1. Metal-Semiconductor Junction
March-28, 2016
Contents:
1. Ideal metal-semiconductor junction outside equilibrium

2. Key questions
In a metal-semiconductor junction under bias, is there current flow? If so, how exactly does it happen?
What are the key dependences of the current in a metal- semiconductor junction?

3. 1. Metal-semiconductor junction outside TE
I-V Characteristics
Few minority carriers anywhere  𝑚𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟 𝑑𝑒𝑣𝑖𝑐𝑒
Bottleneck: transport through SCR

4. In forward bias, 𝐽 ∝ 𝑒 𝑞𝑉/𝑘𝑇
In reverse bias, 𝐽 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑒𝑠 𝑤𝑖𝑡ℎ 𝑉

5. Balance between electron drift and diffusion in SCR:
 TE: perfectly balanced
 forward bias: ℰ ↓ ⇒𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛>𝑑𝑟𝑖𝑓𝑡
 reverse bias: ℰ ↑ ⇒𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛<𝑑𝑟𝑖𝑓𝑡
Net current due to imbalance of drift and diffusion
Drift-diffusion model
Start with electron current equation:
𝐽 𝑒 =𝑞 𝜇 𝑒 𝑛ℰ+𝑞 𝐷 𝑒 𝑑𝑛 𝑑𝑥 =𝑞 𝐷 𝑒 (− 𝑞𝑛 𝑘𝑇 𝑑𝜙 𝑑𝑥 + 𝑑𝑛 𝑑𝑥 )
Multiply by exp(- 𝑞𝜙 𝑘𝑇 ):
𝐽 𝑒 𝑒𝑥𝑝 − 𝑞𝜙 𝑘𝑇 =𝑞 𝐷 𝑒 − 𝑞𝑛 𝑘𝑇 𝑑𝜙 𝑑𝑥 𝑒𝑥𝑝 − 𝑞𝜙 𝑘𝑇 + 𝑑𝑛 𝑑𝑥 𝑒𝑥𝑝 − 𝑞𝜙 𝑘𝑇 =𝑞 𝐷 𝑒 𝑑 𝑑𝑥 𝑛𝑒𝑥𝑝 − 𝑞𝜙 𝑘𝑇

6. ∅ 𝑥 =−( ∅ 𝑏𝑖 −𝑉)( 𝑥 2 𝑥 𝑑 2 − 2𝑥 𝑥 𝑑 +1) for 0 ≤𝑥 ≤ 𝑥 𝑑
Integrate along the depletion region:
 left-hand side: 𝐽 𝑒 ≈𝐽 (negligible hole contribution), and J independent of x:
0 𝑥 𝑑 𝐽 𝑒 exp − 𝑞∅ 𝑘𝑇 𝑑𝑥=𝐽 0 𝑥 𝑑 exp − 𝑞∅ 𝑘𝑇 𝑑𝑥
 right-hand side:
0 𝑥 𝑑 𝑞 𝐷 𝑒 𝑑 𝑑𝑥 𝑛𝑒𝑥𝑝 − 𝑞∅ 𝑘𝑇 𝑑𝑥=𝑞 𝐷 𝑒 𝑛𝑒𝑥𝑝(− 𝑞∅ 𝑘𝑇 )
For left-hand side, use ∅(𝑥) obtained earlier:
∅ 𝑥 =−( ∅ 𝑏𝑖 −𝑉)( 𝑥 2 𝑥 𝑑 2 − 2𝑥 𝑥 𝑑 +1) for 0 ≤𝑥 ≤ 𝑥 𝑑

7. For right-hand side, use boundary conditions:
 At 𝑥=0:
∅ 0 =−( ∅ 𝑏𝑖 −𝑉)
𝑛 0 = 𝑁 𝐷 𝑒𝑥𝑝 −𝑞( ∅ 𝑏𝑖 −𝑉) 𝑘𝑇 = 𝑁 𝐶 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇 𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇
 At 𝑥= 𝑥 𝑑 :
∅ 𝑥 𝑑 =0
𝑛 𝑥 𝑑 = 𝑁 𝐷
Do integral, substitute boundary conditions and get:
𝐽= 𝑞 2 𝐷 𝑒 𝑁 𝑐 𝑘𝑇 2𝑞 ∅ 𝑏𝑖 −𝑉 𝑁 𝐷 𝜖 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇 (𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1)

8. Total current, multiply 𝐽 by area 𝐴 𝑗 :
𝐼= 𝐼 𝑆 (𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1)
𝐼 𝑆 ≡𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 (𝐴)
Think about slope!

9. Key dependencies of drift-diffusion model:
 𝐼 ∝𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1
 𝐼 𝑆 ∝𝑒𝑥𝑝 𝑞 𝜑 𝐵𝑛 𝑘𝑇
 𝐼 𝑆 weakly dependent on 𝑉

10. Temperature dependence of 𝐼 𝑆 :
𝐼 𝑆 ∝ 𝑇 1/2 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇
Not seen in practice!
What one finds experimentally is:
𝐼 𝑆 ∝ 𝑇 2 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇
Made implicit assumption: 𝑞𝑢𝑎𝑠𝑖−𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 across SCR
⇒ drift/diffusion balance only slightly broken.
Quasi-equilibrium assumption is good if:
𝐽 ≪ 𝐽 𝑒 𝑑𝑟𝑖𝑓𝑡 , 𝐽 𝑒 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛

11. Thermionic-emission theory
𝐶𝑙𝑜𝑠𝑒𝑟 𝑡ℎ𝑎𝑛 𝑎 𝑚𝑒𝑎𝑛 𝑓𝑟𝑒𝑒 𝑝𝑎𝑡ℎ 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒, 𝑎𝑟𝑔𝑢𝑒𝑚𝑒𝑛𝑡𝑠
𝑜𝑓 𝑑𝑟𝑖𝑓𝑡 𝑎𝑛𝑑 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑑𝑜 𝑛𝑜𝑡 𝑤𝑜𝑟𝑘
In the last mean free path,
 electrons do not suffer any collisions,
 only those with enough 𝐸 𝐾 get over the barrier
 actually, only half of those with enough 𝐸 𝐾 do
 this is bottleneck: 𝑡ℎ𝑒𝑟𝑚𝑖𝑜𝑛𝑖𝑐 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑡ℎ𝑒𝑜𝑟𝑦

12. In steady state:
𝐽 𝑡 ≈ 𝐽 𝑒 =−𝑞𝑛(𝑥) 𝑣 𝑒 (𝑥)
Focus on bottleneck at 𝑥=0:
𝐽=−𝑞𝑛(0) 𝑣 𝑒 (0)

13. First compute 𝑛(0):
𝑛(0) is only a fraction of the carriers present at 𝑛( 𝑙 𝑐𝑒 ):
𝑛 0 = 𝑛( 𝑙 𝑐𝑒 ) 2 𝑒𝑥𝑝 𝑞[∅ 0 − ∅ 𝑙 𝑐𝑒 ] 𝑘𝑇
If rest of SCR is in 𝑞𝑢𝑎𝑠𝑖−𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚
𝑛 𝑙 𝑐𝑒 ≈ 𝑁 𝐷 𝑒𝑥𝑝 𝑞∅( 𝑙 𝑐𝑒 ) 𝑘𝑇
Also:
∅ 0 =−( ∅ 𝑏𝑖 −𝑉)
Then:
𝑛 0 = 𝑁 𝐷 2 𝑒𝑥𝑝 −𝑞( ∅ 𝑏𝑖 −𝑉) 𝑘𝑇 = 𝑁 𝐶 2 𝑒𝑥𝑝 −𝑞( 𝜑 𝐵𝑛 −𝑉) 𝑘𝑇

14. Compute 𝑣 𝑒 (0):
Over the last mean free path, carriers basically travel at 𝑣 𝑡ℎ .
But, velocity pointing at different angles. After taking care of statistics:
𝑣 𝑒 0 =− 𝑣 𝑡ℎ 2 =− 2𝑘𝑇 𝜋 𝑚 𝑛
(minus sign indicates that carriers are traveling against 𝑥)

15. Finally, compute electron current:
𝐽= 𝐴 ∗ 𝑇 2 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇 𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇
with:
𝐴 ∗ = 4𝜋𝑞 𝑚 𝑛 𝑘 2 ℎ 3
𝐴 ∗ ≡𝑅𝑖𝑐ℎ𝑎𝑟𝑑𝑠𝑜 𝑛 ′ 𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Still must subtract electron injection from metal to semiconductor in TE, so that when V  0, J  0:
𝐽= 𝐴 ∗ 𝑇 2 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇 (𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1)
Valid in forward and reverse bias.

16. 𝐽= 𝐴 ∗ 𝑇 2 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇 (𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1)
In terms of current (I),
𝐼= 𝐴 𝑗 𝐴 ∗ 𝑇 2 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇 𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1 = 𝐼 𝑆 (𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1)
𝐼 𝑆 = 𝐴 𝑗 𝐴 ∗ 𝑇 2 𝑒𝑥𝑝 −𝑞 𝜑 𝐵𝑛 𝑘𝑇
What is the meaning of 𝐼 𝑆 / 𝑇 2 ?

18. Key conclusions
Minority carriers play no role in I-V characteristics of MS junction.
Energy barrier preventing carrier flow from S to M modulated by V, barrier to carrier flow from M to S unchanged by V  rectifying behavior:
𝐼= 𝐼 𝑆 (𝑒𝑥𝑝 𝑞𝑉 𝑘𝑇 −1)
𝐷𝑟𝑖𝑓𝑡−𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑡ℎ𝑒𝑜𝑟𝑦 of current: small perturbation of balance of drift and diffusion inside SCR.
𝐷𝑟𝑖𝑓𝑡−𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑡ℎ𝑒𝑜𝑟𝑦 of current exhibits several dependences observed in devices, but fails temperature dependence.
𝑇ℎ𝑒𝑟𝑚𝑖𝑜𝑛𝑖𝑐 𝑒𝑚𝑚𝑖𝑠𝑖𝑜𝑛 𝑡ℎ𝑒𝑜𝑟𝑦 of current: bottleneck is flow of carriers over energy barrier at M-S interface. Transport at this bottleneck is of a 𝑏𝑎𝑙𝑙𝑖𝑠𝑡𝑖𝑐 𝑛𝑎𝑡𝑢𝑟𝑒.

19. Homework
Refer to the I-V data of a Schottky diode that are posted in ABEEK.
Plot I-V in linear-linear and log-linear scales.
Model measured I-V characteristics of the Schottky diode using either drift-diffusion theory or thermionic-emission theory.
Due by 3 PM of April-4.

20. Student Presentations
Read a paper below and present a summary.
“Low Ohmic Contact to Silicon with a Magnesium/Aluminum Layered Metallization”, Journal of Applied Physics 59, no. 5, pp , 1986, by Akiya Masahiro et al.
When April-4.
Any volunteer?

21. Homework: Model I-V data of the Schottky diode. Due by March-28.