Solid-State Electronics

Transcription

1 Ve 311 Electronic Circuits: Solid-State Electronics Chapter 1 Jón Tómas Guðmundsson tumi@raunvis.hi.is 1. week summer 2011 Semiconductors The development of solid-state materials and the technology for integrated circuit (IC) fabrication have revolutionalized electronics The miniaturization achievable through IC technology has made it possible to perform complex electronic functions with high performance at low cost 1 2 Semiconductors Semiconductors The conductivity of a semiconductor is sensitive to temperature, illumination, magnetic field and the amount of impurity atoms This sensitivity makes the semiconductor one of the most important materials for electronic applications Microelectronic circuits are based on structures build from semiconductors Solid-state materials can be grouped into three classes: insulators semiconductors conductors Elemental semiconductors are formed from a single type of atom (column IV of the periodic table of elements) Compound semiconductors can be formed from combinations of elements from columns III and V or columns II and VI 3 4

2 Semiconductors Semiconductors The elemental semiconductors silicon and germanium are found in column IV of the periodic table of elements A silicon (or germanium) atom residing in isolation contains four valence electrons, requiring another four to complete its outermost shell Single-crystalline silicon is a highly regular three dimensional array of atoms Silicon and germanium have diamond lattice - wherein each atom is surrounded by four neighbors Each atom shares one valence electron with its neighbors - a bond is formed between atoms referred to as covalent bond 5 6 Semiconductors Orbits - Energy bands For an isolated atom, the electrons can have discrete energy levels The energy levels for an isolated hydrogen atom are given by the Bohr model E H = m eq 4 8ɛ 2 0 h2 n 2 = 13.6 n 2 m e is the free-electron mass q is the electronic charge ɛ 0 is the free-space permittivity h is the Planck constant n is a positive integer, principal quantum number The discrete energies are ev for the ground state energy level (n = 1), -3.4 ev for the first excited-state energy level (n = 2) etc. The figure shows an isolated silicon atom that has 14 electrons Of the 14 electrons, 10 occupy deeply lying enery levels The 4 remaining valence electrons are relatively weakly bound and can be involved in chemical reactions 7 8

3 Orbits - Energy bands Orbits - Energy bands Thus, we only need to consider the outer shell (n = 3) for the valence electrons, as the two inner shells are completely full The 3s subshell (i.e. n = 3 and l = 0) has two allowed quantum states per atom. This subshell will contain two valence electrons at T = 0 K The 3p subshell (i.e. n = 3 og l = 1) has six allowed quantum states per atom and will contain two valence electrons at T = 0 K As N isolated atoms are brought together to form a solid the orbits of the outer electrons of different atoms and interact with each other As the interatomic distance decreases, the 3s and 3p subshell of the N silicon atoms will interact and overlap 9 10 Orbits - Energy bands Orbits - Energy bands At the equilibrium interatomic distance, the bands will again split, with four quantum states per atom in the lower band and four quantum states per atom in the upper band At absolute zero temperature the electrons occupy the lowest energy states, all the states in the lower band, valence band, will be full All the states in the upper band, conduction band, will be empty The bandgap energy E g between the bottom of the conduction band and the top of the valence band is the width of the forbidden energy gap E g is the energy required to break a bond in the semiconductor to free an electron to the conduction band and leave a hole in the valence band 11 12

4 Energy gap - Energy bands Energy gap - Energy bands The energy of a free electron is E = p2 2m e where p is the momentum and m e is the free-electron mass Because the periodic potential of the nuclei the free-electron mass has to be replaced by an effective mass The electron effective mass depends on the properties of the semiconductor ( d m 2 ) 1 E e = dp 2 Similar expression can be written for holes The narrower the parabola, corresponding to a larger second derivative, the smaller the effective mass The spacing at p = 0 between these parabolas is the bandgap E g E = p2 2m e Energy gap - Energy bands Energy gap - Energy bands The energy E v corresponds to the top of the valence band and represents the highest permissible energy for a valence electron The energy E c corresponds to the bottom edge of the conduction band and represents the lowest available energy in the conduction band The difference between E c and E v is called the bandgap energy E g E g = E c E v The figure shows a simplified energy-momentum relationship of a of a semiconductor with m e = 0.25m e in the conduction band and effective mass of a hole m h = m e in the valence band The actual energy-momentum relationships (energy-band diagram) for silicon and GaAs are much more complex 15 16

5 Energy gap Energy gap At room temperature the bandgap is 1.12 ev for silicon and 1.42 ev for GaAs The bandgap varies with temperature according to E g = 1.17 ( )T 2 (T +636) for silicon and (a) A conductor (partially filled conduction band or overlapping bands) (b) A semiconductor (c) An insulator for GaAs E g = 1.52 ( )T 2 (T +204) For both silicon and GaAs de g /dt is negative and the bandgap decreases with increased temperature Energy bands Energy bands The energy-band diagram for silicon and GaAs for two crystal directions The bandgap E g is between the bottom of the conduction band and the top of the valence band For silicon the maximum in the valence band occurs at p = 0, but the minimum in the conduction band occurs along the [100] direction When an electron makes a transition from the maximum point in the valence band to the minimum point in the conduction band, it requires not only energy change (> E g ) but also some momentum change 19 20

6 Energy bands Intrinsic semiconductor E c E g h ν > Eg E v A semiconductor that has no impurities or lattice defects is referred to as intrinsic semiconductor Due to this silicon is referred to as indirect bandgap semiconductor GaAs is a direct bandgap semiconductor At absolute zero temperatue all the electrons reside in the covalent bonds shared between the atoms in the array - no electrons are free for conduction As the temperature increases, thermal energy is added to the crystal and some bonds break, freeing a small number of electrons for conduction Intrinsic semiconductor Intrinsic semiconductor The density of these free electrons is equal to the intrinsic carrier density ( n 2 i = BT 3 exp E ) g kt where E g is the bandgap energy h ν > Eg The term intrinsic refers to the generic properties of pure materials E c E g E v k = ev/k T is the temperature B is a material dependent parameter K 3 cm 6 for Si As an electron which has a charge of q equal to C, moves away from the covalent bond it leaves behind a vacancy The vacancy is left with an effective charge of +q 23 24

7 Intrinsic semiconductor An electron from an adjacent bond can fill this vacancy, creating a new vacancy in another position Intrinsic semiconductor This moving vacancy behaves like a particle with charge +q and is called a hole Hole density is represented by the symbol p The intrinsic carrier density at room temperature is n i = cm 3 For an intrinsics semiconductor n = p = n i and the product of the electron and hole concentrations is for silicon and for GaAs n i = cm 3 np = n 2 i Intrinsic semiconductor The main advantage of semiconductors emerge as impurities are added to the material in minute and well controlled amounts This is referred to as impurity doping or doping Impurity doping allows us to change the resistivity over a wide range and determine wheter the electron or the hole population controls the transport properties of the material The intrinsic carrier density versus the reciprocal temperature for silicon and GaAs 27 28

8 Semiconductors Donor impurities in silicon are from column V having five valence electrons in the outer shell The impurities used to dope silicon are most often from columns III and V of the periodic table The figure shows a donor atom replacing a silicon atom in the crystal lattice Four of the five out shell electrons fill the covalent bond structure It takes very little thermal energy to free the extra electron for conduction At room temperature, essentially every donor atom contributes (donates) an electron for conduction Acceptor impurities in silicon are from column III and have one less electron than silicon in the outer shell The primary acceptor impurity in silicon is boron Since boron has only three electrons in its outer shell a vacancy exists in the bond structure This vacancy represents a hole that can move through the lattice 31 32

9 In the doped semiconductor, the electron and hole concentrations are no longer equal If n > p the material is called n type, and it p > n the material is called p type The semiconductor material must remain charge neutral or The carrier with the larger population is called the majority carrier, and the carrier with the smaller population the minority carrier The donor and acceptor concentrations are denoted by N D donor impurity concentration N A acceptor impurity concentration Also n+n A = p+n D np = n 2 i For n-type material and n n = 1 2 For a p type material and p p = 1 2 [ N D N A + [ N A N D + ] (N D N A ) 2 +4n 2 i p n = n2 i n n ] (N A N D ) 2 +4n 2 i In practical situations N D N A n i such that n n N D N A if N D > N A p p N A N D if N A > N D = Example 1.1. n p = n2 i p p 35 36

10 Charged particles move in response to an applied electric field This movement is termed drift and the resulting current is known as drift current The conductivity can be written Ohms law states that the current density is proportional to the electric filed where σ = nq2 τ m n is conductivity due to electrons J = σe [Ωcm] 1 where σ = qnµ n µ n = qτ c m n is the electron mobility, and describes the drift of electrons The mobility can be expressed as the average particle drift velocity per unit electric field µ n = v E If both holes and electron participate J = q(nµ n +pµ p )E = σe Note that mobility is determined by the effective mass and the m and the mean free time τ c We expect that m is small in the strongly curved minimum of the conduction band of GaAs and the mobility is thus high τ c depends on temperature and the impurity concentration The drift velocity versus electric field for Ge, GaAs and Si At low electric field the drift velocity increases linearly with increased electric field and the slope of this curve is the mobility 39 40

11 For large electric fields (> V/cm) the conductivity σ changes with the electric field The drift velocity is then comparable to the thermal velocity of the charge carriers v th 10 7 m/s The added energy imparted by the field is transferred to the lattice rather than increasing the carrier velocity - scattering limited velocity or saturated drift velocity v sat 10 7 cm/s This velocity saturation phenomena ultimately places an upper limit on the frequency response of solid-state devices The two basic scattering mechanisms that influence the electron and hole mobility are lattice scattering The role of lattice scattering increases with increased temperature and the mobility decreaes as temperature is increased impurity scattering At low temperature impurity scattering dominates 43 44

12 GaAs:Si µh [Vs/cm 2 ] T [K] The mobility versus temperature for silicon doped GaAs = Example 1.2. The mobility of electrons and holes in silicon at room temperature Resistance is a physical property of a circuit element or a device For lightly doped semiconductor the lattice scattering dominates and the mobility decreases as the temperature increases and is independent of the doping of cm 3 and lower R = ρl Wd = L 1 Wdσ and A = Wd is the cross sectional area, ρ is the resistivity and L is the length For heavily doped semiconductor, the effect of impurity scattering is most pronounced at low temperature For a given temperature the mobility decreases with increasing impurity concentration because of enhanced imprity scatterings Material Resistivity [Ω cm] Silicon Carbon Aluminum Copper Polystyrene

13 Resistivity versus impurity concentration for silicon and GaAs at 300 K At this temperture all the shallow donor and acceptor levels are ionized = Example 1.3. = Example 1.4. Impurity concentration as a function of the resistivity at 296 K for silicon doped with boron and phosphorus σ = q(nµ n +pµ p ) Transport of Carriers Diffusion Current Transport of Carriers Diffusion Current We have assumed that the doping is uniform in the semiconductor, but this need not be the case To calculate the diffusion current, we will determine the net flow of electrons per unit area crossing the plane at x = 0 Changes in doping are encountered often in semiconductors, and there will be gradients in the electron and hole concentrations Gradients in these free carrier densities give rise to a second current flow mechanism, called diffusion The free carriers tend to move (diffuse) from regions of high concentration to low concentration If the distance λ is the mean-free path of an electron, that is, the average distance an electron travels between collisions, then on the average, electrons moving to the right at x = λ and electrons moving to the left at x = +λ will cross the x = 0 plane 51 52

14 Transport of Carriers Diffusion Current Transport of Carriers Diffusion Current This can be written F n = 1 2 v th(n( λ) n(+λ)) = v th λ n(+λ) n( λ) λ ( λ) v th λ dn dx The electron diffusion current density is given by One half of the electrons at x = λ will be traveling to the right at any instant of time and one half of the electrons at x = +λ will be traveling to the left at any given time j diff n = qf n = v th λ dn dx = qd dn n dx where D n is the electron diffusivity or diffusion coefficient, with units cm 2 /s The net rate of electron flow, F n, in the +x direction at x = 0 is given by F n = 1 2 n( λ)v th 1 2 n(+λ)v th where v th is the random thermal velocity (the velocity for E = 0) There is a similar result for hole diffusion current density j diff p = qd p dp dx where D p is the hole diffusivity or diffusion coefficient Transport of Carriers Diffusion Current Diffusivity and mobility are related by D p µ p = kt q = D n µ n Transport of Carriers Total Current Generally, currents in a semiconductor have both drift and diffusion components which is referred to as Einstein s relationship The quantity is called the thermal voltage kt q = V T V T = 25.8 mv at room temperature The total electron and hole current densities can be found by ( jn T dn = qµ n ne +qd n dx = qµ nn E + V ) T dn n dx ( jp T dp = qµ p ne qd p dx = qµ pp E V ) T dp p dx 55 56

15 References [1] Richard C. Jaeger and Travis N. Blalock, Microelectronic Circuit Design, 4th edition, McGraw Hill, 2011, Chapter 2 [2] S. M. Sze, Semiconductor devices: Physics and Technology, John Wiley & Sons, 2ed., 2002, Chapters og 3 [3] R. S. Muller og T. I. Kamins, Device Electronics for Integrated Circuits, 2nd ed., John Wiley & Sons, 1986, Chapter 1 [4] Ben G. Streetman og Sanjay Banerjee, Solid State Electronic Devices, 5th ed., Prentice Hall, 2000, Chapter 3 [5] J. Bourgoin og M. Lannoo, Point Defects in Semiconductors II, Springer Series in Solid-State Sciences, Vol. 35, Springer-Verlag, Berlin, Heidelberg