a = 0.5 μm, b = 1 μm, and various values of 4Dt. Again, at 4Dt ≅ 1, the diffusion profile is reasonably flat for over 1 μm making it ideal for forming a p-n junction.
11.7 FINITE SOURCE AND RANDOM WALK
11.7 Finite Source and Random Walk 401
p x t x t
N Dt
x
, , Dt
e .
( )
=η( )
= −π 1 4
2
4 (11.27)
And the mean position from the origin, x, for the N atoms is simply,
x Dt x dx
x
= Dt
−∞
∞ −
1
∫
4
2 4
π e .
In addition, making the usual substitution of y x= 4Dt, the integral becomes 4Dt ye dyy2 Dt e y2 0
π π
−
−∞
∞ −
−∞
= − ∞ =
∫
(11.28)which is no big surprise because Equation 11.27 is an even function of x.* Equation 11.28 says that the mean position of an atom is zero, which is also not surprising because the atoms can move either in the positive or in the negative x-direction and on the average, there will be as many to the left of the origin as there is to the right.
11.7.3 m
eans
quared
Istance Ino
ned
ImensIonOn the other hand, the mean square distance covered by atoms moving only in the positive and negative x-directions is given by
x Dt x dx
x
2 1 2 4Dt
4
2
=
−∞
∞ −
∫
π e (11.29)
where means the average of and again making the usual substitution y x= 4Dt this becomes x2 Dt y e dy2 y Dt y e dy2 y
0
4 2 8 2
= =
−∞
∞ − ∞
∫ ∫
−π π (11.30)
because it is an even function of y. To evaluate the integral, let y2= z, then y e dy2 y z e dzz z e dz soz
0
1 2 0
3 1 1 0
2 1
2
1 2
∞ − ∞
− ∞ ( −) −
= =
∫ ∫
/∫
( / )y e dy2 y
0
2 1
2 3 2
1
2 2 4
∞ −
=
=
=
∫
Γ π π (11.31)where Γ
( )
3 2/ is the gamma function of 3/2 where the gamma function is just a number and is defined by (see Appendix A.3)Γ α
( )
=∫
0∞x e dxα−1 −x .Therefore,
Γ 3
2 2
3 2 1 0
=
∫
∞x(( / )−)e dx−x = π.So combining Equations 11.30 and 11.31, the mean square displacement is related to the diffusion coefficient and is given by
x2 =2Dt. (11.32)
* An even function is one for which f(−x) = f(x) such as cos(x). An odd function is one for which f(−x) = −f(x) such as sin(x).
11.7.4 m
eans
quared
IstanceInt
hreed
ImensIonsIn three dimensions, the displacement of an atom from the origin is a vector
R that can be written as
R xi yj zk= + + and because R2=x2+y2+z2, then
R2 = x2 + y2 + z2 .
And in the discussion in Section 11.7.3, there was no distinction made about the three directions x, y, and z, so the argument there holds for the other two mean square distances as well, so that
R Dt Dt Dt
R Dt
2
2
2 2 2
6
= + +
= . (11.33)
11.7.5 r
andomw
alkandd
IffusIon11.7.5.1 Example with 15 Steps
Consider the two-dimensional lattice of Figure 11.18. After 15 random atom jumps* of equal dis- tance a in the ±x or ±y directions, a random walk, it has moved to position
R15 from the origin, and this is the vector sum of all of the 15 individual atom jumps,
R15= + +r1 r2 r15
where the length of each ri is always a. Then the square of the displacement, R152
, is just the dot product of the vector with itself
* In Section 9.3.5, it was shown that interstitial atoms can take about 108 jumps per second, so this random walk of 15 steps takes place in less than a microsecond!
x y
R15
a
11 12
10
13
9 14
7
6 5
4
0 1
2 15 8
3
→R1→3
→R13→15
FIGURE 11.18 Two-dimensional random walk on a square grid of 15 steps each of distance a. R15 is the resulting distance vector of the 15 steps from the origin.
R1 3→ =R13 15→ are two shorter three-step paths.
11.7 Finite Source and Random Walk 403
R R r r r r r r
r r r r
15 15 1 2 15 1 2 15
1 1 1 2
⋅ =
(
+ +)
⋅(
+ +)
= ⋅ + ⋅ +
r r r r r r r r
r r r r
1 15
2 1 2 2 2 15
15 1 15 2
⋅ + ⋅ + ⋅ + ⋅
+ ⋅ + ⋅ + rr r15⋅15
(11.34)
Equation 11.34 is a sum of 225 terms! However, this can be simplified by grouping terms
R15⋅R15=
(
r r1⋅ + ⋅ + ⋅1 r r2 2 r r3 3 +r r15⋅15)
+(
r r1⋅ + ⋅⋅ + ⋅2 r1 + ⋅)
+ ⋅ + ⋅ + ⋅ + ⋅
r r r r r r r r r r r r r
3 1 4 1 15
2 1 2 3 2 4 2 155 15 1 15 2 15 3 15 14
( )
+(
r r⋅ +r r⋅ +r r⋅ +r r .⋅)
(11.35) Grouping the diagonal terms separately in the first sum, it follows that (if you spend the time and look very carefully at Equation 11.34 [Shewmon 1989]):
R R r ri i r r
i
i i j i
j
j 15 15
1 15
1 15
1 14
⋅ = ⋅ +2 ⋅
=
+
=
−
∑ ∑
=∑
. (11.36)The first term in Equation 11.36 is obvious, while the second term is harder to see. However, the 2 in front of the second term occurs since there are two dot products for each term in the sum that are the same, that is, r r1⋅2 and r r2⋅1 underlined in Equation 11.35. The dot product can be written as
r r1⋅ =2 r r1 2cosθ1 2,
where ri are the magnitudes of each of the vectors, ri = a in this case, and θi j, is the angle between any two vectors. Therefore, Equation 11.36 can be written:
R a a i i j
i j
j
152 2 2
1 15
1 14
15 2
= + +
=
−
=
∑
∑
cosθ, .For the random walk of Figure 11.18, cosθi j, =0 1 1, ,− , because the vectors are all at right angles, are parallel, or are antiparallel, respectively. Note that in Figure 11.18, R152
≠ 0, and by inspection it is R152 =x2+y2=( )2a2+a2=5a2. This could be shown by taking all 225 of the dot products above—
not an exciting project!
11.7.5.2 Example with Three Steps
However, the principle can be demonstrated by taking a shorter path of only three vectors, say
R1 3→ =
(
r1+ +r2 r3)
, so that
R R r r r r r r
r r r r
1 3 1 3 1 2 3 1 2 3
1 1 1
→ →
( )
⋅( )
=(
+ +)
⋅(
+ +)
= ⋅ + ⋅ 22+ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ r r1 3 r r2 1 r r2 2 r r2 3 r r3 1 r r3 2 r r3 3
==
(
r r1⋅ + ⋅ + ⋅1 r r2 2 r r3 3)
+ r r1⋅ + ⋅ + ⋅ + ⋅2 r r1 3 r r2 1 r r2 33 3 1 3 21 1 2 2 3 3 2 1 2
+ ⋅ + ⋅
( )
=
(
⋅ + ⋅ + ⋅)
+ ⋅ +
r r r r r r r r r r r r r11 3 2 3
2 2
1 2 1 3 2 3
1 3 2
3 2
⋅ + ⋅
( )
= +
(
+ +)
( )
→ =
r r r
a a
R
cosθ, cosθ, cosθ,
33a2+2a2
(
0 1 0− +)
=a2.Because θ1 3, =180°, cosθ1 3, = −1 and the other two angles are 90° so that their cosines are zero. Note that in Figure 11.18, the length of the vector
R1 3→ is indeed
R1 3→ =a and points in the −y direction from the origin to where vectors 3 and 4 connect.
11.7.5.3 General Result
For any general value of n, Equation 11.36 becomes
R Rn n r ri i r r
i n
i i j i n j
j n
⋅ = ⋅ + ⋅
=
+
=
−
=
∑ ∑
−∑
1 1 1
1
2 (11.37)
and following the procedures above, Equation 11.37 can be written
R n a
n n i i j
i n j
j n
2 2
1 1 1
1 2
= +
+
=
−
=
−
∑
∑
cosθ, (11.38)where Rn is the path taken by a single diffusing atom. To get the mean position of all the diffusing atoms, the average of Equation 11.38 must be taken so,
R n a
n n i i j
i n j
j n
2 2
1 1 1
1 2
= +
+
=
−
=
−
∑
∑
cosθ, (11.39)where the brackets designate the average value. As was shown, some of the cosine values may be positive and some may be negative, and on the average, they will cancel out when summed over a large value of n jumps and the double sum in Equation 11.39 equals zero! Therefore, in general, the mean square distance, Rn2
, in a random walk is just
Rn2 =na2. (11.40)
Both the 15-step and the 3-step examples in Sections 11.7.5.1 and 11.7.5.2 are off by a factor of 5 simply because only a very small number of steps were taken in each case.
11.7.6 d
IffusIonc
oeffIcIentThe mean distance between jumps is a as is shown on the square lattice in Figure 11.18, then R2 =n a2, where n is the number of jumps necessary to get to position
R, Equation 11.40. Therefore, Equation 11.33 can be rewritten as
R2 =n a2=6Dt or
D n
t a
D a
=
= 1 6 1 6
2
Γ 2
(11.41)
where Γ is the number of jumps per second. Equation 11.41 is the same as Equation 9.11 found in Section 9.3 for interstitial diffusion in solids. However, here Equation 11.41 has been developed from a statistical argument rather that the more concrete concept of atoms moving in a cubic lattice.
Appendix 405