The Law of the Junction Revisited

Mark Lundstrom Network for Computational Nanotechnology and Purdue University

Consider a one-sided, short base diode like that shown in Fig. 1. We usually analyze the I-V characteristics by assuming the so-called Law of the Junction,

n 2 n(0) n eqVA kBT 1 i eqVA kBT 1 ! = po( " ) = ( " ). (1) NA

We find the current by assuming that electrons diffuse across the p-region, so

!n(0) D n 2 J qD q n i eqVA kBT 1 n = n = ( " ). (2) WP WP NA

(If the diode has a long p-type region, then WP is replaced by Ln , the minority carrier diffusion length.) To justify the Law of the Junction, we assume low level injection and that the quasi- Fermi levels are constant across the depletion region, as sketched in Fig. 1.

EC Fn

qVA Fp

EV

xp xp WP+xp

Fig.1 An NP homojunction under forward bias

The Law of the Junction generally works well, but under high bias, the assumptions of constant quasi-Fermi levels across the space-charge region and low level injection in the p-region may begin to lose validity.

Lundstrom 1 2/27/13 The metal-semiconductor (MS) diode sketched in Fig. 2 is a different kind of junction. The Law of the Junction cannot be used in this case, so its I-V characteristics must be derived differently.

q(Vbi ! VA ) EC φBn

qV Fn A EFM

x 0

Fig. 2 A metal-semiconductor (MS) diode under forward bias.

Most MS junctions obey the thermionic emission theory. In the N-type bulk, we have ND/2 carriers traveling in the +x direction at the thermal velocity,

2k T ! = B = 2! , (3) T " m* R

where !R is the so-called Richardson velocity. The probability that electrons can get over a barrier of height ΔE is e!"E kBT , so the current from the n-region to the metal is

! N $ (q V (V k T J + = q D ' e ( bi A ) B . (4a) n "# 2 %& T

There is also a current from the metal to the semiconductor, which at zero bias must be the same as the current from the semiconductor to metal at zero bias. The barrier height from the metal to the semiconductor does not change with bias, so we conclude that the current from the metal to the semiconductor is always

" N % qV k T J ! = q D ( e! bi B . (4b) n #$ 2 &' T

Lundstrom 2 2/27/13 The net current for the MS diode is

qV k T qV k T J = J + ! J ! = qN " e! bi B e A B !1 , (5) n n D R ( ) where !R = !T 2 is the Richardson velocity. The I-V characteristic of an MS diode is often expressed in terms of the Schottky barrier height by using the expression for the built in potential of an MS diode,

qVbi = !Bn + kBT ln(ND NC ) (6)

"#bn kBT qVA kBT J == qNC!R e (e " 1) (7) Heterojunctions offer even more possibilities. Figure 3 shows two different kinds of Np heterojunctions. (The capital letter denoted the wider bandgap of the heterojunctin pair.) Heterojunction pairs are characterized by how the bands line up at the junction. The parameters,

!EC and !EV are material-dependent (and sometimes fabrication dependent) parameters that are ! + ! + known for common heterojunctions. When EC (0 ) > EC (0 )and EV (0 ) < EV (0 ) , we obtain the energy band diagram on the left in Fig. 3, In this case, there is a conduction band “spike” but ! + ! + none in the valence band. When EC (0 ) < EC (0 )and EV (0 ) < EV (0 ) , we obtain the energy band diagram on the right in Fig. 3. In the second case, there is no spike in either the conduction or valence band. What is the Law of the Junction for these junctions? The answer is not obvious by inspection, but one might expect the junction with the conduction band spike to behave as a metal-semiconductor junction for electron injection. Even when there are no band spikes, we will still need to deal with the fact the there are two different intrinsic carrier concentrations (because there are different bandgaps and effective masses on the two sides of the junction) so we need to decide which one to use in eqn. (1).

!EC !E C EC E E C F Fn

!EV !E V

E V EV

xp xp x x p p Fig. 3 Two hypothetical hetero-Np junctions. For both junction, then n-type region has a wider bandgap than the p-type region, but the band offset are different in the two cases on the ! + left, EC (0 ) > EC (0 ), which produces a conduction band spike. For the case on the ! + right EC (0 ) < EC (0 )and no conduction band spike occurs. For both cases, ! + EV (0 ) < EV (0 ) , so no valence band spike occurs.

Lundstrom 3 2/27/13 Generalized Law of the PN Homojunction Let’s begin by deriving the familiar Law of the Junction from a different approach, and then see if we can apply that approach to heterojunctions. The goal is to avoid any assumptions about how the quasi-Fermi levels vary across the junction, because, as we will show, they are not constant when a band spike occurs. In the standard theory of the PN junction, we treat minority carrier diffusion away from the junction, but we ignore the thermionic emission process that is responsible for the injection across the barrier. In the MS diode, we treat only the thermionic emission process. Let’s revisit the PN junction and treat both processes as indicated in Fig. 4.

+ ! J1 J1 J2

EC

F Fn p

E V

xn 0 xp W P

Fig. 4 The NP junction under forward bias showing the thermionic emission processes across + ! the junction, J1 and J1 , and the diffusion current of minority carriers across the quasi-

neutral P-region, J2

The thermionic emission current across the junction from the left to the right is

N "q V "V k T J + = q D ! e ( bi A ) B . (8) 1 2 T

Electrons injected into the p-type region are assumed to be thermalized by strong scattering which produces a near-Maxwellian distribution. At the edge the depletion region on the p-side, the corresponding electron density is n(xp). Since the electrons are distributed in a thermal equilibrium Maxwellian velocity distribution, one-half of them have negative velocities and can return to the N-region, so

n(x ) J ! = q p " . (9) 1 2 T

Lundstrom 4 2/27/13 Let’s pause here for a brief digression. The electrons at the beginning of the N-region cannot be distributed exactly in a thermal equilibrium Maxwellian, or the current would be zero. We assume that the negative half is a little smaller than the positive half, which produces a small positive average velocity that represents the average velocity of the minority carrier electrons diffusing across the P-region. The assumption of strong scattering could lose validity for short p- type regions. It’s an interesting exercise to repeat the analysis below assuming ballietic transport across the p-type region.

Once the electrons are in the P-type bulk region, they diffuse across it, so

!n(xp ) J2 = qDn = q!n(xp )"Dp , (10) WP where

Dn !Dp = (11) WP or !Dp = Dn Ln for a long base diode.

The currents must balance, so

+ ! J1 = J1 ! J1 = J2. (12) By using eqns. (8) – (10) in eqn. (12), we can solve for

$ 1 '* n2 - i qVA kBT . (13) !n(0) = n(0) " no (0) = & ), / (e " 1) %&1+ #Dp #R ()+ N A . Equation (13) is the conventional Law of the Junction, eqn. (1), multiplied by a factor that is less than 1. We can use eqns. (10) and (13) to find the electron current as

! n2 $ ( 1 + i qVA /kBT . (14) Jn = q# & * -(e . 1) " N A % )*1 'Dp + 1 'R ,-

Let’s examine the results for two limits. First, assume that !Dp << !R , which means that electrons diffuse away from the junction in the quasi-neutral P-region much more slowly than they are injected across the junction by thermionic emission. In this case, R !1, and eqn. (13) reduces to the Law of the Junction. Equation (14) reduces to eqn. (2). So conventional PN junction theory assumes that the rate limiting process is the diffusion of injected carriers away from the junction – not their thermionic injection across the junction.

Consider next the case where !R << !D , which means that injected carriers leave the p-region region very quickly. (This case corresponds to the MS diode where electrons injected into the metal are removed within a dielectric relaxation time. For this case, eqn. (13) becomes

" n2 % ( i R qVA kBT , (15) !n(xp ) = $ ' (e ) 1) # N A & (Dp

Lundstrom 5 2/27/13 and eqn. (14) becomes n 2 J q i eqVA kBT 1 = !R ( " ), (16) NA which describes the PN junction with a thermionic emission expression. We have succeeded in generalizing the Law of the Junction so that it makes no assumption about whether or not the quasi-Fermi level is constant across the depletion region, but we may be interested in knowing how much the Fermi level droops. If we define the droop, !Fn , as in Fig. 5, when we can compute it as follows. In the N-type bulk, we have

(Fn (!")!Ei (!")) kBT ND = nie , (17a) and at the beginning of the P-region, we have

(Fn (xp )! Ei (xp )) kBT n(xp ) = nie . (17b) Equations (17a) and (17b) can be used with our generalized Law of the Junction, eqn, (13) to find

!Fn = kBT ln(1+ "Dp "R ) . (18) Equation (18) tells us that when diffusion in the quasi-neutral regions is the limiting process

(!Dp << !R ), as it typically is for a PN junction, then the change in the quasi Fermi level across the junction can be ignored. When thermionic emission is the limiting process, however, then the quasi-Fermi level changes significantly across the depletion region.

EI EC !F n F Fn p

EV

xp

Fig. 5 Definition of “droop” in the quasi-Fermi level.

Lundstrom 6 2/27/13 Law of the PN Heterojunction Heterojunctions offer even more possibilities. The N and P regions may have different bandgaps, and the bands may line up in different ways at the interface (sometimes producing band spikes and sometimes not). In some cases, the heterojunction diode behaves like a homojunction diode (when there are no band spikes), and sometimes the heterojunction diode behaves like a MS diode (when a large band spike is present). In the first case, a Law of the Junction applies, but not in the second case. It would be useful to have a general way to treat junctions – one that does not assume the conventional Law of the Junction, but reduces to it under appropriate conditions.

To draw an energy band diagram for a heterojuction, we conceptually begin with two isolated semiconductors as shown in Fig. 6. To draw an energy band diagram, we need a reference energy. The reference is usually omitted for homojunctions, because it is the same for both sides of the junction, but for heterojunctions, it is critical. The vacuum level is commonly used as a reference. We can, however, simply specify the conduction and valence band offsets, ΔEC and

ΔEV , as shown in Fig. 6. For the band lineups on the left, we get the Np heterojunction energy band diagram in Fig 3 (left), which displays a conduction band spike but no valence band spike. Foe the band lineups on the right, we get the Np heterojunction energy band diagram in Fig. 3 (right), which has no spikes in either the conduction or valence band. Let’s derive a Law of the Junction for electrons in each of these two cases.

EVAC

!EC ECp ECN ECN !E C E Cp EFp EFN E FN E E Fp Vp

!EV !E EVp EVN V EVN

Fig. 6 Two different wide bandgap /narrow bandgap Np heterojunctions. The same bandgaps are assumed in both cases, but different band offsets are assumed.

Lundstrom 7 2/27/13 Law of the heterojunction with a band spike Figure 7 shows the energy band diagram of Fig. 3 (left) with the various fluxes and potential drops labeled.

+ − J1 J1 J2

!E VjN C

Vjp EC

Fp FN

!EV

EV

xp xp

Fig. 7 Np heterojunction with a conduction band spike. The total potential drop across the junction is V V V . In equilibrium, V o V o V o V . j = jN + jp j = jN + jp = bi

The thermionic emission flux across the heterojunction from the left to the right is

N J + = q D υ e−qVjN kBT (19) 1 2 Tp which is analogous to eqn. (8) for the homojunction. Electrons injected into the p-type region are thermalized by scattering into a near-Maxwellian distribution. At the edge the depletion region on the p-side, the corresponding electron density is n(xp). Since the electrons are distributed in a thermal equilibrium Maxwellian velocity distribution, approximately one-half of them have negative velocities and can return to the N-region, if they can surmount the energy barrier, so

Lundstrom 8 2/27/13 n(x ) E qV /k T J − = q p υ e−(Δ C − jp ) B , (20) 1 2 Tp which is analogous to eqn. (9) for the homojunction. In equilibrium, detailed balance demands + − that J1 = J1 ., so

−qVbi /kBT −ΔEC /kBT N DυRN e = no (xp )υRpe . (21)

Since nothing in this expression changes out of equilibrium (assuming low-level injection), we can make use of it later to simply the expressions under an applied bias.

Finally, carrier diffuse across the p-region, so

Δn(xp ) J2 = qDn = qΔn(xp )υDp , (22) WP which is analogous to eqn. (10) for the homojunction.

As for the homojunction, the currents must balance, so

+ − J1 = J1 − J1 = J2 . (23) By using eqns. (19) – (22) in eqn. (23), we find (after a little algebra),

1 qVA kBT Δn(xp ) = n(xp ) − no (xp ) = no (xp )(e − 1) , (24) ⎣⎡1+ υDp υems ⎦⎤ where E qV /k T −ΔEC′ /kBT −(Δ C − jp ) B υems = υRpe = υRpe . (25)

The term, !EC" = !EC # qVjp , is just the height of the conduction band spike above the conduction band in the p-type region (see Fig. 7). The parameter, !ems , is an effective thermal velocity for emission of electrons over the conduction band spike and back into the N-region.

We can write eqn. (24) as

2 2 1 ' nip * nip !n(x ) = eqVA kBT - 1 = R eqVA kBT - 1 , (26) p ) N , ( ) N ( ) $#1+ "Dp "ems &% ( A + A which looks much like the generalized Law of the Junction, eqn. (13), except that a thermal velocity have been replaced by an effective thermal velocity for emission over the conduction band spike.

Lundstrom 9 2/27/13 Let’s consider eqn. (26) in two limits, When !EC << kBT , eqn,. (26) reduces to the generalized Law of the Junction for homojunctions, eqn. (13), except that we need to be careful to use the appropriate ni . When !EC >> kBT , eqn. (26) reduces to

" e#!EC$ /kBT % n2 ( Rp ip qVA kBT !n(xp ) = ' * (e # 1) . (27) "Dp & N A )

For a thin p-region with a reasonably high diffusion coefficient, !Dp can be roughly comparable to !R , so the conclusion is that the presence of a band spike suppresses the injected carrier density by a roughly factor of e!"EC# /kBT .

To find the electron current, we use eqn. (22) with eqn. (26) to find

2 ! nip $ 1 J = q eqVA kBT , 1 , (28) n # N & ( ) " A % )(1 'Dp + 1 'ems +* which clearly shows that it is the smaller of the two velocities that matters. For a large band spike,

" n2 % ip )*EC /kBT qVA kBT Jn ! q$ ' (Rpe (e ) 1) . (29) # N A &

Recall that for a homojunction, we have,

" n2 % ip qVA kBT Jn ! q$ ' (Dp (e ) 1) . (30) # N A &

Assuming that !D and !R are roughly comparable, we conclude that the presence of a band spike suppresses the injected current by roughly a factor of e!"EC /kBT .

Finally, it is interesting to ask about the drop in the electron quasi-Fermi level across the junction. Proceeding as in eqns. (17) and (18), we find,

!Fn = kBT ln(1+ "Dp "ems ) . (31)

When !EC >> kBT , we find

F k T ln $1 e!EC# /kBT & E . (32) ! n = B % + ("Dp "Rp ) ' ( ! C

Lundstrom 10 2/27/13 For the Al0.3Ga0.7As / GaAs hetrerojunction, !EC " 0.3 eV, so the change in quasi-Fermi level can be very large when a band spike is present.

Law of the heterojunction without a band spike Having treated homojunctions and heterojunctions with a band spike, we next examine an Np homojunction without a bandspike as in the energy band idagram on the right in Fig. 3. In thiis case, the analysis proceeds as it did for the herterojunction, except that eqn. (20) is replaced by

n(x ) J ! = q p " = qn(x )" . (31) 1 2 Tp p Rp

The result is that we get

2 1 ' nip * !n(x ) = eqVA kBT - 1 , (32) p ) N , ( ) $#1+ "Dp "Rp &% ( A + which is just like the homojunction result, except that we use the thermal velocity and intrinsic carrier concentration appropriate to the p-side of the junction. When there is no band spike, the magnitude of !EC does not matter.

Generalized Law of the Junction for Hetero- and Homojunctiona We can summarize this discussion with equations that describe NP homo- or heterojunctions, with or without a band spike. The generalized Law of the Junction is

2 1 ' nip * !n(x ) = eqVA kBT - 1 (33) p ) N , ( ) $#1+ "Dp "eff &% ( A +

2 ⎛ nip ⎞ 1 J = q eqVA kBT − 1 (34) n ⎜ N ⎟ ( ) ⎝ A ⎠ ⎣⎡1 υDp + 1 υeff ⎦⎤ where ! ! = ! = Tp (homojunctions or heterojunctions with no band spike) (35) eff Rp 2

E qV /k T "#EC$ /kBT "(# C " jp ) B !eff = !Rpe = !Rpe (heterojunctions with a band spike) (36)

The presence of a band spike is often deleterious because it increases the bias (and therefore power) that much be applied to achieve a given current. Note also that !EC" is bias dependent, (see Fig. 7) so it is possible for a diode to make a transition from eqn. (35) to (36) as the forward bias increases.

Lundstrom 11 2/27/13