1 Esaki Diode hen the concentation of impuity atoms in a pn-diode is vey high, the depletion laye width is educed to about 1 nm. Classically, a caie must have an enegy at least equal to the potential-baie height in ode to coss the junction. Howeve, quantum mechanics indicates that thee is a nonzeo pobability that a paticle might penetate though a baie as thin as that indicated above. This phenomenon is called tunneling, and because of this, these high-impuity density p-n devices ae called tunnel diodes, o Esaki diodes. The condition that the baie be less than 1 nm thick is a necessay but not a sucient condition fo tunneling. It is also equied that occupied enegy states exist on the side fom which the electons tunnel and that allowed empty states exist on the othe side. The enegy-band diagam fo a heavily - doped diode unde open cicuit and evese bias conditions is shown below. Figue 1: Esaki diode 1
(a) Unde zeo-bias conditions, thee ae no lled states on one side of the junction which ae at the same enegy as the empty allowed states on the othe side. Hence, thee can be no ow of chage in eithe diection acoss the junction, and the cuent is zeo, an obviously coect conclusion fo an open - cicuited diode. (b) If a evese bias is a applied, the height of the baie is inceased above the open-cicuit value E. Hence, the n-side levels must shift downwad with espect to the p-side levels. In this case, thee ae some enegy states in the valence band on the p-side that lie at the same level as allowed empty states in the conduction band of the n-side. Hence, these electons might tunnel fom the p to the n - side, giving ise to the evese diode cuent. As the magnitude of the evese bias inceases, the heavily - shaded ae gows in size, causing the evese cuent to incease, as shown by section (1) in the Figue below. Figue 2: IV chaacteistics. Conside now the fowad bias case when the potential baie is deceased below E. The n-side level shifts upwad with espect to those on the p-side, and the enegy-band pictue fo this situation is indicated in Figue 3. 2
Figue 3: Fowad Bias 3
It is clea that thee will be occupied states in the conduction band on the n-side which ae at the same enegy as the allowed empty sates (holes) in the valence band on the p-side. Hence electons will tunnel fom the n to the p mateial giving ise to the fowad cuent of section (2). As the fowad bias inceases futhe, the condition in (b) is eached, giving ise to maximum cuent value. If moe fowad bias is applied, the situation in (c) is obtained and the tunneling cuent deceases, giving to section (3). At even lage bias, thee will not be empty allowed states on one side of the junction at the same enegy as the occupied states on the othe side and the tunneling cuent dops to zeo. In addition to the quantum-mechanical cuent descibed above, the egula pn-junction cuent is being collected at lage voltages, giving ise to section (4). Mathematical analysis of the tunneling cuent is based on the esults descibed in the pevious sections. In othe wods, one can use the KB appoximation to calculate the tunneling coecient (o pobability) and then use the esult into the Landaue expession fo the cuent. To calculate the tunneling coecient, we conside the case when E? = and E z 6=, and we appoximate the baie height bye G : Figue 4: Tunneling baie V (Z) =E G 1, Z (1) The attenuation of the baie is thus Z Z 2m 2 = 2 ~ [V (Z), E Z]dZ 2 4
Z s Z 2m = 2 Z Z = 2 Z Z = 2 s 2mE G Z Z = (,2 ) = (,2 ) =, 4 3 E G 1, Z, E Z dz 1, Z, E Z dz s 1, Z, E Z d s 1, Z, E Z d 1, Z, E Z " 1, Z, E Z, Z (, ) Z,, Z 1, E Z # (2) whee 1, Z, E Z = 1,, E Z, E Z =, E Z,, E Z (3) Hence = 4 3 4 3 1, E Z 1, 3 2 E Z (4) povided that E Z =E G 1. Substituting the last expession into the KB esult fo the tansmission coecient gives whee we have dened T (E Z ) = e,2 4 = e, 2 3 ~ {z 2 E } e Z T = T e E Z=E (5) 5
s E = 2 2m s Eg 2 1 = 2m 2 1 ~ = 2 2 2m The expession fo T (E Z ) suggests that the lage the enegy E Z, the lage the pobability fo tansmission, which should be expected. The next task is to calculate the cuent unde fowad bias conditions, fo which we use the peviously deived expession (6) Figue 5: Fowad bias conditions. J = J L!R, J R!L =, em whee Z 1 Z 1 de Z T (E Z ) de t [f L (E Z + E t ), f R (E Z + E t + ev a )] (7) Using the appoximation (valid at T! ); E = E Z + E t = E Z1 + E t, E cn (8) f L (E Z + E t ), f R (E Z + E t + ev a )= 6
@f = f(e Z + E t ), f(e Z + E t ), ev a @E E Z+E t = =,ev a (E Z + E t, E F ) (9) we get fo the fowad cuent J =, em (,ev a)t whee = e2 m V a T = e2 m V a T Z E max t =, e2 m V a T E Z 1 Z 1 de t de Z e EZ=E (E Z + E t, E F ) (,E )e,et=e de t e (E F%, E t )=E E max t h i e,emax t =E, 1 (1) Theefoe E max t = E VP +E cn, ev a (11) i.e., e,emax t =E, 1 = e,(ev P+Ecn,eVa)=E, 1. 1, E VP +E cn, ev. a, 1 E J =, e2 m V a T E = e2 m V a T = e2 m T E t V a, E VP +E cn, ev a E VP +E cn {z } E t 1, ev a E t E, ev a! (12) (13) The maximum tansvese enegy is obtained using,e cno + ev a + E max? = E VP (14) E? max = E VP +E cn, ev a (15) 7