HW3 Diffusion Solutions

ELEG 422/622

HW3, Chapter 3, Diffusion

1. Assume that you have been asked to measure the diffusivity of a donor impurity in a new elemental semiconductor. What constants would you need to measure? What experiments would you attempt? Discuss the measurement techniques that you would use to measure the chemical and carrier profiles. What problems are likely to arise?

One approach would be to perform a series of diffusions of the impurity into semiconductor substrates at various times and temperatures and then measure the resulting dopant distributions in the substrates. As discussed in the text, dopant profiles can be characterized by methods including secondary ion mass spectroscopy (SIMS) or spreading resistance measurements. Assuming a predeposition type diffusion process, for example, each of the experimental dopant distributions would then be fit by the complementary error function solution of Fick’s Law to determine the dopant diffusivity. Alternatively, a more sophisticated model could be developed for the diffusion process and then fit to the experimental data to deduce the dopant diffusivity. You would need to measure the diffusivity at various temperatures and then use this data to construct an Arrhenius plot. The activation energy and pre-exponential factor could then be obtained from the slope and y-intercept, respectively, of the resulting plot. According to the author of the textbook, the experiment should be performed with various substrate concentrations to extract charge effects. This refers to the charged vacancy diffusion mechanisms that appear in Fair’s Vacancy Model, each of which has its own activation energy and pre-exponential factor. Some of the challenges that could arise include breakdown of the simplifying assumptions made in arriving at analytical solutions to Fick’s Law; for example, the approximations that the diffusivity is independent of position or concentration. Diffusivity is increased at high dopant concentrations by the field enhancement effect. Also, uncertainty in the solid solubility limit of the dopant in the substrate, the presence of surface layers such as oxides on the substrate that alter the diffusion processes, and the existence of more complex diffusion mechanisms other than vacancy-mediated diffusion, such as interstitial or interstitialcy mechanisms, will create challenges for the first-order diffusion theory. Additionally, the crystalline quality of the substrate, including dislocations and other crystal defects, will have an impact on the diffusion rates.

ELEG 422/622

HW3, Chapter 3, Diffusion

2. Using Fair’s vacancy model, including charge effects, calculate the diffusivity of arsenic (As) in silicon at 1000° C for the following arsenic doping concentrations: (a) 1E15 cm-3 (b) 1e21 cm-3.

Referring to figure 3.4, at 1000° C the intrinsic carrier concentration ni is about 1E19 cm-3. Also, at 1000° C, 𝑘𝑘𝑘𝑘 = 8.617E-5

eV ∙ K

1273 K = 0.110 eV.

a) At an arsenic doping concentration of Nd = 1E15 cm-3, Nd > ni so in this case n = Nd. Thus we have 𝑛𝑛 − −𝐸𝐸− ⁄𝑘𝑘𝑘𝑘 cm2 −3.44eV⁄0.110eV 1E21cm-3 cm2 −4.05eV⁄0.110eV 𝐷𝐷0 𝑒𝑒 𝑎𝑎 = 0.066 ∙ 𝑒𝑒 + ∙ 12.0 ∙ 𝑒𝑒 -3 s 1E19cm s 𝑛𝑛𝑖𝑖 cm2 cm2 cm2 = 1.59E-15 + 1.11E-13 = 1.12E-13 s s s

𝐷𝐷 = 𝐷𝐷0 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 +

ELEG 422/622

HW3, Chapter 3, Diffusion

3. Delta doping is a process used in advanced GaAs fabrication to increase the Schottky barrier height of the gate electrode. This reduces the gate electrode leakage. Delta doping is done by depositing a monolayer of a p-type dopant material directly between the gate electrode and the GaAs. Assume that the atomic surface coverage is 1.5E15 cm-2 and that the dopant is beryllium (Be). After the gate patterning, the source/drain is annealed at 800° C for 10 min to activate the impurity. (a) If the gate material prevents any out-diffusion from the wafer, use first-order diffusion theory to calculate the junction depth if the channel is doped 1E17 cm-3 n-type. (b) What surface concentration of Be will result? (c) Sketch the profile that you calculated using this simple theory and the profile that might actually be expected. Briefly list two reasons for the difference. Delta doping simply refers to an extremely thin, high concentration layer of dopant atoms, perhaps an atomic monolayer deposited as part of an epitaxial crystal growth process. This situation corresponds to the boundary conditions for a drive-in diffusion: we have a thin layer of dopant at the surface that is prevented from escaping from the wafer due to the gate oxide. The total number of dopant atoms, i.e. the dose (which take units of number per square cm), is fixed. Consequently, this is a drive-in type diffusion, the dopant will diffuse further into the wafer and form a Gaussian profile. The dose of the p-type beryllium is given as QT = 1.5E15 cm-2, the background doping (the doping level of the n-type region where the p-n junction will be formed) is CB = 1E17 cm-3, and the drive-in diffusion process conditions are 800° C for 600 sec. a) Referring to table 3.2 for Be in GaAs, D0 = 7E-6 cm2/s and Ea = 1.2 eV. At 800° C, 𝑘𝑘𝑘𝑘 = 8.617E-5 1073 K = 0.092 eV. Thus

𝐷𝐷 = 𝐷𝐷0 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 = 7E-6

eV ∙ K

cm2 −4.16eV⁄0.092eV cm2 ∙ 𝑒𝑒 = 1.62E-11 s s

and so the characteristic diffusion length √𝐷𝐷𝐷𝐷 = �1.62E-11

cm2 s

∙ 600s = 9.85E-5cm, which is the

average distance the impurities will diffuse into the wafer during a drive-in. According to equation 3.21 for drive-in conditions (which is obtained by setting the expression for the drive-in impurity profile equal to the background concentration, then solving for position), the p-n junction will be located at 𝑥𝑥𝑗𝑗 = �4𝐷𝐷𝐷𝐷 ∙ ln �

𝑄𝑄𝑇𝑇

𝐶𝐶𝐵𝐵 √𝜋𝜋𝜋𝜋𝜋𝜋

� = 2√𝐷𝐷𝐷𝐷�ln �

𝑄𝑄𝑇𝑇

𝐶𝐶𝐵𝐵 √𝜋𝜋 ∙ √𝐷𝐷𝐷𝐷

= 4.16E-4cm = 4.16µm

� = 2 ∙ 1.10E-11cm∙�ln �

1.5E15cm-2

� 1E17cm-3 √𝜋𝜋 ∙ 9.85E-5cm

b) The resulting surface concentration decreases with time (equation 3.19). At the end of the diffusion,

c)

𝐶𝐶𝑠𝑠 = 𝐶𝐶(0, 𝑡𝑡) =

𝑄𝑄𝑇𝑇

√𝜋𝜋 ∙ √𝐷𝐷𝐷𝐷

=

1.5E15cm-2

√𝜋𝜋 ∙ 9.85E-5cm

= 8.59E18cm-3

ELEG 422/622

HW3, Chapter 3, Diffusion

ELEG 422/622

HW3, Chapter 3, Diffusion

4. A silicon wafer was doped in a 1000° C pre-deposition diffusion with phosphorous to its solid solubility limit. The process time was 20 min. After the pre-deposition, the surface of the silicon was sealed and an 1100° C drive-in was done. Find the drive-in time necessary to obtain a junction depth of 4.0 µm. Assume a substrate concentration of 1e17 cm-3. What is the surface concentration after the drive-in?

a) According to figure 2.4, at 1000° C the solid solubility limit of P in Si is about Cs = 1E21 cm-3. Also at 1000° C, 𝑘𝑘𝑘𝑘 = 8.617E-5

eV ∙ K

1273 K = 0.110 eV.

For intrinsic diffusion (the problem set says to

assume intrinsic diffusivity unless otherwise stated), n = ni and the diffusivity is given by −

2−

𝐷𝐷 = 𝐷𝐷0 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 + 𝐷𝐷0− 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 + 𝐷𝐷02− 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 cm2 −3.66eV⁄0.110eV cm2 −4.0eV⁄0.110eV cm2 −4.37eV⁄0.110eV = 3.9 ∙ 𝑒𝑒 + 4.4 ∙ 𝑒𝑒 + 44 ∙ 𝑒𝑒 s s s 2 2 2 cm cm cm cm2 = 1.26E-14 + 6.41E-16 + 2.20E-16 = 1.34E-14 s s s s

while the characteristic diffusion length is √𝐷𝐷𝐷𝐷 = �1.34E-14 deposition diffusion therefore results in a dose of

𝑄𝑄𝑇𝑇 (𝑡𝑡) =

2

√𝜋𝜋

𝐶𝐶(0, 𝑡𝑡)√𝐷𝐷𝐷𝐷 =

2 ∙1E21 cm-3 √𝜋𝜋

cm2 s

∙ 20 ∙ 60s = 4.02E-6cm.

The pre-

∙ 4.02E-6cm = 4.54𝐸𝐸15 cm-2 .

For the drive-in, the diffusivity at 1100° C is 1.56E-13 cm2/s. The drive-in produces a junction depth of 𝑥𝑥𝑗𝑗 = �4𝐷𝐷𝐷𝐷 ∙ ln �

𝑄𝑄𝑇𝑇

𝐶𝐶𝐵𝐵 √𝜋𝜋𝜋𝜋𝜋𝜋

-3

� , where CB = 1e17 cm . Note that it is not possible to sole explicitly for the time.

Therefore, we can plug in values for t and then calculate x, iterating until convergence. Alternatively, plotting xj versus t results in

ELEG 422/622

HW3, Chapter 3, Diffusion -4

6

x 10

5

4

3

2

1

0

0

1

2

3

4

5

6

7

8

9

10 4

x 10

Therefore to achieve a junction depth of 4 microns = 1E-4 cm requires about 4.5E4 s or 12.5 hours.

The surface concentration after the drive in is then 𝐶𝐶𝑠𝑠 = 𝐶𝐶(0, 𝑡𝑡) =

𝑄𝑄𝑇𝑇 √𝜋𝜋𝜋𝜋𝜋𝜋

=

4.54𝐸𝐸15 cm-2

2 �𝜋𝜋∙1.56E-13𝑐𝑐𝑐𝑐 𝑠𝑠

∙4.5E4 s

= 3.03E19cm-3 .

ELEG 422/622

HW3, Chapter 3, Diffusion

5. For deeply scaled MOSFETs, it is necessary to make very shallow source/drain junctions. Assume that one needs a p+/n junction that is 0.05 µm deep. The wafer is implanted with boron at extremely low energy (Rp > ND), the diffusivity of boron at 1000° C is given by cm2 −3.46eV⁄0.110eV cm2 −3.46eV⁄0.110eV ∙ 𝑒𝑒 +∙ 0.41 ∙ 𝑒𝑒 s s cm2 cm2 cm2 = 7.41E-16 + 8.21E-15 = 8.95E-15 s s s +

𝐷𝐷 = 𝐷𝐷0 𝑒𝑒 −𝐸𝐸𝑎𝑎⁄𝑘𝑘𝑘𝑘 + 𝐷𝐷0+ 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 = 0.037

As in the previous problem, the junction depth is given by 𝑥𝑥𝑗𝑗 = �4𝐷𝐷𝐷𝐷 ∙ ln �𝐶𝐶

𝑄𝑄𝑇𝑇

𝐵𝐵 √𝜋𝜋𝜋𝜋𝜋𝜋

and CB = 2e17 cm-3. Plotting xj versus t results in -5

1

x 10

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

50

100

150

200

250

-2

� , where QT = 5E15 cm

300

Therefore to achieve a junction depth of 0.05 microns = 5E-6 cm requires about 70 s.

ELEG 422/622

HW3, Chapter 3, Diffusion

6. A two-step diffusion is performed in a p-type silicon wafer with 1e17 cm-3 uniform background doping. A pre-deposition is done at 1100° C for 1 hour using arsenic, followed by a drive-in at 900° C for 6 hours. What is the final junction depth? Make a plot of dopant concentration versus depth into the wafer following each of the diffusion processes. Compare the characteristic diffusion lengths for the pre-dep and the drive-in and comment on implications for the accuracy of the diffusion model.

At 1100° C, 𝑘𝑘𝑘𝑘 = 8.617E-5

eV ∙ K

1373 K = 0.118 eV, while at 900° C, 𝑘𝑘𝑘𝑘 = 8.617E-5

eV ∙ K

1173 K =

0.101 eV. Also from figure 2.4, the solid solubility of arsenic in silicon at 1100° C is about 2E21 cm-3.

According to table 3.2, the diffusion coefficient for arsenic in silicon depends predominantly on neutral and singly ionized vacancy diffusion mechanisms. Therefore, from equations 3.7 and 3.11, the diffusivity of arsenic at 1100° C (corresponding to the pre-dep conditions) is given by 𝑛𝑛 − −𝐸𝐸− ⁄𝑘𝑘𝑘𝑘 𝐷𝐷 𝑒𝑒 𝑎𝑎 𝑛𝑛𝑖𝑖 0 cm2 -3.44 eV⁄0.118 eV 2E21 cm-3 cm2 -4.05 eV⁄0.118 eV = 0.066 ∙ 𝑒𝑒 + ∙ 12.0 ∙ 𝑒𝑒 s 2E19 cm-3 s cm2 cm2 cm2 = 1.556E-15 + 100 ∙ 1.631E-15 = 1.647E-12 s s s

𝐷𝐷 = 𝐷𝐷0 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 +

In the previous calculation, we used 𝑛𝑛𝑖𝑖 ≈ 2E19 cm-3 from figure 3.4 at 1100° C and 𝑛𝑛 ≈ 𝐶𝐶𝑠𝑠 = 2E21 cm-3 since 𝑁𝑁𝐷𝐷 = 𝐶𝐶𝑠𝑠 ≫ 𝑛𝑛𝑖𝑖 (see example 3.1). For the pre-dep, the characteristic diffusion length √𝐷𝐷𝐷𝐷 = �1.647E-12

cm2 s

∙ 3600s = 7.70E-5cm, and so

the dose of arsenic introduced into the wafer during the pre-deposition is given by equation 3.16 as 𝑄𝑄𝑇𝑇 (𝑡𝑡) =

2

√𝜋𝜋

𝐶𝐶(0, 𝑡𝑡)√𝐷𝐷𝐷𝐷 =

2

√𝜋𝜋

∙2E21 cm-3 ∙ 7.70E-5cm = 1.738E17 cm-2

For the drive-in, the diffusivity of arsenic at 900° C is given by

𝑛𝑛 − −𝐸𝐸− ⁄𝑘𝑘𝑘𝑘 𝐷𝐷 𝑒𝑒 𝑎𝑎 𝑛𝑛𝑖𝑖 0 cm2 -3.44 eV⁄0.101 eV 2E21 cm-3 cm2 -4.05 eV⁄0.101 eV = 0.066 ∙ 𝑒𝑒 + ∙ 12.0 ∙ 𝑒𝑒 s 6E18 cm-3 s cm2 cm2 cm2 = 1.094E-16 + 3.333E3 ∙ 4.761E-17 = 1.588E-13 s s s

𝐷𝐷 = 𝐷𝐷0 𝑒𝑒 −𝐸𝐸𝑎𝑎 ⁄𝑘𝑘𝑘𝑘 +

where 𝑛𝑛𝑖𝑖 ≈ 6E18 cm-3 at 900° C according to figure 3.4. The characteristic diffusion length √𝐷𝐷𝐷𝐷 = �1.588E-13

cm2 s

∙ 6 ∙ 3600s = 5.857E-5cm for the drive-in.

After drive-in the junction depth is given by equation 3.21 as

ELEG 422/622

HW3, Chapter 3, Diffusion

𝑥𝑥𝑗𝑗 = �4𝐷𝐷𝐷𝐷 ∙ ln �

𝑄𝑄𝑇𝑇

𝐶𝐶𝐵𝐵 √𝜋𝜋𝜋𝜋𝜋𝜋

� = 2√𝐷𝐷𝐷𝐷�ln �

𝑄𝑄𝑇𝑇

𝐶𝐶𝐵𝐵 √𝜋𝜋 ∙ √𝐷𝐷𝐷𝐷

= 3.65E-4cm = 3.65µm

� = 2 ∙ 5.857E-5cm∙�ln �

1.738E17 cm-2

1E17cm-3 √𝜋𝜋 ∙ 5.857E-5cm

�

The dopant profile after the pre-dep is given by 𝐶𝐶(𝑧𝑧, 𝑡𝑡) = 𝐶𝐶𝑠𝑠 ∙ erfc �

𝑧𝑧

2√𝐷𝐷𝐷𝐷

� = 2E21 cm-3 ∙ erfc �

While the dopant profile after the drive-in is given by 𝐶𝐶(𝑧𝑧, 𝑡𝑡) =

Which look like this:

𝑄𝑄𝑇𝑇

√𝜋𝜋 ∙ √𝐷𝐷𝐷𝐷

∙ 𝑒𝑒 −𝑧𝑧

2 ��2∙√𝐷𝐷𝐷𝐷�2

=

𝑧𝑧 � 2 ∙ 7.70E-5cm

1.738E17 cm-2 −𝑧𝑧 2⁄(2∙5.857E-5cm)2 ∙ 𝑒𝑒 √𝜋𝜋 ∙ 5.857E-5cm

22

10

pre-dep drive-in 20

10

18

cm-3

10

16

10

14

10

12

10

10

10

0

0.1

0.2

0.3

0.4

0.5 cm

0.6

0.7

0.8

1

0.9

-3

x 10

Under the conditions stated in the problem, the diffusion length for the drive-in is smaller than that for the predep. The solution of Fick’s Law for the drive-in assumes that the entire concentration of dopants is at the surface, which is approximately true if the diffusion length for the drive-in is much larger than that for the pre-dep (see equation 3.20). In the present situation, we violated this condition, and the profile predicted by the expression above exhibits an unexpected result where dopant profile appears to be shallower following the subsequent drivein diffusion process. (This is because we assume all the dopant as at the surface at the start of the drive-in).