# Semiconductor Physics. Lecture 3

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1 Semiconductor Physics Lecture 3

2 Intrinsic carrier density Intrinsic carrier density Law of mass action Valid also if we add an impurity which either donates extra electrons or holes the number of carriers remains constant for a given temperature

3 Doping Donor levels lie just below conduction band and acceptor levels just above the valence band E c Donor ions E c E v Acceptor ions E v Donor levels lie E D below the bottom of the conduction band and acceptor levels E A above the valence band Since E a and E d are smaller than kt at room temperature the dopant sites are fully ionised so the number of majority carrier (those provided by doping, electrons in the case of n-type, holes in the case of p-type) is simply the number of dopant atoms

4 Electron and hole concentrations Recalling the expression for the Fermi energy And the law of mass action

5 Temperature dependence II Initial rise as dopant sites become fully ionised Flat region where majority of carriers are from the dopant (good for devices) Final rise where intrinsic behaviour dominates

6 Conduction If we apply an electric field electrons and holes accelerate until they are scattered We can define an average drift velocity v in terms of the mean time between scattering events This is a completely classical description but we can bring in quantum effects by replacing the electron mass with the effective mass Current density J is then Conductivity is given by

7 Conductivity For a metal Recognise the second part of the RHS is the mobility For semiconductors we have more than one type of carrier with more than one mobility electrons and holes have different effective masses and may have different lifetimes

8 Conductivity II Reason for separating the conductivity into mobility and carrier density In semiconductors both carrier density and mobility are temperature dependent in metals only the mobility is We have generally at least two types of carrier and we may have even more

9 Mobility The mobile carriers in a semiconductor have an average of 3/2 kt of thermal energy V th =10 7 cm/s they undergo regular collisions with defects and phonons mean free path of 10-5 cm lifetime of 1 ps If an electric field is applied then the carriers will begin to accelerate until they are scattered

10 Various contributions Lattice scattering gives mobility a T -3/2 dependence (more phonons at higher T so less mobility) Impurity scattering has a T 3/2 effect carriers spend less time near to impurity sites Also principal impurity sites are the dopant atoms so the more carriers we put in by increasing the doping then the less mobile these carriers are

11 Hall Effect For a conductor passing a current in a magnetic field there is a force on the moving current perpendicular to both the current and the magnetic field directions. Since the carriers cannot escape from the wire a voltage develops which counteracts this force the Hall voltage For a single carrier type

12 But by definition the hall coefficient R H is Hall Effect II Substitution for E y and J x Cancel through and have A simple way to measure the carrier density and type since q includes the sign of the carriers (-ve for electrons +ve for holes)?

13 Hall Effect in Semiconductors Now have to consider two types of carrier consider the y component of the current density j y So But Setting j y =0

14 Hall Effect in Semiconductors II So R H is Hall Effect appears to depend on mobility (counter intuitive) Might expect This is what Sze quotes as the Hall coefficient

15 The general case and limiting cases Our model considered only the carrier velocity produced by the applied electric field In general the carriers have a thermal velocity and will tend to spiral about the magnetic field our model works only if In general For small B the terms in B 2 go to zero and we recover our original result For large B the B 2 terms dominate and we find the naïve simple view of the Hall coefficient holds

16 Hall effect in intrinsic semiconductors Implies that in general intrinsic semiconductors where n=p will have a Hall effect But at high fields this will go to

17 Carrier Diffusion Up to now we have considered equilibrium thermal properties and the effects of external fields now turn to effects of concentration gradients Imagine a semiconductor whose electron density varies spatially The electron mean free path=l Where Flow of electrons across a plane at x=0 from left Flow of electrons across a plane at x=0 from right

18 Carrier Diffusion II Net flow Replace n(-l) and n(l) by taylor expansion about n(0) Each carrier carries a charge and so carrier diffusion gives rise to a current

19 Einstein relation Equipartition for a 1-D case implies and given Einstein relationship relating two transport properties Diffusivity (freedom of electrons to move under concentration gradients) and mobility (freedom of electrons to move under electric field)

20 Current density equations In general we have currents flowing because of both concentration gradients and electric fields For electrons For holes The total conductivity

21 Carrier injection So far we have considered cases where the semiconductor is in thermal equilibrium (at least locally) and the law of mass action holds Non-equilibrium case We can inject extra carrier by various methods by shining light on the material by biasing a pn junction Magnitude of the number of carriers determines the level of injection

22 Carrier injection by light Since a photon creates an electron hole pair p= n Case of n-doped silicon where N D =10 15 cm -3 and n i =10 10 cm -3 n=10 15 cm -3, p= 10 5 cm -3 If we inject p= n=10 12 cm -3 Increase p by 7 orders of magnitude Increase n by 1% Low level injection affects only minority carrier concentration If p= n=10 17 cm -3 Overwhelm the equilibrium majority carrier concentration High level injection

23 Direct recombination Injected carriers are a non-equilibrium phenomenon and are removed by recombination of electron hole pairs In a direct gap (like GaAs semiconductor this occurs directly electrons and holes simply combine and annihilate (sometimes with the emission of a photon) Recombination rate is proportional to the concentration of holes and electrons For thermal equilibrium case where carrier concentrations are constant

24 Direct recombination II Rates of recombinatioon and generation are thus Rate of change of hole concentration is given by For steady state define U the net recombination rate U=R-G th =G l Recalling n= p For low level injection where p and p no are small compared to n no Recombination rate is proportional to excess minority carrier concentration

25 Direct recombination III Like a first order chemical reaction rate depends on one reactant Also because Minority carrier lifetime is controlled by majority carrier concentration

26 Indirect recombination For indirect band gap semiconductors like silicon direct recombination is rare because have to lose crystal momentum as well as energy instead recombination occurs indirectly via trapping states (defects and impurities) Four possible processes (a) Trapping of an electron (b) Emission of a trapped electron (c) Combination of a trapped electron and a hole (d) Formation of a trapped electron and free hole pair

27 Indirect recombination II The rate at which electrons are trapped is proportional to the number of electrons n and the number of non-occupied trapping states. The probability that a state occupied is given by the Fermi function Rate will then be proportional to Or Where the constant of proportionality is given as the electron thermal velocity times a cross section for the trapping atom (of order cm 2 ) Can imagine this as the volume swept by the electron in unit time. If an unfilled trapping state lies in this volume the electron is trapped

28 Indirect recombination III Rate of emisson of the electron from the trapped state For thermal equilibrium R a =R b so the emission probability e n But And so

29 Hole annihilation and creation The rate of hole annihilation by a filled trapping state is analogously And the rate of hole emission is Where e p is the emission probability which can be obtained from the thermal equilibrium condition R c =R d

30 Net recombination rate For steady state number of electrons leaving and entering the CB are equal Principle of detailed balance Similarly for holes in VB Combining Substituting for R a etc gives

31 Net recombination rate Solve this for F and then substitute back to get a net recombination rate Horrible expression but can simplify since for low-level injection n n >> p n and since the trapping states are near the centre of the gap Same form as for direct gap semiconductors but Minority carrier lifetime is now controlled by density of trapping states not majority carrier concentration

32 Energy dependence of recombination Can also simplify this equation by assuming that the electron and hole cross sections are equal Under low injection conditions this approximates to Where the recombination lifetime

33 Carrier depletion Can also perturb from equilibrium by removing carriers Setting p n and n n < n i Becomes With the generation lifetime

34 Continuity equation We have seen the ways carriers can move under applied field and concentration gradients and that the can be created and destroyed but in general all these processes occur together If we consider a slice of semiconductor of width dx then rate of change of the carrier density will be the sum of the currents entering from each surface and the overall generation and recombination rates.

35 Continuity equation II Expand currents as Taylor series Substitute in for various forms of current and for minority carriers this becomes

36 Continuity equation III We must also satisfy Poisson s equation Where the charge density is the sum of the hole, electron, and ionised donor and acceptor densities taking into account their relative charges. In general the continuity equation is difficult to solve analytically but it can be done for some special cases

37 Light falling on a semiconductor Steady state with no electric fields For the infinite case pn(0)=constant and pn( )= pno diffusion length

38 Extraction case If we extract all minority carriers at a distance W the boundary condition becomes pn(w)= pno Diffusion current density at x=w

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