Constant Surface Concentration Research Articles

Studies of the Push‐Out Effect in Silicon: I . Comparison of Sequential Boron‐Phosphorus and Gallium‐Phosphorus Diffusions

The push‐out effect in silicon has been investigated by comparing the effect of similar phosphorus emitter diffusions on previously diffused boron and gallium base diffusions, using junction depth and electrical measurements. The closely similar behavior of the two base dopants suggests that both diffuse by the same mechanism in silicon, despite their different covalent radii. For both dopants the amount of push‐out increases with emitter diffusion time for constant base diffusion and emitter surface concentration, and decreases as base depth increases for constant emitter diffusion and base surface concentration, and it is concluded that push‐out occurs during the emitter diffusion rather than during cooling. The results are discussed with reference to an approximate theory, and agreement between experiment and this theory strongly suggests that push‐out is due to a uniform enhancement of the base diffusivity throughout the emitter diffusion period. This study, using junction depth and electrical profiling techniques only, is the first part of this investigation, the second of which will report results of a radiotracer study of push‐out.

Water absorption by swelling seeds. I. Constant surface boundary condition

Recent analyses of water absorption by seeds have used a non-swelling model with constant diffusivity and constant surface concentration boundary condition. A physically realistic modification of such analyses is developed, where the spherical seed has the properties of normal swelling, moisture content dependent diffusively, and a moisture characteristic which is described by the double layer theory relevant to colloids under mechanical restraint. Analysis of this model by a finite difference approximation produces a relationship between linearized moisture content and dimensionless time which is appropriate to all spherical swelling materials. A good match was obtained between this relationship and experimental absorption data for wheat seed. The relationship given graphically in dimensionless form is applicable to any swelling system, e.g. soil aggregates, which satisfy normal swelling and constrained double layer boundary conditions. The analysis gave values for the diffusivity of the seed similar to, but larger than, those calculated from non-swelling models. These are much smaller than most values of diffusivity of soils in the available water range. The magnitude of the maximum rate at which a seed can absorb water relative to that which the soil can supply, predicts that for seeds embedded in most soils, the constant surface concentration boundary condition used by most authors is inappropriate. In terms of the swelling behaviour of most soils, the swelling model, unlike the non-swelling model, predicts that imbibing seed embedded in soil will be subject to a mechanical constraint.

Diffusion coefficients varying with a power of the concentration: Convenient solutions and a reexamination of Zn in GaAs

Diffusion, at constant surface concentration, with coefficient varying to the positive nth power is considered. It is shown that the diffusion equation for these cases has a unique solution whose leading term is concentration ∝Z1/n, where z = 1−(x/x+), x being the distance from the surface and x+ the distance at which the concentration just reaches zero. Numerical solutions can be obtained by standard methods of solution; explicit expressions are given for n = 1, 2, and 3. It is also shown that plots of log(concentration) versus logz give rapid reliable estimates of n from usual diffusion profiles. These methods are applied to previously analyzed data for diffusion of Zn in GaAs at 1000 °C. This reanalysis suggests that (i) the diffusion coefficient for Zn in GaAs is accurately given by the ``interstitial-substitional'' mechanism at the surface of GaAs, (ii) for surface concentrations of Zn⪞5×1019 cm−3, the diffusion is influenced by a limited flux of gallium vacancies from the surface, and (iii) the diffusion coefficient for gallium vacancies is roughly 5×10−9 cm2/sec at 1000°C. It is suggested that ``double'' profiles, often found for surface Zn concentration ⪞1020 cm−3, are due to dual vacancy sources: the free surface and a dislocation net occurring at positions of large Zn gradient.

The rotating cylinder electrode

Printed in Great Britain. Constant Surface and bulk concentrations Diffusion coefficient Characteristic dimension Faradaic equivalence Shear force Friction factor (Diffusion) current (Limiting) current density Mass flux (see Equation 24) Constant Length of cylinder Gap between cylinder and cell bottom Segment length Index, integer Radii of inner and outer cylinders Radial distance RDE roughness factor (see Equation 4)

Cooperative effects between arsenic and boron in silicon during simultaneous diffusions from ion implanted and chemical source predepositions

Simultaneous diffusions of As and B from predeposited layers (chemical source or ion implantation) have been used in order to fabricate the emitter and base regions, respectively, of microwave transistors. Mathematical simulations of the doping profiles in these transistors have shown that the cooperative diffusion effects that occur in sequentially diffused As-B structures (chemical sources) are noticeably absent in the predeposited-diffused structures. The result is that transistors that are fabricated via this technique show no base retardation, and the active base doping concentrations are higher than predicted by previously established diffusion equations. In order to determine the extent to which cooperative effects are important, transistor doping profiles were measured and compared with calculated profiles. By including the electric field interaction, the vacancy undersaturation condition due to [ V SiAs 2] complex formation, and ion pairing, it was possible to estimate the significance of each effect upon the B diffusion. It is shown that the electric field interaction is two- to three-times smaller in a predeposited-diffused structure than in a constant surface concentration diffusion (non-depleting source). More importantly, a negligible undersaturation of vacancies occurs during simultaneous As-B diffusions from predeposited layers. In the case of a chemical source As predeposition, this is due to the fact that a quasi-equilibrium concentration of [ V SiAs 2] complexes is achieved during the predeposition (before the simultaneous diffusion with B). In the case of an As implantation predeposition, complexes appear to be formed either during implantation or very rapidly during annealing for doses ≲ 3 × 10 15 cm −2. Data is presented which suggests that no inactive As complexes are formed in As implanted-annealed structures in which the maximum solubility of As + ions is approached (3·8 × 10 20 atoms/cm 3 at 1000°C, or a dose of 5–8 × 10 15 cm −2). This result pertains to the As dose used in this study. Since this result has not been observed in As-doped layers that were diffused from chemical sources, further work is needed in order to explain this anomaly.

Heat transfer fouling through growth of calcareous film deposits

The fouling of heat-transfer equipment surface area results in the equipment being overdesigned some 30 per cent. This added metal and size costs the United States about $400 million dollars per year. Consequently, there is a great incentive to better predict the growth of deposits or films on surfaces. The growth of calcareous deposits on metal surfaces is predicted through the application of fundamental principles of fluid mechanics and forced convective mass transfer of dissolved oxygen in water. The film thickness was the single most important variable affecting surface transport rate and growth distribution. The idealized case of a bare cylinder maintained at a constant surface concentration with a diffusion barrier series with the concentration boundary layer is taken as the model. Experimentally the growth of the diffusion barrier with time and the complicating effects of hydrodynamics were studied in the laboratory using spherical electrodes placed in a flowing sea water system. The model is used to predict the experimentally measured current decay.

An approximate evaluation of surface concentration during one-dimensional diffusion with surface flux a function of time for concentration-dependent diffusivity

An approximate expression describing the surface concentration at a given time is derived for a diffusivity D dependent on the concentration c when diffusion occurs in columns with the surface flux a prescribed function of the time. Estimates of the surface concentration during sorption into semi-infinite columns initially at zero concentration, for the cases of D constant, D is proportional to c and D is proportional to c2 when the flux is constant and when it varies linearly with time, are compared with exact values obtained from the concentration profiles calculated after applying a transformation used by Barenblatt. Estimates have also been compared with exact values for these diffusivities and for D is proportional to 10c and D is proportional to 1000c for the case of constant surface concentration when the flux is inversely proportional to the square root of time. These illustrative examples show that the error in using the approximate expression is often small, especially in sorption cases in which the diffusivity increases with concentration.

Assymmetry of temperature and concentration profiles in porous catalyst particles

The character of asymmetric solutions in porous catalysts involving internal and external transport limitations is analysed. Whenever uniqueness of the solution holds, symmetry exists. When solutions have multiplicity, a sufficient condition for symmetry is that the Sherwood and Nusselt numbers must be equal. This includes the simple case of uniformly constant surface concentrations and temperature. However, in the more general and realistic case of unequal Sherwood and Nusselt numbers, it is suggested that some of the solutions may be highly asymmetrical. It is proposed that the difference between these numbers may be used as a measure of asymmetry.

Hydrodynamic Potentiometry and Amperometry at Ring‐Disk Electrodes

An electronically regulated speed control unit has been programmed to produce an relation for disk and ring‐disk electrodes. Besides automatic hydrodynamic amperometry (Levich plots), one can perform two new techniques of hydrodynamic potentiometry: (a) recording of disk potential ( vs. ω½ at constant disk current ; and (b) recording of vs. at constant surface concentrations of all species, obtained by holding constant while scanning . The equations governing (a) are derived and the implications of (b) for measuring rate constants and/or uncompensated solution resistance are discussed.

Analysis of Double-Stream Diffusion with Variable Trapping and Releasing Rates and Variable Diffusion Coefficient

An analytical method has been developed for solving the partial differential equations that govern double-stream diffusion under very general conditions. The diffusion coefficient of the slow stream is set equal to zero. The trapping and releasing rates and the diffusion coefficient of the fast stream are assumed to be functions of distance and time rather than constants. General solutions of the equations which satisfy the boundary and initial conditions for the two important cases of a thin planar source and a constant surface concentration are obtained in terms of the diffusivity of the fast stream D1(x). The expressions obtained for the trapping and releasing rates are found to be physically realizable. Expressions for the concentration of the diffusing substance N(x, t) are obtained by considering suitable forms of D1(x). By varying the parameters in the expressions for the function N(x, t), concentration profiles are obtained which give a reasonable fit to the experimental data. A method is also developed to define the nominal Fick's diffusivity from the expressions of the apparent diffusivity. The values computed are of the same order-of-magnitude, but not in complete agreement, with those calculated using other methods.

Vacancy Transients During Impurity Diffusion in Semiconductors

Vacancy transients associated with the diffusion of donor atoms in semiconductors are examined for a model of strong vacancy-impurity association. The causes for the vacancy transients are the drift of charged vacancies in the internal field and the shifting of the local thermal-equilibrium condition of vacancy concentration. The impurity is considered to diffuse into a semi-infinite solid with a constant surface concentration. Simultaneous equations of continuity for the impurity and the vacancy are obtained from phenomenological flux expressions, and their Boltzmann transforms are solved numerically. For the example studied, which is typical of donor diffusions in silicon, the maximum vacancy undersaturation was found to be 4.17\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$. An inequality estimation gives $\ensuremath{\sigma}\ensuremath{\lesssim}\frac{8{D}_{A}(0)}{{D}_{\ensuremath{\nu}}}$, where $\ensuremath{\sigma}$ is the vacancy supersaturation (negative in the present case), ${D}_{A}(0)$ is the impurity diffusivity at the surface, and ${D}_{v}$ is the vacancy diffusivity. It is concluded that, during impurity diffusion, the departure of the vacancy concentration from its equilibrium value is entirely negligible for the model concerned. It is also pointed out that the cross coefficients of the phenomenological flux expressions are dependent on an atomistic model. For example, it is shown that if Seitz's chemical-pump model is assumed, then there will be significant vacancy nonequilibrium.

Diffusion et solubilite du plutonium dans le nickel

The diffusion coefficients at infinite dilution, and the solubility of plutonium and nickel were determined by α. autoradiography. The technique of welded couples was used in conjunction with a polyphase system (Ni saturated with Pu+precipitates of Pu 2Ni 17 attached to pure nickel). The diffusion coefficients were determined from examination of the precipitates existing in contact with the interface after annealing, using Pick's equation for a semi-infinite medium with constant surface concentration. For a sufficiently large proportion of precipitates one can take it that throughout the anneal the interface remains at fixed concentration, which is the solubility limit; this is obtained by extrapolating the concentration/penetration curve. The activation energy, 51 ±5 kcal/mole, is slightly lower than that for the self-diffusion of nickel, 65 ± 5 kcal/mole (the atomic radius of plutonium, 1.69–1.82 Å according to allotropie form, is larger than that of nickel, 1.38 Å). The frequency factor is in the range 0.14–1.14 cm 2/sec. Between 1025° and 1125°C, the solubility of plutonium in nickel is in the range 130–320 × 10 −6 wt %.

Controlled phosphorus diffusion into silicon from P 2O 5 vapour using a red phosphorus source

A single-stage method for low concentration diffusion of phosphorus into silicon using a red phosphorus source is described. When accurately controlled vapour of red phosphorus is combined with oxygen in the diffusion furnace to form a P 2O 5 vapour, accurate control of surface concentration on silicon slices from 10 17 atoms cm −3 to greater than 3 × 10 20 atoms cm −3 can be achieved. Results of sheet resistivity and surface concentrations are presented as a function of partial pressure of oxygen and source temperature. It is concluded from these results that the surface concentration is dependent on the oxide thickness grown on the slice during the diffusion. Results on surface concentration as a function of diffusion time at constant pressures of oxygen are also presented. From these results a definite relationship between surface concentration and the rate of oxidation and diffusion time has been demonstrated. A mechanism for this system of diffusion is described in terms of the diffusion rate of the phosphorus through the oxide and the rate of oxidation of the silicon slice. A condition for constant surface concentration for all diffusion times has been deduced and from this condition a value for the diffusion coefficient of phosphorus in the oxide has been calculated. The shape of the impurity profiles of diffused layers has been studied and show that at low concentrations, impurity distribution characterized by erfc can be obtained. Finally the application of this method in two stage diffusion is described to indicate that it can replace the conventional P 2O 5 diffusion system.

On the Concentration Gradient across a Spherical Source Washed by Slow Flow

A model has been numerically analyzed to help interpret the orienting effects of flow upon cells. The model is a sphere steadily and uniformly emitting a diffusible stuff into a medium otherwise free of it and moving past with Stokes flow. Its properties depend primarily upon the Peclet number, Pe, equal to a . v(infinity)/D, i.e., the sphere's radius, a, times the free stream speed, v(infinity), over the stuff's diffusion constant, D. As Pe rises, and washing becomes more effective, the average surface concentration, C(s a) falls (Figs. 2 and 5) and the residual material becomes relatively concentrated on the sphere's lee pole (Figs. 2 and 4). Specifically, as Pe rises from 0.1 to 1, the relative concentration gradient, G, rises from 0.7 to 5.0 per cent and to the point where it is rising at about 8 per cent per decade; by Pe 1000, G = 22.1 per cent. From Pe 1 through 1000, G/(1 - C(s a)), or the gradient per concentration deficiency remains at about 26 per cent suggesting that G approaches a ceiling of about 26 per cent. Also from Pe 1 through 1000, the average mass transfer co-efficient nearly equals that previously calculated for spheres maintaining constant surface concentration instead of flux. The complete differential equation without approximations, the Gauss-Seidel method, and an approximation for the outer boundary condition were used.

Diffusion‐Induced Imperfections in Silicon

Diffusion‐induced dislocations and precipitates have been studied through electron microscopy as a function of the diffusion depth. High concentrations of phosphorus and arsenic were diffused in (111) Si wafers to a shallow depth under the condition of constant solute surface concentration. Extensive dislocation networks were observed in phosphorus diffused wafers, whereas only a few single dislocations were detected in arsenic‐diffused wafers. The results are shown to be in general agreement with Prussin's model of dislocation generation by relief of the stress due to solute lattice contraction of P and As.

Studies on Mixture Dyeing. II. The Kinetics of a Mixture of Chrysophenine G and Sirius Red 4B

Abstract 1) The single and mixture dyeing of Chrysophenine G (C.I. Direct Yellow 12) and Sirius Red 4B (C. I. Direct Red 81) have been examined kinetically. 2) The affinity and the diffusion coefficient of each dye at 90, 85 and 80°C have been evaluated. The apparent activation energy of diffusion at a constant surface concentration has been calculated. 3) In the mixture dyeing, although the apparent diffusion coefficient of each dye became larger than that in the single dyeing, whose surface concentration was similar to that in the mixture dyeing, the dye with the smaller diffusion coefficient had a more profound effect on the diffusion of the dye with the larger diffusion coefficient than in the reverse case.

Concentration-dependent diffusion of boron and phosphorus in silicon

A calculation is made of the tail of the impurity concentration profile resulting from concentration-dependent diffusion from a constant surface concentration into a semi-infinite medium. The calculation predicts that if the concentration dependence at low impurity concentrations is negligible, the low concentration portion of the doping profile should still take the familiar form, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C = C'_{s} erfc (x/2 D_{i^{\frac{1}{2}}}t^{\frac{1}{2}})</tex> . D <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> is the commonly known diffusion coefficient at low impurity concentrations, while <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C'_{s}</tex> is the "apparent" surface concentration. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C'_{s}</tex> depends on the actual surface concentration and also depends on how the diffusion coefficient varies with impurity concentration at high concentrations. It is a constant for a given diffusion system but could be orders of magnitude higher than the actual surface concentration. Empirical data have been obtained for boron and phosphorus diffusions in silicon and found to be in good agreement with this prediction.

Diffusion with Interstitial-Substitutional Equilibrium. Zinc in GaAs

The diffusion of an impurity existing in a substitutional-interstitial equilibrium in an extrinsic semiconductor is considered. The dependence of the effective diffusion coefficient $D$ on the impurity concentration can be simplified by assuming that the concentrations of the substitutional species and electrical carriers are nearly equal and that the diffusion of the substitutional species can be neglected. Then $D$ is shown to vary as the first, second, or third power of the impurity concentration depending on the charge states of the substitutional and interstitial species. Universal calculated results are presented for these three cases for a constant surface concentration and semi-infinite medium. The results are used to explain the anomalous diffusion of zinc in GaAs, for which the model predicts that $D$ should vary as the square of the concentration. Six available diffusion profiles at 1000\ifmmode^\circ\else\textdegree\fi{}C can be fit using a single parameter, and the small temperature dependence of available diffusion profiles is in accordance with the theory. The interstitial zinc concentration is estimated to be several orders of magnitude below that of the substitutional zinc.

Diffusion of Phosphorus into Silicon

The phosphorus diffusion into silicon under the condition of constant surface concentration has been studied by means of radiotracer technique and conductivity measurements. The anormalies of apparent diffusion coefficient calculated from usual p-n junction method could be explained by assuming the existence of an extra fast diffusion mechanism, corresponding to the distribution profile in the high impurity concentration region of a diffused layer. The activation energy of the fast diffusion was about 1.8 eV, this diffusion was considered to be an interstitial migration of unionized phosphorus. A maximum value of conductivity lies at the doped phosphorus concentration of about 5×10 20 /cc. Electron mobility decreases with the increase of phosphorus concentration in the high concentration range above 5×10 20 /cc.

Studies on Mixture Dyeing. I. The Kinetics of the Mixture of Benzopurpurine 4B and Sky Blue 6B

Abstract 1. The single and mixture dyeing of Benzopurpurine 4B (C. I. 23500) and Sky Blue 6B (C. I. 24410) were examined kinetically. 2. The diffusion coefficient of each dye decreased with a decrease in the surface concentration. The apparent activation energy of diffusion at a constant surface concentration was evaluated. 3. In mixture dyeing, the dye distributions in the substrate were considerably different from that in the single dyeing, but there was good agreement between the dye distributions experimentally determined and those theoretically calculated in terms of the thermodynamics of irreversible processes. 4. The interaction between two dyes in the substrate increased with an increase in the sodium chloride concentration in the dye bath and with a decrease in the temperature.