Weak discontinuity waves in n-type semiconductors with
defects of dislocation
M. P. Mazzeoa and L. Restucciaa
aUniversity of Messina, Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, Italy
Aim and motivation The propagation of weak discon-tinuities is investigated in an isotropic, homogenous and elastic n -type semiconductor with defects of dis-location. To this aim we introduce a new variable re-lated to the surface across which the solutions or/and some of their derivatives undergo a jump. Following a Boillat’s methodology for quasi-linear and hyper-bolic systems of the first order, we obtain Bernoulli’s equation governing the propagation of weak disconti-nuities.
Following A. Jeffrey in [2], the solution hypersurfaces of systems of PDEs are referred to as waves because they may be interpreted as representing propagat-ing wavefronts. Some of the solutions present various types of discontinuities, when some surface is crossed, the solution or/and its derivatives undergo a jump. In this case it is said that the solution presents a shock, or it is a shock wave or that we are in presence of a discontinuity waves (jumps of the first order deriva-tives) [1, 2].
Equation governing the evolution of electronic and dis-location fields We apply the theory developed in [3] for a semiconductor with defects of dislocation (see Fig.1) of n -type to a problem of a propagation of electronic-dislocation discontinuity waves in a n -type Ge.
Equations governing the electronic and dislocation fields evolution can reduce to the following form (see [3]): ρ ˙n + jk,kn = ρnτ+ − κa, τnjkn − αna,k + ρDnn,k = −jkn, ˙a + Vk,k = 0 τaV˙k + Daa,k − αυn,k = −Vk.
where the superimposed dot denotes partial time derivative, the interaction between the electron and dislocation fluxes is disregarded and αn = αn(a), αυ = αυ(n) are coupling functions reflecting some new cross-kinetic effects during electronic-dislocation interac-tions. Furthermore, τ+ denotes the life time of elec-trons, τn is the relaxation time of electrons, Dn and Da are the diffusion coefficients on electrons and dis-locations, respectively, τa denotes the relaxation time of dislocations, k is the recombination constant due to dislocations and the recombination function gn has the form depending on the dislocation field. More-over, we consider the relaxation semiconductor, so that τ+ = τn, αn and αυ are constant
One-dimensional case Now, we consider the one-dimensional case and we apply Boillat metodology for quasi-linear and hyperbolic systems of the first order, we obtain Bernoulli’s equation governing the prop-agation of weak discontinuities. Assuming that the electronic-dislocation discontinuity wave propagation is along the x axis, the involved quantities depend on x1, denoted by x, x2 = x3 = 0, we have: ∂n ∂t
+
1 ρ ∂j1n ∂x=
n τn−
κa ρ,
∂j1n ∂t−
αn τn ∂a ∂x+
ρDn τn ∂n ∂x= −
j1n τn,
∂j2n ∂t= −
j2n τn,
∂j3n ∂t= −
j3n τn,
∂a ∂t+
∂V1 ∂x= 0
∂V1 ∂t+
Da τa ∂a ∂x−
αυ τa ∂n ∂x= −
V1 τa,
∂V2 ∂t= −
V2 τa,
∂V3 ∂t= −
V3 τa.
where αn = αn(a) and αv = αv(n). From the above system we have j2n = j2n0e−τ n1 t + f1(x), jn 3 = j3n0e− 1 τ nt + f2(x), V2 = V20e−τ a1 + f 3(x), V3 = V30e− 1 τ a + f 4(x).
Then, the remained system reads
Ut + A(U)Ux = B(U) where U = (n, j1n, a, V1)T, and B = n τ+ − κa ρ , − j1n τn, 0, − V1 τa, T and A = 0 1ρ 0 0 ρDn τn 0 − αn τn 0 0 0 0 1 −αv τa 0 Dτaa 0 ,
In our case, being n = (n1, 0, 0), An(U) = An1 having the form An1 = 0 1ρn1 0 0 ρDn τn n1 0 −ατnnn1 0 0 0 0 n1 −αv τan1 0 Da τa n1 0
The matrix An1 admits the following simple eigenval-ues: λ(±)1 = ±√1 2 s ρDnτa + ρDaτn − G ρτnτa , λ(±)2 = ±√1 2 s ρDnτa + ρDaτn + G ρτnτa , G = q (ρDnτa − ρDaτn)2 + 4ραnαvτnτa.
The eigenvalues λ(±)1 are real when the condition ρDnτa + ρDaτn − G ≥ 0 is valid, i.e. αnαv ≤ DnDa. The eigenvalues λ(±)2 are always real. The left eigenvectors l(±)1 , l(±)2 , and the right eigenvectors r(±)1 , r(±)2 corre-sponding, to eigenvalues λ(±)1 , λ(±)2 , have the form
l(±)1 = ρ n1λ (±) 1 , 1, λ(±)1 S 2n1αvτn, S 2αvτn , l(±)2 = ρ n1λ (±) 2 , 1, − λ(±)2 R 2n1αvτn, − R 2αvτn , r(±)1 = τnλ(±)1 R ρn1C , 1, 2αv(τn)2λ(±)1 n1C , 2αv(τn)2 λ(±)1 2 C , r(±)2 = − τnλ(±)2 S ρn1L , 1, 2αv(τn)2λ(±)2 n1L , 2αv(τn)2 λ(±)2 2 L , with R = (−ρDnτa + ρDaτn + G) S = (ρDnτa − ρDaτn + G) C = (−1) hρDn2τa + 2αvαnτn − Dn (ρDaτn + G) i L = (−1) hρDn2τa + 2αvαnτn − Dn (ρDaτn − G)i
They are linearly independent, so the system Ut + A(U)Ux = B(U) is hyperbolic. The discontinuity waves which are propagating with the velocity given by λ(±)1 and λ(±)2 are not exceptional waves in the sense of Lax-Boillat [1], because ∇λ(±)1 · r(±)1 = −αnτ nR∂αv ∂n − 2ρ(αvτn)2∂α∂an 2ρn1GC 6= 0 ∇λ(±)2 · r(±)2 = −αnτ nS∂αv ∂n + 2ρ(αvτn)2∂α∂an 2ρn1GL 6= 0, with ∇ ≡ ∂ ∂n, ∂ ∂j1n, ∂ ∂a, ∂ ∂V1, .
We fix our attention on λ = λ(+)2 , which corresponds to a progressive fast wave traveling to the right. Analo-gous results are valid for the waves propagating with the other velocities.
The eigenvectors left and right l = l(+)2 and r = r(+)2 , corresponding to λ(+)2 satisfy the following relation
l(+)2 ·r(+)2 = τn 1 √ 2 q ρDnτa+ρDaτn+G ρτnτa 2 (3ρDnτa − 3Daτn − G) L +1.
The characteristic rays are dt dσ = 1 dxi dσ = ∂Ψ0 ∂Φα = λ(+)2 0 , dΦα dσ = 0,
Now, we consider an uniform unperturbed state in which U0, solution of the system Ut + A(U)Ux = B(U), has the form U0 = (n0, 0, a0, 0), with n0 and a0 constants. In U0 we have λ(+)2 0 = 1 √ 2 s ρ0Dnτa + ρ0Daτn + G0 ρ0τnτa , l(+)2 0 = ρ0 n01 λ(+)2 0 , 1, − λ(+)2 0 R 0 2n01α0vτn , − R0 2αv0τn , r(+)2 0 = − τn λ(+)2 0 S 0 ρ0n01L0 , 1, 2αv(τn)2 λ(+)2 0 n01L0 , 2αv(τn)2 h λ(+)2 0 i2 L0 , l(+)2 · r(+)2 0 = τn h λ(+)2 0 i2 3ρ0Dnτa − 3Daτn − G0 L0 + 1 and ∇λ(+)2 · r(+)2 0 = −α0nτnS0 ∂αv ∂n 0 + 2ρ0(α0vτn)2 ∂αn ∂a 0 2ρ0n01G0L0 , where the symbol “ 0 00 indicates that the quantities are calculated in U0. The radial velocity along the charac-teristic rays is Λ0(U0, n0) = λ(+)2 0 n 0 = √1 2 s ρ0Dnτa + ρ0Daτn + G0 ρ0τnτa n 0.
By integration of the characteristic rays one obtain x0 = σ = t, x = (x)0 + λ(+)2 (U0)σ = (x)0 + γ0t,
and the wave front in the first approximation is ϕ(t, x) = ϕ0 x(t) − γ0t , where γ0 = γ(U0) = λ(+)2 0 = 1 √ 2 s ρ0Dnτa + ρ0Daτn + G0 ρ0τnτa . The amplitude π(x, t) satisfies the following equation:
l(+)2 · r(+)2 0 dπ dσ + (|ϕx|)0 ∇λ(+)2 · r(+)2 0 π 2 = c0 π, where c0 = h ∇ l(+)2 · B · r(+)2 i 0
Taking into account that
l(+)2 · B = ρλ (+) 2 n1 n τn − κa ρ − j n 1 τn + V1R 2αvτnτa, ∇ l(+)2 · B = ∂ l(+)2 · B ∂n , ∂ l(+)2 · B ∂j , ∂ l(+)2 · B ∂a , ∂ l(+)2 · B ∂V , where ∂ l(+)2 · B ∂n = ραn G ∂αv ∂n 1 2λ(+)2 n1 n τn − κa ρ + V1 αv + ρλ n1τn + V1R 2α2vτnτa ∂αv ∂n , ∂ l(+)2 · B ∂j = − 1 τn, ∂ l(+)2 · B ∂a = ραv 2n1λ(+)2 G ∂αn ∂a n τn − κa ρ − λ(+)2 κ n1 + ρV1 G ∂αn ∂a , ∂ l(+)2 · B ∂V = R 2αvτnτa.
From the above results we obtain
h(σ) = π0exp − Z σ 0 c0 l(+)2 · r(+)2 0 dτ , Φ(σ) = 1+ Z σ 0 ∇λ(+)2 · r(+)2 0 hdτ.
Relation above gives Φ(σ) = 1+ Z σ 0 h 2ρ0n01G0L0 " −αn0τnS0 ∂αv ∂n 0 + 2ρ0(α0vτn)2 ∂αn ∂a 0# dτ.
In the case in which there exists a critical time σc where Φ(σc) = 0, then π → ∞, and this may correspond to a shock wave [1].
References
[1] Boillat, G., La propagation des ondes, Gauthier-Villars, Paris, 1965.
[2] Jeffrey, A., Quasilinear hyperbolic systems and waves, Pit-man, London, 1976.
[3] L.Restuccia and B.Maruszewski, ”Interactions between electronic field and dislocations in a deformable semiconduc-tor”, Int. Journal of Applied Electromagnetics and Me-chanics, 6, 139-153 (1995).
[4] M.P. Mazzeo, L. Restuccia, ”Thermodynamics of ex-trinsic semiconductors with dislocations”, Communication to SIMAI Congress, ISSN: 1827-9015, DOI: 10.1685/CSC06113, ISSN: 1970-4429 (2006).
