) = F (E ) ≡ f 0 ( x, p. 1 1 + eθ. with θ =. ) − E F E tot ( x, p , kT. (4.4). with Boltzmann’s constant k, the (lattice) temperature T , the Fermi level E F and the carrier’s total energy E tot which is the sum of potential and kinetic ) = E pot ( energy E tot ( x, p x ) + E kin ( p ). In this case we can write Eq. (4.3) as v·. ∂ f0 ∂f v · 0 ∇x θ + F =0 ∂θ ∂θ kT. (4.5). v /(kTL ) because of Eq. (4.2). If we permit E pot , E F and T to vary since ∇p θ = with position Eq. (4.3) becomes E pot + E kin − E F F + 0 = ∇x T T . E 1 F 1 pot + E kin − E F ∇x + 0 = ∇x E pot − E F + T T T T E pot + E kin − E F 1 1 0 = − ∇x (E F ) + ∇x T T T is a conservative field with F = −∇x E pot . This equation has to hold for if F so each of the terms has to vanish independently and we can conany p clude that ∇x E F = ∇x T = 0 meaning that both the Fermi-level and temperature are constant in equilibrium.. 38.

(232) 4.1.3 Uniform electric field with RTA In order to obtain an analytical solution of the BTE we have to make further 3 simplifications. In the case of an applied electric field E = qE F. (4.6). we will use the relaxation time approximation (RTA) f − f0 ∂f · ∇p f = − + v · ∇x f + q E . ∂t τf. (4.7). ) depends only on the nature of the scattering The relaxation time τf ( x, p process and is the characteristic time describing how the system relaxes. The RTA assumes that the collision probability during the time interval dt ) is equal to dt /τf and that some for an electron at phase space point ( x, p time after scattering has occurred, the electron distribution does not depend on the non-equilibrium distribution just before the scattering. This means that the information about the non-equilibrium state is completely lost due to the scattering processes. A more detailed study of the RTA shows (see e.g. [37]) that Eq. (4.7) is a good approximation of the BTE in the case of low fields when the scattering is elastic and/or isotropic. The quasi-equilibrium distribution function f 0 has the same form as F (E ) but with the Fermi-level E F replaced by the quasi-Fermi level E F ) = f 0 ( x, p. 1 1 + eθ. with θ =. x ) + E kin ( p ) − E F ( x) E pot ( kT ( x). .. (4.8). The function f 0 cannot be the solution to the BTE because it is symmetric in momentum , so the average velocity is zero, and no current flows. But we expect the solution to be f 0 plus a correction term and make the guess q ∂ f0 ∂ f0 . + v · ∇x f 0 + v ·E (4.9) f = f 0 − τf ∂t kT ∂θ Inserting our guess into the BTE yields

(233) ∂ q ∂ f0 ∂ f0 · ∇p f 0 − τf + v · ∇x + q E + v · ∇x f 0 + v ·E ∂t ∂t kT ∂θ ∂ f0 q ∂ f0 = + v · ∇x f 0 + v ·E ∂t

(234) kT ∂θ q ∂ f0 ∂ ∂ f0 ⇔ + v · ∇x + q E · ∇p −τf + v · ∇x f 0 + v ·E ∂t ∂t kT ∂θ q ∂ f0 − qE · ∇p f 0 . = v ·E kT ∂θ denotes the electric field while the energy is given by the Throughout Sec. 4.1 the field E scalar E .. 3. 39.

(235) Using Eq. (4.9) we see that our guess is correct if . ∂ q ∂ f0 − qE · ∇p f 0 . · ∇p f − f 0 = + v · ∇x + q E v ·E ∂t kT ∂θ. (4.10). only through E kin The right side of Eq. (4.10) vanishes since θ depends on p · ∇p f 0 = q qE. ∂ f0 ∂f 1 q ∂ f0 · ∇p θ = q 0 · ∇p E kin = · E E E v ∂θ ∂θ kT kT ∂θ. which means that the left side of Eq. (4.10) has to vanish, too. Assuming that changes in carrier concentration are much slower than the typical time between two scattering events and that the concentration does not vary significantly over the mean free path, the left side of Eq. (4.10) can be approximated by

(236) . q ∂ f0 ∂ ∂ f0 · ∇p f − f 0 ≈ q E · ∇p −τf + v · ∇x + q E + v · ∇x f 0 + v ·E ∂t ∂t kT ∂θ q ∂ f0 · ∇p τf v ·E ≈ −q E kT ∂θ. 2 | = 0 + O |E and it vanishes under the assumptions above if we are dealing with small |. electric fields, i.e. we can neglect second and higher order terms in |E 4 The carrier concentration in our semi-classical approach is the momentum integral of the distribution function divided by twice the unit volume (2πħ)3 since it can be occupied by two fermions 1 1 3 x , t ) = 3 3 f d p ≈ 3 3 f 0 d3p , (4.11) ρ c ( 4π ħ 4π ħ. and the approximation is valid in our case considering small perturbations. For the current density we can write 1 q j( x , t ) = 3 3 q v ( f − f 0 )d3p , (4.12) v f d3p = 3 3 4π ħ 4π ħ since the integral of v f 0 vanishes. Using our guess Eq. (4.9) we obtain q q ∂ f0 ∂ f0 j( x, t ) = − 3 3 + v · ∇x f 0 + v · E d3p v τf 4π ħ ∂t kT ∂θ q ∂ f0 q v · E d3p v · ∇x f 0 + v τf =− 3 3 4π ħ kT ∂θ where the last equal sign holds because f 0 is time-independent. 4. The approach is semi-classical because carriers are treated as classical particles obeying Newton’s laws. Quantum mechanics is used only to describe the collisions.. 40.

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