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ELEKTRONIKOS PAGRINDAI 1 2008 CONDUCTIVITY OF SOLIDS Objectives: Revelation on what, why and how conductivity of solids and currents in solids depend. Content: 1. Conduction process in solids 2. Mobility of charge carriers in semiconductors 3. Conductivity of semiconductors 4. Strong field effects • • • Carrier velocity saturation The Gunn effect Strong field effects caused by variation of carrier density 5. Photoconductivity 6. Conductivity of metals 7. Superconductivity 8. Josephson effects 9. Hall effect 10. Diffusion of charge carriers in semiconductors 11. Total current flow in semiconductors VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2 2008 Conduction process in solids Let us consider a solid containing conduction electrons. If the external electrical field is not applied, electrons move randomly colliding with crystal lattice defects. After application of the field, electrons began to drift in the direction opposite to the direction of the field. The drift velocity is dependent on the strength of the field. At applied field, force F = qE acts on an electron. During the collision with the lattice, the electron losses its velocity. The action of a crystal lattice on a drifting electron can be accounted by a restraining force. It is given by Fp = − 1 τr mn v E Thus, the total force acting on an electron is given by 1 d vE FΣ = F + Fr = qE − mn vE = ... = mn a = mn τr dt VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 3 2008 Conduction process in solids d vE qE − mn v E = mn τr dt 1 ... When the external electrical field is applied, the drift velocity increases. At the same time the restraining force also increases. The drift velocity becomes constant when the restraining force becomes equal to the accelerating force. Then vE = qτ r E = µn E mn µn = qτ r mn The coefficient µn is called mobility. If E = 1, we have that v E = µ n . … The mobility can be defined as the incremental average electron velocity per unit of electric field. … The mobility of charge carriers in a solid is limited by the scattering of the carriers by the crystal lattice. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 4 2008 Conductivity Let us consider a parallelepiped. V = vE N Σ = nV = nv E j = qnv E j = qnµ n E = σE … The drift velocity in a solid is proportional to the strength of the electric field. The last formula can be rearranged to I = U / R (Ohm’s law). σ is conductivity. σ = qnµ n σ = q(nµ n + pµ p ) … Conductivity is dependent on charge carriers density and their mobility. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 5 2008 Relaxation time and mobility Now let us suppose that after some initial application of the field it is suddenly reduced to zero. d vE qE − mn v E = mn τr dt 1 t v E (t ) = v E (0) exp − τr d v E (t ) 1 = − v E (t ) τr dt ... The last equation indicates that the drift velocity relaxes back to zero from the initial value exponentially. The process during that the system returns to equilibrium after its excitation is called relaxation. The time constant τr is the electron relaxation time. If we know relaxation time, we can find the average distance between two collisions. λ = vτ r ... Charge carriers mobility: VGTU EF ESK kλ = vτ r µ n ,p = τ r = kλ v q kλ mn , p v [email protected] v ≅ vT ELEKTRONIKOS PAGRINDAI 6 2008 Mobility of charge carriers in semiconductors Charge carriers are microparticles. It is necessary to take into account their dual nature. The lengths of de Broglie’s waves corresponding to the free electrons and holes are much longer than crystal lattice constants. Then free electrons and holes do not reflect from the nodes of a perfect crystal lattice. They are reflected only by lattice defects caused by thermal vibrations, impurity atoms, dislocations and other lattice imperfections. An intrinsic semiconductor at 0 K is transparent (clear) for electronic waves. There are no impurities in intrinsic semiconductors. Scattering of carriers is caused by phonons. With the increase of temperature thermal vibrations of the crystal lattice arise. These vibrations cause thermal defects that cause scattering of electrons and holes. µ n ,p = q kλ mn , p vT λ ~ 1/ T µ n , p ~ T −3 / 2 k ≅1 vT ~ T VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 7 2008 Mobility of charge carriers in semiconductors µ n , p ~ T −3 / 2 ... The carrier mobility in an intrinsic semiconductor decreases with temperature. In a heavily doped semiconductor the impurity scattering predominates in the low temperature range. As the temperature is increased the thermal velocity of charge carriers increases and the impurity scattering decreases. Impurity density does not depend on temperature, and the mean free path of a carrier does not depend on temperature. The mean velocity increases with temperature. The change of the carrier motion direction near an ion is less at higher velocity. For this reason the number of collisions, after what the electron losses its velocity in some direction, increases with temperature. λ = const k ~ (v T ) 4 ~ T 2 VGTU EF ESK vT ~ T µn,p kλ T 2 ~ ~ ~ T 3/ 2 vT T [email protected] ELEKTRONIKOS PAGRINDAI 8 2008 Mobility of charge carriers in semiconductors µn,p kλ T 2 ~ ~ ~ T 3/ 2 vT T ... In the low temperature range the mobility in the doped semiconductors increases with temperature. In the high temperature range the lattice scattering predominates and mobility in a doped semiconductor decreases with temperature In a heavily doped semiconductor the maximum mobility appears in the middle temperature range (usually at the temperature lower than 300 K). Mobility is also dependent upon impurity density. At room temperature mobility in a doped semiconductor is constant for low impurity concentrations because it is limited by the lattice scattering. For impurity densities greater than 1016 cm-3 mobility decreases as a result of impurity scattering. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 9 2008 Mobility at normal temperature Mobility is also dependent upon impurity density. µ n ,p q kλ = mn , p v At room temperature mobility in a doped semiconductor is constant for low impurity concentrations because it is limited by the lattice scattering. For impurity densities greater than 1016 cm-3 mobility decreases as a result of impurity scattering. Electron mobility in silicon is about 3 times greater than that of holes. Collisions of charge carriers with neutral impurity atoms, dislocations and other imperfections of lattices also limits the mobility. It is possible to estimate the influence of various factors using relationship 1 µ VGTU EF ESK ≅ 1 ∑µ i i [email protected] ELEKTRONIKOS PAGRINDAI 10 2008 Mobility of charge carriers in semiconductors. Problem The mobility of electrons in silicon is around 1500 cm2/(V·s). Estimate the mean free path of an electron. Compare it with the lattice constant a = 0,543 nm. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 11 2008 Conductivity of semiconductors Two types of carriers exist in semiconductors. According to this σ = q(nµ n + pµ p ) In the intrinsic semiconductor densities of electrons and holes are equal. Then σ i = qni ( µ n + µ p ) ni ~ T 3 / 2 exp(−∆W / 2kT ) µ n , p ~ T −3 / 2 σ i = σ 0 e − ∆W / 2 kT ln σ i = ln σ 0 − ∆W 2kT ... Conductivity of an intrinsic semiconductor is strongly dependent upon the gap energy and temperature because the density of intrinsic carriers strongly depends on the gap energy and temperature. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 12 2008 Conductivity of semiconductors The specific conductivity of a doped semiconductor depends on the majority carrier density and their mobility. σ n = qnµ n σ p = qpµ p The conductivity of the doped semiconductor varies with temperature for two reasons. One is the dependence of the free charge carrier density on temperature which is felt either at very low temperatures (when not all impurity atoms are ionised) or at very high temperatures (when thermal generation of intrinsic carriers begins). The second reason is the dependence of mobility on temperature. It determines the character of conductivity in the intermediate range: the conductivity slowly decreases with increasing temperature. At room temperature conductivity depends on the impurity density. If the impurity density is higher, the density of majority carriers is higher and the conductivity of the doped semiconductor is also higher. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 13 2008 Strong field effects If the electric field is low, the conductivity of a semiconductor is constant. Then the current density linearly depends on the strength of the field. j = σE. The proportionality between j and E is gradually lost at higher and higher fields. It is because conductivity becomes dependent on the strength of the field. This phenomenon is the strong field effect. In the strong field region j = σ (E)E σ = q(nµ n + pµ p ) ... The strong field effects may be caused by carrier mobility or density variation in the strong field. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 14 2008 Mobility versus strength of electrical field. Carrier velocity saturation µ n ,p q kλ = mn , p v At small strength of the field the mean charge carrier velocity equals the mean thermal motion velocity. The mobility is constant. If the strength of the field increases, the drift velocity also increases. This causes an increase of the mean velocity and decrease of the mobility. The decrease of the mobility causes the saturation of the drift velocity. If the condition that the drift velocity is small is no longer satisfied, we have that the mean velocity of a charge carrier is greater than it mean thermal velocity. Then the carriers are not in thermal equilibrium with the crystal lattice. The carriers with energies greater than 3kT/2 are called hot carriers (hot electrons and hot holes). VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 15 2008 Mobility in compound semiconductors. The Gunn effect The energy of a free microparticle is given by In the case of conduction electrons we must consider the effective mass. The effective mass is dependant on energy. For this reason the graph W(k) becomes complicate. There are two local minima or two valleys in the conduction band of GaAs. h2 2 h2 p2 Wk = k = = 2 2m 2m 2 mλ In low electric fields, electrons reside in the lower central valley. Then the electron mobility in GaAs is high (about 8000 cm2/(V⋅s)). v E1 = µ n1E j = qnµ n1E As the field is increased, electrons transfer to the upper valley and their mobility is considerably reduced (to 0,01 m2/(V⋅s)) vE 2 = µ n 2 E VGTU EF ESK j = qnµ n 2 E [email protected] 2nd valley 1st valley ELEKTRONIKOS PAGRINDAI 16 2008 Mobility in compound semiconductors. The Gunn effect The reduction of the mobility causes a reduction of the drift velocity and current density. If the current density decreases with E, the differential conductivity is negative: σ d = d j /d E < 0, ρ d < 0 ... The reduction of mobility leads to the differential negative resistance effect. It was suggested by Ridley and Watkins in 1961 and discovered by J. B. Gunn in 1963 after whom the effect is named. Devices based on the Gunn effect are called either Gunn diodes or transferred-electron devices (TEDs). They can be used in amplifiers or oscillators. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 17 2008 Strong field effects caused by variation of carrier density The sufficient increase in the conductivity of semiconductors may appear due to the ionization of semiconductor atoms. As a result of semiconductor atoms ionization, densities of charge carriers increase, current strength increases rapidly with increase of applied voltage and the strength of the field, and breakdown in semiconductor begins. Electrons may be liberated when host atoms are struck by other free electrons accelerated by the strong electric field. This mechanism is called avalanche breakdown. In the very strong electric field electrons are stripped from their host atoms by the strength of the electric field. This mechanism is called Zener breakdown. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 18 2008 Photoconductivity Light with the wavelength less than some critical value can cause a change in conductivity of a semiconductor. Such optically induced conductivity is called photoconductivity. It is possible due to the internal photoelectric effect. If photon energy exceeds the forbidden energy gap, the photon can release an electron from a covalent bond. In that way electron-hole pairs can be generated by light. The generated excess carriers cause intrinsic photoconductivity. W > ∆W ν min = ∆W / h VGTU EF ESK λmax = c /ν min = hc / ∆W [email protected] ELEKTRONIKOS PAGRINDAI 19 2008 Photoconductivity W > ∆W Besides intrinsic photoconductivity extrinsic photoconductivity is possible. In this instance the carrier is generated from either a donor or acceptor level. Clearly, necessary for generation energy in these cases is always less than half the semiconductor band-gap energy. Therefore extrinsic photoconductivity is possible in the range of longer wavelengths. The extrinsic photoconductivity is possible only in the low temperature or ionisation range because impurity atoms are ionised at higher temperatures. Photoconductivity is dependant on the wavelength. If the wavelength decreases, photoconductivity increase, becomes maximal, and after that decreases. Light can also cause change of mobility of charge carriers. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 20 2008 Conductivity of metals ... The conductivity of a metal depends on the density of free electrons and their mobility. When atoms form a metal, the valence electrons become free. Thus the density of free electrons in a metal depends upon the density of metal atoms and the valence of the element. The density of the free electrons in metals is about 1022–1023 cm–3. Conductivity of metals is dependant on electron mobility. The conductivity of metals is high. Therefore the strength of the electric field in metals usually is low. In the low field only those electrons near to the Fermi level can change their energy and participate in the conduction process. According to this the mobility of electrons in conductors is given by µn = VGTU EF ESK q λF k F mn v F [email protected] ELEKTRONIKOS PAGRINDAI 21 2008 Conductivity of metals and alloys The Fermi energy depends upon the density of free electrons. Because the density of free electrons in a metal does not vary with temperature, the Fermi energy is almost constant. Then the mean velocity of electrons is also almost constant. Usually electrons in a metal are scattered by phonons. Then µn ~ 1 / T σ ~ 1/ T ρ = 1/ σ ~ T ρ ≅ ρ 0 [1 + α (T − T0 )] ... The resistivity of a metal is proportional to temperature. ... The resistivity of a metal is the sum of a thermal part and a residual resistivity. The residual resistivity at low temperatures is caused by lattice defects (impurity scattering). Increasing impurity concentration in a metal causes the increase of the residual resistivity and resistivity. Lattice defects caused by other atoms are particularly large in disordered alloys such as nichrome. Hence such alloys are used when high resistance combined with a low temperature coefficient of resistance are required. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 22 2008 Superconductivity The residual electrical resistance of many materials drops abruptly to an unmeasurably small value when the material is cooled below a sharply defined transition temperature. This phenomenon was discovered by H. Hamerlingh Onnes in Leiden in 1911 and was called superconductivity. An electric current induced in a superconducting lead ring can persist (without any battery) for several years without significant decay. The modern theory of superconductivity was created only in 1957. This theory was created by J. Bardeen, L. Cooper and J. Schrieffer and is called BSC theory. According to the theory free electrons form electron pairs called Cooper pairs at the transition temperature. We can imagine that Cooper pairs appear in this way. A negative electron attracts positive ions. When they approach the electron a positive charge appears. This attracts the other electron having different spin quantum number. When two electrons having different spin quantum numbers form a Cooper pair the spin quantum number becomes 0. So Cooper pairs have properties of bosons. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 23 2008 Superconductivity When two electrons having different spin quantum numbers form a Cooper pair, their energy decreases. As a consequence a forbidden energy gap appears in the energy diagram of a superconductor. The width of the gap is some milielectronvolts. The energy levels below the gap are occupied by electrons and Cooper pairs. The levels above the gap are unoccupied.. At low temperatures near to 0 K the lattice vibrations are not intense. The energy of the vibrations is not enough to break Cooper pairs. Therefore at low temperatures Cooper pairs move in the crystal without any collisions and scattering. Due to the long free path of charge carriers conductivity is also great (infinite) and a superconductor allows electricity to pass freely, without resistance. The superconducting material exhibits perfect diamagnetism in weak magnetic flux densities (the flux inside the material is zero). If the value of the applied flux density rises to a value greater than a critical value, superconductivity is destroyed. The value of the transition flux density is a function of the temperature of the material and its nature. A superconducting current itself can produce the magnetic flux density greater than the critical value; therefore there is an upper limit of the current density that may be sustained by the material in the superconducting state. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 24 2008 Superconductivity Until 1986, physicists had believed that BCS theory forbade superconductivity at temperatures above about 30 K. In that year, Bednorz and Müller discovered superconductivity in a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K (Nobel Prize in Physics, 1987). It was shortly found by Paul C. W. Chu of the University of Houston and M.K. Wu at the University of Alabama in Huntsville [1] that replacing the lanthanum with yttrium, i.e. making YBCO, raised the critical temperature to 92 K, which was important because liquid nitrogen could then be used as a refrigerant (at atmospheric pressure, the boiling point of nitrogen is 77 K.) This is important commercially because liquid nitrogen can be produced cheaply on-site with no raw materials, and is not prone to some of the problems (solid air plugs, etc) of helium in piping. Many other cuprate superconductors have since been discovered, and the theory of superconductivity in these materials is one of the major outstanding challenges of theoretical condensed matter physics. As of March 2007, the current world record of superconductivity is held by a ceramic superconductor consisting of thallium, mercury, copper, barium, calcium, strontium and oxygen (Tc=138 K). Also a patent has been applied for a material which becomes superconductive at an even higher temperature (up to 150 K).[2] http://en.wikipedia.org/wiki/Superconductivity VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 25 2008 Superconductivity Superconductors are used to make some of the most powerful electromagnets known (superconducting magnets)… They can also be used for magnetic separation, where weakly magnetic particles are extracted from a background of less or nonmagnetic particles, as in the pigment industries. Superconductors have also been used to make digital circuits (e.g. based on the Rapid Single Flux Quantum technology) and microwave filters for mobile phone base stations. Superconductors are used to build Josephson junctions which are the building blocks of SQUIDs (superconducting quantum interference devices), the most sensitive magnetometers known… The cryogenic switching devices called cryotrons have very simple construction. A thin film cryotron consists of two insulated crossing strips made of superconductive materials with different critical field curves such as tin and lead. The presence of a current in one of the strips changes the superconductivity of the other element and hence switches it off or on. It is important that the switching process is very fast (it lasts only some picoseconds). VGTU EF ESK Substrate Tin strip [email protected] SiO2 Lead strip ELEKTRONIKOS PAGRINDAI 26 2008 Josephson effects Two superconductive layers with a thin dielectric layer between them (Fig 5.12(a)) form a Josephson structure or Josephson junction. The terms are named eponymously after British physicist Brian David Josephson, who predicted the existence of the effect in 1962. The superconducting current can flow across the junction in the absence of an applied voltage. This is the direct-current Josephson effect. The direct-current Josephson effect occurs due to the tunnelling current. The Cooper pairs can penetrate through the thin potential barrier without change in their energy. Then there is no voltage drop across the junction. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 27 2008 Josephson effects The current-voltage characteristic of the Josephson junction If the current exceeds critical value, the voltage drop, corresponding to the forbidden energy gap, arises. … When a small direct voltage is applied, the alternating current Josephson effect occurs. The superconducting current across the junction becomes an alternating current and the junction radiates electromagnetic waves. Cooper pairs penetrating the dielectric layer occur over the gap. After that they jump to the energy level below the gap. The dissipation of energy is in the form of electromagnetic waves. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 28 2008 Josephson effects The dissipation of energy is in the form of electromagnetic waves. I = I 0 sin ϕ ∆WC = 2qU ∆ν = ∆WC / h = 2qU / h 2 qU 2qU I = I sin t = I 0 sin(2π f t ) 0 ϕ = 2π∆νt = 2π t h h 2q f = U ... Frequency of oscillation is proportional to the voltage drop on the junction. h If the voltage change is 1 mV, the frequency change is 483,6 MHz. The field of electronics holds great promise for practical applications of superconductors. The use of new superconductive films may result in more densely packed chips which could transmit information more rapidly by several orders of magnitude. Superconducting electronics have achieved impressive accomplishments in the field of digital electronics. Logic delays of 13 picoseconds and switching times of 9 picoseconds have been experimentally demonstrated. Through the use of basic Josephson junctions scientists are able to make very sensitive microwave detectors, magnetometers, SQUIDs and very stable voltage sources. The Josephson junction quantum computer was demonstrated in April 1999 by NEC Fundamental Research Laboratories in Tsukuba, Japan. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 29 2008 Josephson effects The definition of the volt in the International System of Units (SI) is as follows: "The volt is the electromotive force between two points of a conductor carrying a current of 1 ampere when the power dissipated between the two points is 1 watt" [3]. Realization of the volt in the SI system rests on experiments comparing an electrostatic force with a mechanical force, but the uncertainties obtained by this method are much too great to meet the requirements of modern instrumentation. Conversely, the stability of voltage references based on the Josephson effect depends solely on frequency stability, which can easily reach 10-12. For this reason, National Metrology Laboratories started using the AC Josephson effect as a representation of the volt and adopted KJ, KJ-90= 483 597,9 GHz/V as a true value for the Josephson constant. This value was accepted by international agreement at the 18th General Conference on Weights and Measures and came into application on 1st January 1990. VGTU EF ESK http://www.lne.eu/en/r_and_d/electrical_metrol ogy/josephson-effect-ej.asp [email protected] ELEKTRONIKOS PAGRINDAI 30 2008 Hall effect ... Let us consider that a current is passed through a semiconductor and a magnetic field is applied at a right angle to the direction of the current flow. Then an electric field is induced in the direction mutually perpendicular to I and B. This phenomenon is known as the Hall effect, discovered in 1879 by E. H. Hall. The electrons flowing with some drift velocity experience the Lorentz force F L = −q [v E × B] It tends to drive electrons towards the right face D of the bar. The electrons moving to the right leave positive ions and as a consequence the electric field arises in the bar. This produces the electrical force FE = qE E = U H / d In equilibrium UH = FL = FE, qv E B = qE 1 jBd = RH jBd qn RH = 1 / qn E = vE B σ = qnµ n j = qnv E RH σ = µ n ... Simultaneous measurement of σ and RH can lead to the experimental value for the carrier drift mobility. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 31 2008 Diffusion of charge carriers in semiconductors It is also possible for current to flow in a semiconductor even in the absence of the field. It can flow due to a carrier concentration gradient in the crystal. The diffusion current can flow as a result of non-uniform densities of either electrons or holes. In neutral gas the flow or diffusion of particles occurs in the direction from the high density (high pressure) to the low density (low pressure) region. The particle flow depends on concentration gradient. It the instance when density depends on one coordinate, concentration gradient is given by grad N = VGTU EF ESK dN dx [email protected] ELEKTRONIKOS PAGRINDAI 32 2008 Diffusion of charge Krūvininkų carriers difuzija in semiconductors The existence of a gradient implies that if an imaginary surface (for example, indicated by dashed line) is drawn, the density of particles on one side of the surface is greater than the density on the other side. The particles are in a random motion. Accordingly particles move back and forth across the surface. Then in a given time interval more particles cross the surface from the side of greater density than in the reverse direction. Thus, the flow or diffusion of particles occurs. FΣ = − A dp ∆x dx FD = − kT VGTU EF ESK 1 dN N dx NV = NA∆x FD = FΣ 1 dp =− NV N dx p = N kT ... The diffusion force is proportional to temperature and concentration gradient. [email protected] ELEKTRONIKOS PAGRINDAI 33 2008 Diffusion of charge carriers in semiconductors ... As a result of diffusion the density of neutral particles becomes uniform. We will see that electrical field can appear in semiconductors as a result of charge carriers diffusion. ... The diffusion force acts in a way entirely analogous to that in which the force due to an electric field acts on electrons in a solid. The flow of particles is limited by collisions. FE = qE v En τr τr qτ r = µn E = E= qE = FE mn mn mn In the case of diffusion force: vDn = τr mn FD = − τr mn kT 1 dn 1 dn = − Dn n dx n dx τr qτ r kT kT Dn = kT = = µn mn mn q q The diffusion of electrons causes electron current to flow. jDn = −qnvDn = qDn VGTU EF ESK dn dx [email protected] ELEKTRONIKOS PAGRINDAI 34 2008 Diffusion of charge carriers in semiconductors Considering holes, we have : vDp = − Dp 1 dp p dx Dp = τr mp kT = µ p kT q jDp = qpvDp = −qDp dp dx Notice that the electron diffusion current is in the same direction as the positive gradient. The hole current flows in the opposite direction with respect to the positive gradient. … At any given temperature the diffusion coefficient and the mobility of carriers in a given material are not independent of each other. Dn Dp kT = = µn µp q This equation is known as Einstein’s relation. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 35 2008 Total current flow in semiconductors ... The current flow in a semiconductor is due to the motion of the charge carriers under the influence of applied fields or concentration gradients. It is quite possible to have these two effects occurring simultaneously and the net current flow is then the sum of drift and diffusion currents. jn = jnE + jnD = qnµ n E + qDn dn dx jp = jpE + jpD = qpµ p E − qDp dp dx j = jn + jp = jnE + jnD + jpE + jpD . ... The total current flow in a semiconductor is the sum of electronic diffusion, electronic drift, hole diffusion and hole drift currents. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 36 2008 Conductivity and current in solids. Problems 1. The mobility of electrons in silicon is around 1500 cm2/(V⋅s) at T = 300 K. Estimate the mean free path of an electron. Compare it with the lattice constant a = 0.543 nm. 2. Estimate the conductivity and resistivity of intrinsic silicon at T = 300 K. 3. Find the ratio of the values of electrical conductivity of intrinsic silicon at 40 and 200C. Assume that the forbidden energy gap in silicon is about 1.1 eV. Comment on the result. 4. Estimate the ratio of the values of electrical conductivity of extrinsic silicon at 40 and 200C. 5. Estimate the critical strength of the electric field in silicon assuming that the electron drift velocity in the critical field is equal to the mean-square thermal velocity. Assume that electron mobility in silicon is 0.13 m2/(V·s) at 300 K. 6. The visible light spectrum wavelengths are between 380 and 780 nm. Estimate the maximum gap energy of material suitable for visible light detector. 7. The resistivity of a doped silicon crystal is 9.27⋅10-3 m and the Hall coefficient is 3.84⋅10-4 m3/C. Assuming that conduction is determined by a single type of charge carriers, calculate the density and mobility of the carriers. VGTU EF ESK [email protected]