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Home , Electron mobility

ELECTRONIC MATERIALS

Electronic Materials

Superconductors Conductors Semi-Conductors Dielectrics

Type I Linear Elemental Insulators Dielectrics Hg, MgB2 Ag, Au, Cu, Hg, W Si, Ge, a-Sn, C Porcelain, glasses, Al2O3

Type II Non-Linear Compound Linear YBCO,BSCCO ZnO, diodes GaAs, AlAs Al2O3, SiO2, ZrO2

Organic NonLinear BaTiO3, PZT

Inorganic ITO,YSZ, BaTiO3

Electrical Properties Objectives To understand: Electronic Conduction in materials Band Structure Conductivity Ionic conduction in ceramics Dielectric Behavior Polarization Important Terms

Acceptor: element introduced to to generate free hole (by “accepting” from semiconductor atom and “releasing” hole at the same time); acceptor atom must have one valence electron less than semiconductor; (B) is the most common acceptor in technology; alternatives include indium and gallium. Donor: element introduced to semiconductor to generate free electron (by “donating” a electron to semiconductor); must have one more valence electron than semiconductor; common donors in Si: P, As, Sb and in GaAs: S, Se, Sn. Dopant: element introduced to semiconductor to establish either p-type (acceptor) or n-type (donor) conductivity Electron: negatively charged particle in an atom; carrier of smallest (elemental) charge of 1.6x10- 19C; carrier of negative charge in semiconductors Hole: positive in a semiconductors which materially does not exits; lack of electron moving in the direction opposite to that of electron and carrying a positive charge; features higher effective mass than electron, hence, lower mobility. Fermi level: energy level in solids at which the Fermi-Dirac distribution function is equal to 0.5. Fermi-Dirac distribution function: formula describing the probability of a state being occupied by an electron depending on the state's energy level. Semiconductor: solid-state material in which, in contrast to metals and insulators: (i) electrical conductivity can be controlled by orders of magnitude by adding very small amounts of alien elements known as dopants, (ii) electrical conductivity can be controlled not only by negatively charged , but also by positively charged holes and (iii) electrical conductivity is sensitive to temperature, illumination, and magnetic fields; these properties result from the fact that, due to the nature of interatomic bonds (mostly covalent), in semiconductors, in contrast to metals but similarly to insulators, the valence band and the conduction band are separated by the energy gap; in semiconductors, however, width of the energy gap rarely exceeds about 3.5 eV while in insulators energy gap is typically wider that about 5 eV; semiconductor properties are displayed by the elements from the IVth group of periodic table: carbon, (C) in the form of diamond, silicon (Si), germanium (Ge) and tin (Sn); numerous compound semiconductors can be formed by combining elements from IVth group, e.g. SiGe and SiC, as well as from groups III and V,e.g. GaAs, InP, or GaN, and groups II and Vi, e.g. CdTe, ZnS, etc. Furthermore, select organic materials display semiconductor properties. Semiconductor Diode: two-terminal device in which current is a very strong function of the direction of applied voltage; large current is allowed to flow under forward voltage bias while almost no current will flow under the reverse voltage bias, hence, displays rectifying characteristics; typically implemented by p-n junction. Intrinsic semiconductor: semiconductor materials in which free electrons and holes result only from band-to band generation, and hence, feature the same concentration; undoped; does not display neither n- or p-type conductivity. Extrinsic semiconductor: doped semiconductor featuring either p- or n-type conductivity. Scattering: process responsible for electron in a certain state defined by its crystal momentum suddenly moving into a different state; fundamental effect defining electron transport in semiconductor. : velocity of carriers under ; as opposed to carriers in free space carriers in a semiconductor are not "infinitely" accelerated by the electric field due to scattering - thus they reach a finite velocity regardless of the period of time over which the field is acting; at a given electric field drift velocity is determined the carrier mobility. Electron mobility: measure of electron scattering in semiconductor; proportionality factor between electron drift velocity and electric field as well as carrier concentration and conductivity of semiconductor; unit cm2/V s; the same way as effective mass of an electron, electron mobility is different for different semiconductors; electron mobility at 300 K for three key semiconductors: Si - 1500 cm2/V s, GaAs -7500 cm2/V s, 6H-SiC -400 cm2/V s; higher mobility makes semiconductor better suited for high applications. : applied to semiconductor through which current is flowing causes generation of electric field perpendicular to the direction of current flow and the magnetic field; used to derive some electrical properties of semiconductor. : energy band separating conduction and valence bands in the solid; no electron energy levels are allowed in the forbidden band; no energy gap in metals in which case conduction and valence bands overlap; solids featuring energy gap are defined as either semiconductors or insulators based on the width of energy gap; values of Eg (at 300K) for common semiconductors: InSb - 0.17 eV, Ge - 0.67 eV, Si - 1.12 eV, GaAs - 1.43 eV GaP - 2.26 eV, 6H-SiC - 2.9 eV, GaN - 3.5 eV, and insulators Ta2O5 - 4.2 eV , TiO2 -5 eV, Si3N4 - 5.1, Al2O3 ~5 eV, SiO2 -8.0. Dielectric: a solid displaying insulating properties (energy gap wider than 4 eV); upper most energy band is completely empty, hence, extremely low conductivity; characteristics of material are independent of the applied voltage. Lithography: process used to transfer pattern from the mask/reticle to the layer of resist deposited on the surface of the wafer; kind of lithography depends on the wavelength of radiation used to expose resist: photolithography (or optical lithography) uses UV radiation, X-ray lithography uses X-ray, e- beam lithography uses electron bean, ion beam lithography uses ion beam. Electron beam (e-beam) lithography, EBL: lithography technique using focused beam of electrons to expose the resist; no mask is used as pattern is "written" directly into the resist by very fast scanning of electron beam; pattern transfer resolution below 0.1 µm is possible. : three-terminal in which input signal (voltage or current depending on the type of transistor) controls output current, the most important semiconductor device; performs switching and amplifying functions; early on replaced bulky and inefficient vacuum triode in electronic circuits; invention of transistor (Schockley, Brattain and Bardeen) triggered electronic revolution after the World Word II; can operate as a discrete device or a building cell of integrated circuits; numerous kinds of can be distinguished based on their design and principles of operation; two major types: field-effect (unipolar) transistor and bipolar transistor. Definitions Ohm’s Law V = IR V - Voltage, I - current, R -Resistance Units V - (or W/A (Watts/amp) or J/C Joules/Coulomb)) I – amperes (or C/s - Coulombs/) R - ohms (Ω) The resistance is influenced by specimen configuration Consider current moving through a conductor R = V/I with cross sectional area, A and a length, L Area (A) Resistivity (ρ) and Conductivity (σ) I Geometrically independent forms of Ohm’s Law Length (L) Probe Conductivity Measurements L R = Rsample + Rcontact A R = V/I RA −Ω m 2 ρ == −Ω= m L m V It can give erroneous values if I

contact resistance, Rcontact, is not Voltmeter negligible with respect to R L sample A

I = V1/R1 I

Rsample = V2/I V2 R = (V R )/V sample 2 1 1 R1 Current Source ρ = Rsample (A/L)

Eliminates contact resistance V1 Voltmeter 1 Conductivity (σ) σ ()−Ω= m −1 Conductivity is the “ease of conduction” ρ Ranges over 27 orders of magnitude! Metals 107 (Ω.cm)-1 Semiconductors 10-6 -104 (Ω.cm)-1 Insulators 10-10 -10-20 (Ω.cm)-1 ResistivitiesResistivities ofof RealReal MaterialsMaterials Compound Resistivity Compound Resistivity ( Ω -cm) ( Ω -cm) Ca 3.9 × 10 -6 Si ~ 0.1 Ti 42 × 10 -6 Ge ~ 0.05 Mn 185 × 10 -6 ReO 3 36 × 10 -6 Zn 5.9 × 10 -6 Fe 3 O 4 52 × 10 -6 Cu 1.7 × 10 -6 TiO 2 9 × 10 4 Ag 1.6 × 10 -6 ZrO 2 1 × 10 9 Pb 21 × 10 -6 Al 2O 3 1 × 10 19 (a) Charge carriers, such as electrons, are deflected by atoms or defects and take an irregular path through a conductor. The average rate at which the carriers move is the drift velocity (v). (b) Valence electrons in the metallic bond move easily. (c) Covalent bonds must be broken in semiconductors and insulators for an electron to be able to move. (d) Entire ions must diffuse to carry charge in many ionically bonded materials.

• Electronic conduction: – Flow of electrons, e- and electron holes, h+ • Ionic conduction – Flow of charged ions, Ag+

Electrical Conductivity in Metals If the nuclei are arranged in a perfectly ordered lattice then there should be no resistance to oppose the flow of current. Also, an increase in temperature causes an increase in the thermal population of the higher levels and it should increase the electrical conductivity. BUT, thermal vibrations of the nuclei increase electrical resistance so conductivity actually decreases with temperature.

Movement of an electron through (a) a perfect crystal, (b) a crystal heated to a high temperature, and (c) a crystal containing atomic level defects. Scattering of the electrons reduces the mobility and conductivity. The effect of temperature on the electrical resistivity of a with a perfect crystal structure. The slope of the curve is the temperature resistivity coefficient.

For most metals, resistivity increases approx. linearly with temperature.

ρ ρ o []1 += α ()− TT o ρ is the resistivity at temperature T (measured in Celsius).

ρο is the reference resistivity at the reference temperature To (usually taken to be 20 oC). α is a parameter called temperature coefficient of resistivity. Platinum Resistance Thermometer: A resistance thermometer (conductor with a fixed cross section), which measures temperature by measuring the change in the resistance of a conductor, is made of platinum and has a resistance of 50.0Ω at 20oC. When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8Ω. Find the melting point of Indium. Use α=3.92x10-3(oC)-1 for platinum. Solution:

Ro=50.0 Ω. o To=20 C. R=76.8 Ω. − RR Ω − .. 050876 Ω TT =− o = = 137 oC o −3 o −1 α ×Ro []. ×10923 ()C [].050 Ω× = 157 oCT MicroscopicMicroscopic ConductivityConductivity We can relate the conductivity of a material to microscopic parameters that describe the motion of the electrons (or other charge carrying particles such as holes or ions). From the equations RA V I V = IR and ρ =⇒= ρ L L A If J= (I/A) Ampere/m2; and ξ=electric field intensity (V/L) then 1 σ ()−Ω= m −1 J = σξ ρ We can also determine that the current density is = nqJ ν Where n is the number of charge carriers (carriers/cm3); q is the charge on each carrier (1.6x10-19C); and ν is the average drift velocity (cm/s) at which the charge carriers move. Therefore ν ξ nq = σ ⎞ ⎟ ⎟ ⎠ of electricity in semiconductors. to another or the movement of 2 V.s cm then ⎛ ⎜ ⎜ ⎝ σξ == mobility is a constant.

ν μσ == q μ nq nqJ = ν ξ Electrons are the charge carriers in metals. Electrons and holes are both carriers Electrons that “hop” from one defect The charge ions are both the carriers of electricity in ceramics. Control of Electrical Conductivity By controlling the number of charge carriers in the material (n) By controlling the mobility or “ease of movement” of the charge carriers (μ) σ = nqμ Example: Calculation of Drift velocity of electrons in (valence =1) Assuming that all of the valence electrons contribute to current flow, (a) calculate the mobility of an electron in copper and (b) calculate the average drift velocity for electrons in a 100cm copper wire when 10V are applied. Data: conductivity of copper = 5.98 x 105(Ω.cm)-1 ; q = 1.6x10-19C ; lattice parameter of copper = 0.36151x10-7cm ; copper has a FCC structure

The equation that we need to apply is: σ = nqμ n is the number of carriers. q is the charge . μ is the mobility

Number of carriers (n): The valence of copper is 1. Therefore, the number of valence electrons equals the number of copper atoms in the material.

cellatoms electrons atoms)/)(/( ×14 n = = . ×= 104668 22 /cmelectrons 3 _volumecell .( ×10361510 − cm)37 Part (a)

σ . 10985 Ω× .cm −− 115 cm 2 cm 2 μ == = . = .244244 nq .( ×104668 22 cmelectrons 3 ×1061 −19C).)(/ Ω.C .sV

: ampNote =11 Coulomb / second Part (b) V 10V −1 The electric field intensity is: ξ === 10 .. cmV L 100cm

The equation we need to use is = μν ξ

cm 2 V What does it mean? v = .( = 42410244 /.).)( scm .sV cm The drift velocity is the average velocity that a particle, such as an electron, attains, due to an electric field.

The electron moves at the Fermi speed and it has only a tiny drift velocity superimposed by the applied electric field. In metals, the Fermi energy gives us information about the velocity of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts, so only those electrons very close to the Fermi energy (EF) can participate. 2E v = F F m Pauli Exclusion Principle

Electrons in a single atom occupy discrete levels of energy. No two “energy levels” or “states” in an atom can have the same energy. Each energy level can contain at most two electrons -- one with “clockwise spin” and one with “counterclockwise spin”. If two or more atoms are brought together, their outer (i.e., valence) energy levels must shift slightly so they will be different from one another. If many (e.g., N) atoms are brought together to form a solid, the Pauli Exclusion Principle still requires that only two electrons in the entire solid have the same energy. There will be N distinct, but only slightly different valence energy levels, forming a valence band. When a solid is formed, the different split energy levels of electrons come together to form continuous bands of energies (electron energy band). The extent of splitting depends on the interatomic separation and begins with the outer most electron shells as they are the first to be perturbed as the atoms coalesce.

Band Theory: Two Approaches • There are two approaches to finding the electron energies associated with atoms in a periodic lattice. • Approach #1: “Bound” Electron Approach (Solve single atom energies!) – Isolated atoms brought close together to form a solid. • Approach #2: “Unbound” or Free Electron Approach (E = p2/2m) – Free electrons modified by a periodic potential (i.e. lattice ions).

• Both approaches result in grouped energy levels with allowed and forbidden energy regions. – Energy bands overlap for metals. – Energy bands do not overlap (or have a “gap”) for semiconductors. Band Theory: “Bound” Electron Approach

• For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width ΔE. – Leads to a continuous band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level).

Two atoms Six atoms Solid of N atoms The electrical properties of a solid material are a consequence of its electron band structure, that is, the arrangement of the outermost electron bands and the way in which they are filled with electrons. Four different types of band structure are possible at 0oK:

Outermost band is only partially filled with electrons (eg, Na, Cu) (Cu has a single 4s valence electron, however for a solid comprised of N atoms, the 4s band is capable of accommodating 2N electrons. Thus, only half of the 4s band is filled.)

Overlap of an empty band and a filled band (eg. Mg). (An isolated Mg atom has 2 electrons in its 3s level. When a solid is formed, the 3s and 3p energy levels overlap.) A completely filled valence band is separated from the conduction band by a relatively wide energy band gap (insulators). A completely filled valence band is separated from the conduction band by a relative narrow energy band gap (semiconductors)

Temperature Effect

When the temperature of the metal increases, some electrons gain energy and are excited into the empty energy levels in a valence band. This condition creates an equal number of empty energy levels, or holes, vacated by the excited electrons. Only a small increase in energy is required to cause excitation of electrons. Both the excited electrons (free electrons) and the newly created holes can then carry an electrical charge. Fermi Energy is the energy corresponding to the highest filled state at 0oK. Only electrons above the Fermi energy can be affected by an electric field (free electrons). Band Diagram: Fermi-Dirac “Filling” Function

The Fermi-Dirac function gives the fraction of allowed states, fFD(E), at an energy level E, that are populated at a given temperature. 1 fEFD ()= ()EE− F e kT + 1

where the Fermi Energy, EF, is defined as the energy where fFD(EF) = 1/2. That is to say one half of the available states are occupied. T is the -5 temperature (in K) and kB is the Boltzman constant (kB = 8.62 x10 eV/K) Probability of electrons (fermions) to be found at various energy levels.

1 fE()= For E – EF = 0.05 eV ⇒ f(E) = 0.12 FD ()EE− –129 F For E – EF = 7.5 eV ⇒ f(E) = 10 e kT + 1 Exponential dependence has HUGE effect!

Temperature dependence of Fermi-Dirac function shown as follows:

→ Step function behavior “smears” out at higher temperatures. FermiFermi--DiracDirac FunctionFunction MetalsMetals andand SemiconductorsSemiconductors

f(E) as determined experimentally for Ru metal (note the energy scale)

f(E) for a semiconductor Velocity of the electrons In metals, the Fermi energy gives us information about the velocities of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts (see copper wire example), so only those electrons very close to the Fermi energy can participate. The Fermi velocity of these conduction electrons can be calculated from the Fermi energy.

Speed of light: 3x108m.s-1 Fermi Energies, Fermi Temperatures, and Fermi Velocities

Element Fermi Energy eV Fermi Temperature x10^4 K Fermi Velocity 10^6 m/s

Li 4.74 5.51 1.29 Na 3.24 3.77 1.07 K 2.12 2.46 0.86 Rb 1.85 2.15 0.81 Cs 1.59 1.84 0.75 Cu 7.00 8.16 1.57 Ag 5.49 6.38 1.39 Au 5.53 6.42 1.40 Be 14.3 16.6 2.25 Mg 7.08 8.23 1.58 from N. W. Ashcroft and N. D. Mermin,

The Fermi temperature is the temperature associated with the Fermi energy by solving

EF = kB TF for , where m is the particle mass and kB is Boltzmann's constant.