The kinetics of conductivity type conversion in HgCdTe by ion beam milling

Abstract

It is well established that dry etching of HgCdTe by Ar ion-beam milling (IBM) converts vacancy-doped p-type HgCdTe to n-type. Less well established is the understanding of the observed rate of conversion. This paper shows that the observed variations of the depth of conversion with milling time can be accounted for in terms of two rate processes operating in tandem. The first process is the diffusion of Hg interstitials from the surface to the conversion boundary followed by the second, or conversion, process that is the capture of the Hg interstitials by cation vacancies. Assuming steady-state conditions exist a quadratic equation is derived for the conversion depth and its milling-time dependence for two cases: when the initial p-type layer is electrically intrinsic at the milling temperature and when it is extrinsic. Good agreement is found with the available experimental Ar-ion-milling data. Good agreement is also found for type conversion due to etching with a radio-frequency Ar plasma. The effect of capture of Hg interstitials by traps, in addition to capture by cation vacancies, is also considered.

Appendices

Appendix A

The drift mobility, μ, is related to the interstitial diffusivity, D, through the Einstein relation μ = 2qD/kT [21]: the factor 2 arises because the Hg interstitial is doubly ionised. Equation 12 can then be rewritten as

$$ J/D\,=\,(2q^{2}N_{\rm D}/\varepsilon\varepsilon_{0} kT)(x\,-\,x_{\rm n})C\,-\,\hbox{d}C/\hbox{d}x $$

(A1)

or for simplicity as

$$ A\,=\,ByC\,-\,dC/dy $$

(A2)

where A = J/D, B = 2q ² N _D/ɛɛ₀ kT and y = x − x _n: note that 0 ≤ y ≤ x _J − x _n. Integrating Eq. A2 yields

$$ C_{\rm J}\,=\,C_{\rm n}\exp\left({Bl^{2}/2} \right)\,-\,A\exp\left({Bl^{2}/2} \right)\int_0^l{\exp \left({-By^{2}/2} \right)} \hbox{d}y $$

(A3)

for x _n ≤ x ≤ x _J and l = x _J − x _n. Rearranging Eq. A3 gives

$$ C_{\rm n}\,=\,C_{\rm J}\exp(-Bl^{2}/2)\,+\,AI(l) $$

(A4)

where I(l) denotes the integral $\int_0^l{\exp \left({-By^{2}/2} \right)\hbox{d}y}.$ From Eq. 11 and J = N _Adx _J/dt it follows that D (C ₀ − C _n) = N _A x _ndx _J/dt, which together with k _J C _J = dx _J/dt, J = N _Adx _J/dt (see Eq. 3) permits C _J and C _n to be eliminated in Eq. A4 to obtain

$$ [N_{\rm A}x_{\rm n}+(D/k_{\rm J})\exp(-Bl^{2}/2)+I(l)N_{\rm A}] \hbox{d}x_{\rm n}/\hbox{d}t=DC_{0} $$

(A5)

using dx _J/dt = dx _n/dt as l is constant (because N _A and N _D do not vary with x). Integrating Eq. A5 subject to x _n = 0 at t = 0 yields

$$ x^{2}_{\rm n}+2[(D/k_{\rm J}N_{\rm A})\exp(-Bl^{2}/2) +I(l)]x_{\rm n}=2DC_{0}t/N_{\rm A} $$

(A6)

Alternatively, as x _n = x _J − l and x _J is the experimentally measured quantity Eq. A6 could be recast in terms of x _J. The resulting equation, however, is complicated due to the large number of terms and a simpler approach is to use Eq. A6 with x _n determined from the measured x _J and calculated l. It is easily seen that Eq. A6 has two limiting forms, namely when x _n ≪ or ≫ 2[(D/k _J N _A)exp(−Bl ²/2) + I(l)], which are, respectively, linear and parabolic relationships and qualitatively similar to Eq. 5. It needs to be noted that Eq. A6 is applicable only when the depletion layer on the n-side of the p–n junction is determined by N _D and the depletion layer does not penetrate into the extrinsic n-type surface region i.e. x _n ≥ x _s (see Fig. 2): it is assumed that x _s is negligible relative to l.

Comparing Eqs. 5 and A6 shows that the extrinsic situation introduces the additional quantities l, B and I(l) whose magnitudes will vary with N _D and N _A. l is given by [22]

$$ l^{2}\,=\,(2\varepsilon\varepsilon_{0}kT/q^{2})\{N_{\rm A}/(N_{\rm D}(N_{\rm A}\,+\,N_{\rm D}))\} \ln(N_{\rm A}/n_{\rm i}) $$

(A7)

and as B = 2q ² N _D/ɛɛ₀ kT it follows that

$$ Bl^{2}/2\,=\,2N_{\rm A}(N_{\rm A}\,+\,N_{\rm D})^{-1}\ln(N_{\rm A}/n_{\rm i}) $$

(A8)

The integral I(l) is defined in Eq. A4 and can be recast as

$$ I(l)\,=\,\frac{1}{\beta}\int_0^\beta {\exp\left({-z^{2}} \right)\hbox{d}z} $$

(A9)

where β = (B/2)^1/2 l. The integral term in Eq. A9 is the probability integral so that for $\beta\,\lesssim\,0.05$ I(l) ≅ l and I(l) < l for larger β.

Appendix B

If the in-diffusing Hg interstitials in addition to being captured by cation vacancies can also be trapped at centres to create donors (which are at least singly ionised) the analysis of Sect. 2.1 for the intrinsic case is readily adapted to take account of such trap**. Assume that only one trap is present at a uniform concentration N _T and that at x _J the trap** rate is k _T C _J N _T, where k _T is the capture coefficient for Hg interstitials by the empty trap, then Eq. 2 is modified to

$$ J\,=\,D(C_{0}\,-\,C_{\rm J})/x_{\rm J}\,=\,(k_{\rm J}N_{\rm A}\,+\,k_{\rm T}N_{\rm T})C_{\rm J}\,=\,(N_{\rm A}\,+\,N_{\rm T})\hbox{d}x_{\rm J}/\hbox{d}t $$

(B1)

It then follows that

$$ C_{\rm J}\,=\,DC_{0}/(D\,+\,(k_{\rm J}N_{\rm A}\,+\,k_{\rm T}N_{\rm T})x_{\rm J}) $$

(B2)

and

$$ (D\,+\,(k_{\rm J}N_{\rm A}\,+\,k_{\rm T}N_{\rm T})x_{\rm J})\hbox{d}x_{\rm J}/\hbox{d}t\,=\,DC_{0}(k_{\rm J}N_{\rm A}\,+\,k_{\rm T}N_{\rm T})/(N_{\rm A}\,+\,N_{\rm T}) $$

(B3)

Integrating Eq. B3 subject to x _J= 0 at t = 0 yields

$$ (k_{\rm J}N_{\rm A}\,+\,k_{\rm T}N_{\rm T})x^{2}_{\rm J}\,+\,2Dx_{\rm J}\,=\,2 DC_{0}(k_{\rm J}N_{\rm A}\,+\,k_{\rm T}N_{\rm T})(N_{\rm A}\,+\,N_{\rm T})^{-1} t $$

(B4)

Assuming N _A ≫ N _T Eq. B4 becomes

$$ k_{\rm J}N_{\rm A}(1+\alpha)x^{2}_{\rm J}\,+\,2 Dx_{\rm J}\,=\,2k_{\rm J}DC_{0}(1\,+\,\alpha)t $$

(B5)

where α = k _T N _T/k _J N _A. Although N _A ≫ N _T it is feasible that k _T/k _J ≫ 1 so that α cannot be neglected in Eq. B5. According to [20] N _T < 1E14 cm⁻³ so that the condition N _T ≪ N _A will be met for N _A > 1E15 cm⁻³. For the extrinsic situation it is straightforward, by combining the approaches leading to Eqs. A6 and B5, to obtain

$$ x^{2}_{\rm n}\,+\,2[D\exp(-Bl^{2}/2)/(k_{\rm J}N_{\rm A}\,+\,k_{\rm T}N_{\rm T})\,+\,I(l)]x_{\rm n}\,=\,2 DC_{0}(N_{\rm A}\,+\,N_{\rm T})^{-1}t $$

(B6)

The structures of Eqs. B5 and B6 show that the inclusion of multiple traps is straightforward. Any emission of Hg interstitials from traps in the converted region during IBM has been neglected because emission time constants at room temperature are several hours [20].