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Impurity concentration profile in irradiation enhanced diffusion experiments
Volume 49A, number 2
PHYSICS LETTERS
26 August 1974
IMPURITY CONCENTRATION PROFILE IN IRRADIATION ENHANCED DIFFUSION EXPERIMENTS~ M.A. St. PETERS, D. PEAK and J.W. CORBETT Physics Department, State Universityof New York at Albany, Albany, New York 12222, USA Received 6 June 1974 Experiments dealing with irradiation enhanced diffusion in semiconductors in which the exciting particles have.a range less than the sample thickness are shown to be characterized by nonexponential diffusant concentration profiles as functions of lattice depth.
A number of recent papers have dealt with the ionization and recoil enhanced diffusion of impurities and
in which there is a uniform, enhaRced effective diffusion coefficient, D1 and (ii) the remainder of the sam-
defects in semiconductors [1—7].What apparently has not been appreciated is that in experiments in which the exciting particles have a range less than the sample thickness, the diffusion profile directly reflects this limited penetration depth. To illustratesituation. this point following experimental A we thinconsider sheet ofthe tracer atoms is deposited on a planar surface of a sample material, This planar face is then exposed to light, X-rays, electrons, or other particles, all of which create ionization and, in the case of the particle irradiations, atomic recoils throughout the range of the bombarding particle, Often, as in the case of photons or charged particles, the irradiation will be effectively stopped well within
pie, x > x1, in which the diffusion coefficient is the normal, thermally activated one,D2. If the tracer dif. fusion is assumed to be of simple Fickian form, then in our model the concentration profile at any given time2cis determined by integrating = ~ c D‘ aX I’ where i is either I or 2 indicating the appropriate lat. tice region, D~is the corresponding diffusivity, and c~ the corresponding concentration. Solutions to (1) must satisfy suitable boundary conditions. We have chosen these to be: (i) the tracer concentration to be everywhere continuous, (ii) the diffusion current density ~ = —D~a~c 1) to be everywhere continuous, and (iii) the number of tracer atoms to be conserved. Stan. dard procedures [9] for handling (1) yield the following dimension free result 112 ~ [b’~~ u.(y z) = (irzy 1 n0
the target sample. If the radiation-induced ionization or recoils enhance diffusion, they do so only in that portion of the sample into which their influence penetrates. We treat the full complexities of this problem elsewhere [81, but here illustrate it with a simple model. Consider first only ionization enhancement. Let us assume that the sample has essentially planar symme-
X exp (— { 2n +(2—y)~ 1~ + [1+a(y— 1)] ~
try so that the spatial variations in the problem are completely described in terms of the depth parameter, x, where x = 0 is the incident face on which all of the tracer atoms are initially located~.We divide the sample into two regions: (i) the surface region, 0 ~ X ~ X1, Supported m part by the Metallurgy and Materials Program of the Division of Research, United States Atomic Energy Commission under Contract AT(11-l)-3478. * For our purposes, the crystal can otherwise be assumed to have infinite extent.
/4z)
2
+b exp(— {2ny6~1+[l+a(y—1)] &i2} /4z)] (2) where el,, / is the Kronecker delta. In this expression we have employed the following definitions: (i) u, is a normalized concentration given by u1 = cixi /00, where c1 (x, 0) = o06(x); (ii) y is a dimensionless multiple of x1 ,y x/x1 ; (iii) z is a dimensionless diffusion time, 2
.
1/2
z = D~t/x1(iv) a is given by a = (D1/D2) ; and (v) b by b (a—l)/(a+l). Eq. (2) is plotted in fig. I for y-values near one. Shown are curves for fixed z, z = 0.1, for three differ159
Volume 49A, number 2
PHYSICS LETTERS
26 August 1974
ple under the conditions described above would be neatly partitioned into two regions characterized by different (constant) diffusivities, the qualitative fea-
0.6 -
o I— Li 0
z
0 0
04
ence centration of nonexponential profile, shouldstructure be seen in in the experiments impurity conwhere ionization enhanced diffusion is evident and where the region of ionization is essentially limited.
-
0 Li
Li 0.2 -~ 0 r-4 -j
onephotons, tures Inmay of conclusion, the discover model nonexponential we presented should point here,out profiles namely thatof the if,the instead prestype of the sample is bombarded with particles
oH
depicted in fig. 1 which may have nothing to do with
0 z
08
.0
.2
REDUCED LATTICE COORDINATE,y Fig. 1. The tracer atom concentration u(y,z)=c(x,t)x
1/o0 versus a reduced lattice depth y = x/xi is shown for a fixed diffusion time z2.The = D1t/x~ 0.1 for three diffusivity ratios difa = 1 =curve corresponds to a uniform afusivity = (D1/D2)~ throughout the lattice while the a = 2 and a 10 curves show the effects of an enhanced diffusion in the region .v ~ 1.
ent values of a. We see the clear signature of ionization enhanced diffusion which extends only partially into the sample. The a = I curve corresponds to a lattice with a single, uniform diffusivity and displays a typical exponential fall-off withy2. On the other hand, for a = 2 and a = 10 we see “structure” in the concentration profile at x = x 1. The physical significance of this structure is quite apparent. While the diffusion time is the same for atoms on each of the three curves in region I, it is different for all three in region 2. This is so because the diffusion, or local, time of a tracer atom in region 2 is really D2t/x~.Hence, atoms at y 1 have had only one percent as long to diffuse into region 2 on the a = 10 curve as on the a = 1 curve. The results presented here should be viewed as merely suggestive of what might be actually found in the laboratory. While it seems unlikely that a real sam-
160
upon ionization about collision if the enhancement. particles with deliver tracer These sufficient atoms anomalies torecoil propel can energy come them into the lattice. Suchthe recoil effects produce preferential migration into the sample similar in nature to diffusion in the presence of a drifting force. Our analysis of this
situation shows that nonexponential profiles can which are very much like sees result in sedimentation problems [10].the profiles one
References ~i] V.M. Lenchenko, Fiz Tverd. Tela 11(1969)799 [Soy. Phys. — So!. St. 11(1969)649]. [2] R.E. McKeighen and IS. Koehier, Phys. Rev. B4 (1971)
462. [3] J.W. Corbett and J.C. Bourgoin, IEEE Trans Nuc!. Sd. NS-18, No.6(1971)11. [4] J.C. Bourgoin and J.W. Corbett, Phys. Lett. A38 (1972) 135.
[51 L.N. [6]
[7] [8] [9]
Zyuz, A.E. Kiv, O.R. Niyazova and F.T. Umarova, ZhETF Pis. Red. 17 (1973) 230 [JETP Lett. 17 (1973) 165]. lW. Corbett, J.C. Bourgoin and C. Weigel, in Radiation damage and defects in semiconductors (Inst. of Phys., London, 1973) p. 1. J.C. Bourgoin, J.W. Corbett and H.L. Frisch, J. Chem. Phys. 59 (1973) 4042. M.A. St. Peters, D. Peak and J.W. Corbett, to be published. L.H. Sneddon, in The use of integral transforms
McGraw-Hill, New York, 1972) p. 135. [10] S. Chandieasekhar, Rev. Mod. Phys. 15 (1943) 1.
PHYSICS LETTERS
26 August 1974
IMPURITY CONCENTRATION PROFILE IN IRRADIATION ENHANCED DIFFUSION EXPERIMENTS~ M.A. St. PETERS, D. PEAK and J.W. CORBETT Physics Department, State Universityof New York at Albany, Albany, New York 12222, USA Received 6 June 1974 Experiments dealing with irradiation enhanced diffusion in semiconductors in which the exciting particles have.a range less than the sample thickness are shown to be characterized by nonexponential diffusant concentration profiles as functions of lattice depth.
A number of recent papers have dealt with the ionization and recoil enhanced diffusion of impurities and
in which there is a uniform, enhaRced effective diffusion coefficient, D1 and (ii) the remainder of the sam-
defects in semiconductors [1—7].What apparently has not been appreciated is that in experiments in which the exciting particles have a range less than the sample thickness, the diffusion profile directly reflects this limited penetration depth. To illustratesituation. this point following experimental A we thinconsider sheet ofthe tracer atoms is deposited on a planar surface of a sample material, This planar face is then exposed to light, X-rays, electrons, or other particles, all of which create ionization and, in the case of the particle irradiations, atomic recoils throughout the range of the bombarding particle, Often, as in the case of photons or charged particles, the irradiation will be effectively stopped well within
pie, x > x1, in which the diffusion coefficient is the normal, thermally activated one,D2. If the tracer dif. fusion is assumed to be of simple Fickian form, then in our model the concentration profile at any given time2cis determined by integrating = ~ c D‘ aX I’ where i is either I or 2 indicating the appropriate lat. tice region, D~is the corresponding diffusivity, and c~ the corresponding concentration. Solutions to (1) must satisfy suitable boundary conditions. We have chosen these to be: (i) the tracer concentration to be everywhere continuous, (ii) the diffusion current density ~ = —D~a~c 1) to be everywhere continuous, and (iii) the number of tracer atoms to be conserved. Stan. dard procedures [9] for handling (1) yield the following dimension free result 112 ~ [b’~~ u.(y z) = (irzy 1 n0
the target sample. If the radiation-induced ionization or recoils enhance diffusion, they do so only in that portion of the sample into which their influence penetrates. We treat the full complexities of this problem elsewhere [81, but here illustrate it with a simple model. Consider first only ionization enhancement. Let us assume that the sample has essentially planar symme-
X exp (— { 2n +(2—y)~ 1~ + [1+a(y— 1)] ~
try so that the spatial variations in the problem are completely described in terms of the depth parameter, x, where x = 0 is the incident face on which all of the tracer atoms are initially located~.We divide the sample into two regions: (i) the surface region, 0 ~ X ~ X1, Supported m part by the Metallurgy and Materials Program of the Division of Research, United States Atomic Energy Commission under Contract AT(11-l)-3478. * For our purposes, the crystal can otherwise be assumed to have infinite extent.
/4z)
2
+b exp(— {2ny6~1+[l+a(y—1)] &i2} /4z)] (2) where el,, / is the Kronecker delta. In this expression we have employed the following definitions: (i) u, is a normalized concentration given by u1 = cixi /00, where c1 (x, 0) = o06(x); (ii) y is a dimensionless multiple of x1 ,y x/x1 ; (iii) z is a dimensionless diffusion time, 2
.
1/2
z = D~t/x1(iv) a is given by a = (D1/D2) ; and (v) b by b (a—l)/(a+l). Eq. (2) is plotted in fig. I for y-values near one. Shown are curves for fixed z, z = 0.1, for three differ159
Volume 49A, number 2
PHYSICS LETTERS
26 August 1974
ple under the conditions described above would be neatly partitioned into two regions characterized by different (constant) diffusivities, the qualitative fea-
0.6 -
o I— Li 0
z
0 0
04
ence centration of nonexponential profile, shouldstructure be seen in in the experiments impurity conwhere ionization enhanced diffusion is evident and where the region of ionization is essentially limited.
-
0 Li
Li 0.2 -~ 0 r-4 -j
onephotons, tures Inmay of conclusion, the discover model nonexponential we presented should point here,out profiles namely thatof the if,the instead prestype of the sample is bombarded with particles
oH
depicted in fig. 1 which may have nothing to do with
0 z
08
.0
.2
REDUCED LATTICE COORDINATE,y Fig. 1. The tracer atom concentration u(y,z)=c(x,t)x
1/o0 versus a reduced lattice depth y = x/xi is shown for a fixed diffusion time z2.The = D1t/x~ 0.1 for three diffusivity ratios difa = 1 =curve corresponds to a uniform afusivity = (D1/D2)~ throughout the lattice while the a = 2 and a 10 curves show the effects of an enhanced diffusion in the region .v ~ 1.
ent values of a. We see the clear signature of ionization enhanced diffusion which extends only partially into the sample. The a = I curve corresponds to a lattice with a single, uniform diffusivity and displays a typical exponential fall-off withy2. On the other hand, for a = 2 and a = 10 we see “structure” in the concentration profile at x = x 1. The physical significance of this structure is quite apparent. While the diffusion time is the same for atoms on each of the three curves in region I, it is different for all three in region 2. This is so because the diffusion, or local, time of a tracer atom in region 2 is really D2t/x~.Hence, atoms at y 1 have had only one percent as long to diffuse into region 2 on the a = 10 curve as on the a = 1 curve. The results presented here should be viewed as merely suggestive of what might be actually found in the laboratory. While it seems unlikely that a real sam-
160
upon ionization about collision if the enhancement. particles with deliver tracer These sufficient atoms anomalies torecoil propel can energy come them into the lattice. Suchthe recoil effects produce preferential migration into the sample similar in nature to diffusion in the presence of a drifting force. Our analysis of this
situation shows that nonexponential profiles can which are very much like sees result in sedimentation problems [10].the profiles one
References ~i] V.M. Lenchenko, Fiz Tverd. Tela 11(1969)799 [Soy. Phys. — So!. St. 11(1969)649]. [2] R.E. McKeighen and IS. Koehier, Phys. Rev. B4 (1971)
462. [3] J.W. Corbett and J.C. Bourgoin, IEEE Trans Nuc!. Sd. NS-18, No.6(1971)11. [4] J.C. Bourgoin and J.W. Corbett, Phys. Lett. A38 (1972) 135.
[51 L.N. [6]
[7] [8] [9]
Zyuz, A.E. Kiv, O.R. Niyazova and F.T. Umarova, ZhETF Pis. Red. 17 (1973) 230 [JETP Lett. 17 (1973) 165]. lW. Corbett, J.C. Bourgoin and C. Weigel, in Radiation damage and defects in semiconductors (Inst. of Phys., London, 1973) p. 1. J.C. Bourgoin, J.W. Corbett and H.L. Frisch, J. Chem. Phys. 59 (1973) 4042. M.A. St. Peters, D. Peak and J.W. Corbett, to be published. L.H. Sneddon, in The use of integral transforms
McGraw-Hill, New York, 1972) p. 135. [10] S. Chandieasekhar, Rev. Mod. Phys. 15 (1943) 1.