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Semiconductor Devices A brief review Dr. K. Fobelets Purpose of the course • Study bipolar devices in more detail – Diodes and BJTs – Closer to reality: recombination – What causes the delays in these devices when switching? The most frequently used sentence in this course will be: Excess minority carrier concentration Structure • 1. Lectures : 10 hrs – Basic principles based on Q&A session – Recombination and how does it impact the characteristics – LONG pn diode – correct and approximated solutions – LONG BJT – Switching of pn diodes and BJTs • 2. Classes: solving past exam papers Review • Electrons and holes • Minority and majority carriers • Energy band diagram Free charged carriers in Si Covalent bond Si Intrinsic Si Thermal energy: kT Movement: kT Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Extrinsic Si Extrinsic Si Si Si Si Si Si Si Si Si Si Si Si As Si Si Si Si Si Si Si Si Si Si NA p-type ND n-type Si B Extrinsic Si Obtained by doping B As In semiconductors two types of free charged carriers exist: electrons and holes. Q1: What are holes? Si Si Si + Si Si Si Si a) Spherical voids in a semiconductor b) A positively charged Si atom that has lost its electron c) A positively charged particle that is the result of quantum mechanics Si Si Si Si Si Si Si Si Si Si Si C The two charged particles describe together the conduction in semiconductors. Electron e- with charge q=-e and mass mn = m0 m*n Hole h+ with charge q=+e and mass mp = m0 m*p Intrinsic silicon (Si) has a small number of both free electrons and holes such that ni=pi. In order to increase the free carrier concentration, the semiconductor can be doped. With donors ND more electrons are created, with acceptors NA more holes are generated. Q2: When intrinsic Si is doped with donor atoms, which of the following statements is correct? a) b) c) d) n = p = n i = pi n > ni & p < ni n > p > ni p > n > ni n: electron concentration p: hole concentration ni: intrinsic electron concentration pi: intrinsic hole concentration B n > ni & p < ni in an n-type semiconductor. n-type semiconductor n = ND p = ni2/ND p-type semiconductor n = ni2/NA p = NA By heart The concept of majority carrier and minority carrier is important in semiconductor devices. Majority carrier is the carrier type in a doped semiconductor with the highest concentration. Minority carrier is the carrier type with the lowest concentration. Q3: True or False? The holes are the majority carriers in a p-type semiconductor (doped with acceptor atoms NA). TRUE p-type semiconductor p p > n p p-type p-type semiconductor electron semiconductor hole concentration concentration n-type semiconductor n n > p n n-type n-type semiconductor hole semiconductor electron concentration concentration MAJORITY CARRIERS MINORITY CARRIERS Drift and diffusion • Two types of carrier movement – As a result of an electric field → DRIFT – As a result of a carrier gradient → DIFFUSION Drift of carriers under influence of an electric field: E + E - + E - J q number of carriers v J q number of carriers E Diffusion of carriers due to a carrier gradient x J q diffusion J qD d dx constant concentrat ion gradient number of carriers The purpose of semiconducting devices is to generate a current/voltage in response to an applied voltage/current. Two different types of current can exist in a semiconductor: drift and diffusion current. The expression of the total current that can flow in a semiconductor is given by the drift-diffusion equation: dn ( x ) J n ( x ) e n n ( x ) E ( x ) eD n dx J p ( x ) e p p ( x ) E ( x ) eD p (1) dp ( x ) dx (2) Q4: Which statement is true? a) b) c) d) Term (1) is drift current and (2) diffusion current Term (2) is drift current and (1) diffusion current Only term (1) can exist in a semiconductor Only term (2) can exist in a semiconductor A Drift current is proportional to the carrier concentration and the electric field Diffusion current is proportional to the carrier gradient. E(x) n(x) p(x) Jndrift Jpdrift Jndiff Jpdiff Motion of free charged carriers in a semiconductor. Q5: If a p-type semiconductor at room temperature is conducting carriers due to drift, which of the following motion paths would be followed by the holes? E E + + a) c) (b) (d) + E - + E - B When carriers move in a semiconductor they are scattered along the way. This means that they will be accelerated by the electric field (in this case) and then interact with atoms, impurities, other carriers that makes them lose some of their kinetic energy = scattering. Therefore the carriers will travel with an average velocity in amplitude and direction. v E e m Q6: Solve diffusion processes p+ 1. 2. n p Draw arrows indicating the direction of diffusion of carriers. Identify the type of carriers that is diffusing. Solution p+ Holes Electrons n p p+ n p Q7: Why is there no net current while diffusion is happening? 1. Because hole diffusion and electron diffusion cancel each other. 2. Because an internal electric field is built up across each junction causing drift of holes/electrons that cancel the diffusion of .holes/electrons. 3. Because holes and electrons diffuse automatically back to where they came from. - E + p+ Holes +E n p Holes diffusion Electrons drift Electrons 2. Because an internal electric field is built up across each junction causing drift of holes/electrons that cancel the diffusion of .holes/electrons. Depletion + Si Si Si Si Si Si Si B Si Si - Si Si Si Si B E - ND p-Si - Capacitive effect Capacitive effect NA - B : boron atom ionised n-Si Si Si Si Si Si Si B Si As+ Si Si Si + Si Si Si Si As E + + As : arsenic atom ionised As Q8: True - False Ec EF Ev The position of the Fermi level EF determines the type of the semiconductor. Q9: Multiple choice Ec EF Ev 1. This is the energy band diagram of an n-type semiconductor. 2. This is the energy band diagram of a p-type semiconductor. 3. This is the energy band diagram of an intrinsic semiconductor. Ec EF Bottom of conduction band EG Bandgap. No energy levels in this energy region. Ei Intrinsic “level”. Is the position of the Fermi level EF when the semiconductor is intrinsic. Ev Top of valence band Position of Fermi level is determined by the doping type and density For n-type Si: E c E F n N C exp kT N E E F N C exp c C kT n ND N E c E F k T ln C ND Devices • A combination of n and p type semiconductors plus ohmic contacts to apply the external voltages/currents makes devices • When combining a-similar materials diffusion will occur and as a result an internal electric field will be built up to an amount that opposes diffusion current. Energy band diagram e.g. p-Si – n-Si p-Si – n-Si – p-Si It is possible to start from the knowledge on workfunctions, f and the energy reference: the vacuum level, Evac. The workfunction is dependent on the doping concentration! Evac Evac p-Si e×fn-Si n-Si e×fp-Si EF EF Evac Evac p-Si e×fn-Si n-Si e×fp-Si EF EF Depleted region on both sides Evac e V 0 e f p Si f n Si Evac p-Si e×fp-Si e×fn-Si n-Si Ec EF Ec EF Ev Ev Diffusion and drift can occur at the same time. E A charge packet Both also always occur across junctions A look at the short pn-diode PN diode I p p n n E V p n Short PN diode DIFFUSION I p p n n E V p n Short PN diode DIFFUSION I p p n n E V p n Short PN diode DIFFUSION p p n V n Minority carrier concentration I How do we find the current? E p n distance Linear variation of minority carrier concentration Apply diffusion current formula to the minority carrier variation Short PN diode I p p n n V E p n Only few carriers can contribute to the current Contents of course this year • Long pn diode – Introducing the concept of recombination of carriers. – Switching of the pn diode, where does the delay come from? • Bipolar junction transistor – Internal functioning – Switching delays But what happens in a long pn diode? p n p n Short Long Ln Lp Minority carrier diffusion length In long semiconductors recombination of the minority carriers will occur whilst diffusing Excess holes, p in an n-type semiconductor will recombine with the large amount of available electrons. Loss of both carrier type, but felt most in excess minority carriers. Remember: the amount of majority carriers is much larger than the excess. In long semiconductors recombination of the minority carriers will occur whilst diffusing Excess holes, p in an n-type semiconductor will recombine with the large amount of available electrons. Loss of both carrier type, but felt most in excess minority carriers. Remember: the amount of majority carriers is Injection of carriers much larger than the excess. x Lp • Diffusing minority carriers (e.g. holes) recombine with majority carriers (electrons) within a diffusion length Lp Generation-recombination • Generation of carriers and recombination is continuously happening at the same time such that the equilibrium carrier concentrations are maintained. R=G Charge neutral Recombination - generation • In case there is an excess carrier concentration then the recombination rate R of the excess, will be larger than its generation rate, G: R>G When there is a shortage, then G > R Recombination - generation • Simple model: Recombination/generation rate is proportional to excess carrier concentration. • Thus no net recombination/generation takes place if the carrier density equals the thermal equilibrium value. Recombination of e- in p-type semiconductor h+ Recombination of in n-type semiconductor U n Rn G n U p Rp Gp n p n p0 n p n p n0 p n p n p n p Diffusion, drift and recombination of carriers What is the consequence of this recombination on the characteristics of the pn diode with neutral regions larger than the diffusion lengths of the minority carriers? In the pn diode the carrier gradient determines the current thus we have to find the function p(x) of the minority carrier concentration. • Note, reasoning done for p(x). For n(x) analogous approach. Mathematical description of diffusion and recombination Jp (x+Dx) Jp(x) A x x p t Rate of hole variation x x Dx = x+Dx 1 J p ( x) J p ( x Dx) q Dx p p Variation of hole Recombination concentration in + rate Dx x A/s Mathematical description of diffusion and recombination p t Dx 0 : 1 J p ( x) J p ( x Dx) Dx x x Dx q p ( x, t ) 1 J p t q x p p p p p p 0 p = bulk defined + excess concentration p t 1 J p q x with p p Jp : total current = drift + diffusion Neglect drift current (no electric field applied) 2 p 0 p n0 ni ND Mathematical description of diffusion and recombination J p ( x ) eD p p t 1 J p q x p p p p0 p p t 2 Dp x 2 dx p p = bulk defined+ excess concentration p 2 Dp p dp ( x ) x 2 p p with 2 p 0 p n0 ni ND Solve equation in steady state p t p 0 2 x 2 p p D p p 2 Lp p Boundary conditions: General solution of 2nd Diffusion length contact x X n p 0 0 x 0 p D p order differential equation: p ( x ) Dp Dp Xn sinh L p x X sinh L p n Xn x x p ( x ) C 1 sinh C2 L p Too complicated • Short approximation • Long approximation Xn << Lp Xn >> Lp p ( x ) LINEAR Dp Xn sinh L p x X sinh L p n EXPONENTIAL Short semiconductor • Xn ≤ Lp carriers do not have time to recombine (=∞) ! • Taking linear approximation. pn(x) pn(x) p’n pn(x) Dp pn(x)= pn0+ Dp (1–x/Xn) NO recombination : variation of the excess carrier concentration linear pn0 0 Xn x Contact imposes pn(Xn)=0 Diffusion and recombination • Xn >> Lp carriers do have time to recombine (t<∞) ! • Taking exponential approximations pn(x) pn(x) p’n Dp pn0 pn(x) pn(x)=pn0+ 0 Lp Dp X 1 exp n L p Xn x exp X x exp n L L p p When recombination occurs and Xn >> Lp variation of the excess carrier concentration is exponential Contact imposes pn(Xn)=0 pn still too complex for quick calculations • Take really extreme case • Xn >>> Lp or Xn → ∞ pn(x)= Dp X 1 exp n L p exp X x exp n L L p p x D p exp L p Note: I and Q of both expressions of pn(x) for Xn → ∞ the same I for pn(x)= x D p exp L p same as for linear approximation when Xn=Lp Diffusion and recombination • Xn >>> Lp carriers do have time to recombine (t<∞) ! • Taking exponential approximations pn(x) p’n pn(x)=pn0+Dp e-x/Lp pn0 pn(x) Dp When recombination occurs and Xn → ∞ variation of the excess carrier concentration is exponential 0 Lp ∞ x Imposes pn(Xn)=0 SHORT ↔ LONG approximation pn(x) Lp=200 nm, Xn=20nm Correct solution Exponential solution Linear solution pn(x) Lp=Xn=200nm Boundary of short Short x x pn(x) Lp=200 nm, Xn=400nm pn(x) Intermediate x Lp=200 nm, Xn=1000nm Long x • Calculation of currents in pn diode with neutral regions larger than the diffusion length, using the long semiconductor approximation → • Exponential variation of the excess minority carrier concentration. Carrier injections: forward bias e-diff p • Carrier injection across junction n h+diff -wp 0 wn V n ' p n p 0 exp VT np(-x) pn(x) n’p p’n np0 V p ' n p n 0 exp VT • Creates minority carrier concentration gradients pn0 -x x np0=ni2/NA & pp=NA pn0= ni2/ND & nn=ND Carrier injections: reverse bias e-drift n + h drift p -wp 0 np(-x) wn pn(x) np0 V n ' ' p n p 0 exp V T V p ' ' n p n 0 exp V T pn0 n’’p -x • Minority carriers are swept across junction V<0 p’’n x • Small amount of minority carriers → small current Thus e-diff p n h+diff -wp 0 wn np(-x) n’p np0 -x Dnp pn(x) Dpn p’n pn0 x ( x) n p ( x ) D n p e Ln Dnp = np0 (eeV/kT -1) (x) p n ( x ) D p n e Lp Dpn = pn0 (eeV/kT -1) Two methods to calculate current -wp np 0 wn Slope I pn Qn Dnp -x 0 Qp Dpn 0 1. Gradient excess carrier concentration 2. Re-supply of recombined excess charge x x 1. Excess carrier concentration gradient Maximum diffusion currents at the edges of the transition region np e-x Slope Dnp 0 -wp In = e A Dn dnp/dx = max @ x=0 pn h+ Dpn 0 wn x Ip = -e A Dp dpn/dx = max @ x=0 1. Excess carrier concentration gradient Fill in expression for excess carrier concentration e eV n p ( x ) n p 0 exp 1 e kT In max diff h+ ( x) p n ( x ) p n 0 exp 1 e k T Ln eV dn p 0 exp 1 e kT eAD n dx eV (x) ( x) eV e dp n 0 exp 1 kT Ln Ip eAD max diff Lp p max diff Dn eV exp 1 Ln kT eAn p0 In Lp dx x0 In ( x) x0 Ip max diff eAp n 0 D p eV exp 1 Lp kT Ip Changing gradient! → Changing diffusion current density p Itot n Ip In Itot=In + Ip Ip In In Ip I x diff x drift eAn tot eV p0 D n exp e 1 Ln kT In x diff ( x) Ln eAp n 0 D p eV exp 1 e Lp kT Ip In I x diff x drift tot Ip x diff (x) Lp 2. Re-supply of recombined excess carriers -wp 0 wn np -x x pn Ip np = Dnp e-(-x)/Ln np0 I Dnp Qn In pn = Dpn e-(x)/Lp Dpn Qp 0 0 pn0 x Excess carrier charge Q recombines every seconds (carrier life time). For steady state Q has to be re-supplied every seconds → current 2. Re-supply of recombined excess carriers Charge – minority carrier life time ratio np np = Dnp pn Ip e-(-x)/Ln Qn np0 Dnp In pn = Dpn e-(x)/Lp Dpn Qp -x 0 -wp 0 pn0 x 0 wn Charge = area under excess carrier concentration: integrate -∞ and + ∞ are the contacts: excess charge = 0! 0 Qn = -e A ∫-∞ np dx In = Qn/n = e A Ln Dnp /n ∞ Qp = e A ∫0 pn dx Ip = Qp/p = e A Lp Dpn /p Total current Same equation as short diode with length exactly equal to the minority carrier diffusion lengths • I = Ip(0) + In(0) = e A (Dp pn0 /Lp + Dn np0/Ln )(eeV/kT -1) • I = I0 (eeV/kT -1) • With I0 = e A (Dp pn0/Lp + Dn np0/Ln) Reverse bias current SHORT ↔ LONG approximation error on current calculation: ratio of currents 10 9 Error on linear and exponential approximation same when Xn=Lp 8 Ireal/Iapprox 7 6 Ireal/Iexp 5 Ireal/Ilin 4 3 2 1 0 0 1 2 3 Xn/Lp 4 5 • Non-idealities in the pn diodes Log(I) ideal real c) b) a) V (a) Low voltage: low injection of carriers Log(I) ideal real a) V V (c) High voltage: high injection of carriers Log(I) ideal real n’p ≈ pp p’n ≈ nn c) V I tot eV nkT I s e a) n=2 1 b) n=1 c) n=2 (d) Higher currents Log(I) ideal real d) Current determined by resistance V Switching of p-n diodes • When a p-n diode is forward biased, excess carrier concentrations exists at both sides of the depletion region edge. • To switch the diode from forward to off or reverse bias, this excess carrier concentration needs to be removed. • The transients resulting from the time it takes to remove the excess carriers will lead to the equivalent capacitance. np pn p -wp 0 wn n np p eh+ pn n Switching off i on Steady state snap shots off -wp 0 wn 0 t p To this? How do we go from this: -wp p+pno Excess carrier concentration Dpn pno Off: NO current flows!!! x n 0 wn Variation of the excess carrier concentration as a function of time. p(x,t) p t 1 J p q x p p Relationship for charge Qp contact eA p ( x, t ) t 0 dQ p ( t ) dt dQ p ( t ) dt eA e Ip dx eA contact e Jp contact Q p (t ) p J p x 0 eA e Jp 0 contact dx 0 Q p (t ) p eA p p dx Transient during switching off Recombination term i(t)= I + dQ/dt = Q/ + dQ/dt Charge depletion term (or buildup) Excess charge due to charge injection at any instance of time Average lifetime of minority carriers For switch from on to off: At t<0 → Ion=Ion (Von) At t≥0 → Ioff = 0 (Voff = 0) And at t=-0 Q(0)=Ion At t→∞ Q(∞)=0 t>0 0 = Q/ + dQ/dt Q(t)=Ion e-t/ Since no current in “off”, charge has to disappear by recombination! Transient during switching off variation of the excess carrier concentration as a function of time Qp(t)=eA∫p(x,t)dx=Ippe-t/p p Dpn Variation in time gradient→ i≠0 i=0→gradient=0 t=0 x A voltage, vd will exists across the diode as long as charge remains p(x,t)=Dp(vd(t)) e-x/Lp Revision • When a pn diode switches, the excess minority carrier concentration needs to change. The removal of the excess minority carrier concentration causes the delay in the pn diode. • The variation of the excess carrier concentration as a function of time given by: i p (t ) Q p (t ) p dQ p ( t ) dt ON-OFF (open circuit) take: p+n → Itot ≈ Ip p+ vd i p (t ) n Q p (t ) Ip dt p @ t 0 ; i p ( 0 ) I ON R dQ p ( t ) V R @ t 0; i p ( 0 ) 0 t=0 V 0 Q p (t ) p dQ p ( t ) dt t Q p ( t ) I ON p exp p Q p (0 ) p OFF (open circuit) → ON take: p+n → Itot ≈ Ip @ t 0; i p (0 ) 0; Q p (0 ) 0 @ t 0 ; i p ( 0 ) I ON p+ vd I ON n Ip Q p (t ) dQ p ( t ) p p I ON Q p ( t ) dt Q p ( t ) p I ON ln Q p ( t ) p I ON t=0 V dt dQ p ( t ) R R dQ p ( t ) p i p (t ) V Q p (t ) t 0 p dt integrate p t p ln Q p ( t ) p I ON ln p I ON Q p ( t ) p I ON ln p I ON t p t p t I Q p ( t ) p I ON p I ON exp p ON p 1 exp t p dQ p ( t ) dt Reverse recovery transient Switch the diode from forward to reverse bias ei h+ np on Steady state snap shots pn n p -wp 0 wn 0 e- t h+ off To this? How do we go from this: p Excess carrier concentration -wp 0 Dpn Dpn 0 x Reverse bias current flows!!! wn Transients when switching to reverse bias e(t) i(t) R E e(t) p n If≈E/R If I t Ir≈-E/R -E V p If → gradient≠0 Ir → gradient≠0 -Ir i(t) t t x v(t) t -E Storage delay time: tsd If i(t) v(t) t -Ir tsd Time required for the stored charge to disappear tsd = minority carrier ln(1 + If/Ir) Calculate storage delay time: tsd IF i p (t ) i(t) Q p (t ) dQ p ( t ) dt p @ t 0 ; i(0 ) I F Q (0 ) p I F v(t) t @ t 0 ; i(0 ) I R X Q ( 0 ) p I R ! @ t t sd ; Q ( t sd ) 0 -IR tsd 0 t t sd i p ( t ) I R IR Q p (t ) p dQ p ( t ) dt Calculated storage delay time: tsd IR dt v(t) t p t dQ p ( t ) dt p dt p dQ p ( t ) dQ p ( t ) pIR Q p (t ) ln p I R Q p ( t ) integrate t 0 p t tsd p I R Q p (t ) IF i(t) -IR Q p (t ) ln p I R Q p ( t ) ln p I R Q p ( 0 ) p t exp p I Q p (t ) p R pIR pIF Q p ( t ) p I R p I R p I F t exp p Calculated storage delay time: tsd IF i(t) Q p ( t ) p I R p I R p I F v(t) t t sd t 0 p I R p I R p I F IR t sd p ln I R I F -IR tsd t exp p t exp sd p I I F p ln R IR After: tsd IF i(t) Q p ( t sd ) 0 vd 0 v(t) t Build-up of depletion region t bu RC depl -IR vd E tsd Small signal equivalent circuit • Junction capacitance • Diffusion capacitance • Due to depletion region • Due to charge storage effects p n w • Cj = e A/w • w function of bias → C voltage variable capacitance • Important in reverse bias np p pn n -wp 0 wn • Cd = dQ/dV = d (I )/dV = e/kT I • Important in forward bias Equivalent conductances • Diffusion conductance • Series resistance rs • gd = dI/dV = e/kT I0 ≈ e/kT I • Due to n and p region + contact resistance eeV/kT • Vd = Vappl – rs I • Slope of the current voltage characteristic in forward bias rd rs Cj Cd Only linear circuit elements present Large signal equivalent circuit Rs C Reverse bias: depletion capacitance Forward bias: diffusion capacitance Non-linear circuit elements present Conclusions • The characteristics in a pn diode are based upon excess minority carrier diffusion. – Excess carrier concentrations are being formed by injection of carriers across the junction. – The gradient of the excess minority carrier concentration at the junction determines the magnitude of the current. – Delay times are due to the storage of excess minority charge in the layers. Revision • When recombination is taken into account, the excess minority carrier concentration reduces while diffusing through the neutral regions of the diode. • The variation of the excess carrier concentration is then given by: p t p 2 Dp x 2 p p Lifetime of minority carrier holes Revision • The steady state solution for the excess minority carrier concentration is then: p ( x ) Dp Xn sinh L p x X sinh L p n • This is considered too complex for quick calculations and approximations are used in the case of a short or long neutral region. Revision • Short: Xn ≤ Lp linear pn(x) pn(x) p’n pn(x) Dp pn(x)= pn0+ Dp (1–x/Xn) pn0 0 Xn x Contact imposes pn(Xn)=0 Revision • Long: Xn >>> Lp exponential pn(x)=pn0+ pn(x) Dp X 1 exp n L p exp X x exp n L L p p Dp pn0 pn(x) p’n 0 pn(x)=pn0+Dp e-x/Lp Lp ∞ x Imposes pn(Xn)=0 Revision • These approximation make some errors in the calculation of the current and the charge stored in the neutral regions. • However we will see that: 1. I and Q for simplified and non-simplified exponential variation of pn(x) for Xn → ∞ is the same 2. I for pn(x) when Xn=Lp x = D p exp Lp is same as for linear approximation Errors on current 1400 Lp=20 nm Current (a.u.) 1200 1000 Series1 Correct Exponential Series2 Linear Series3 800 600 400 200 0 101 20 2 40 3 2004 Xn (nm) Short = good approximation up to Xn = Lp Long = good approximation up to Xn > 5 ×Lp