Upload
dongwook-go
View
227
Download
3
Tags:
Embed Size (px)
Citation preview
- Schottky Barrier and the Tunneling Current Calculation
Dongwook Go
SEM (Scanning Electron Microscope)
Source : http://lamp.tu-graz.ac.at/~hadley/sem/index/index.php
4-point Measurement EBIC (Electron Bean Induced Current)
Source : http://lamp.tu-graz.ac.at/~hadley/sem/index/index.php
Source :
Alexander Schnabel, Stephan Stonica,
Experimental laboratory exercise report, TU Graz
Theory and Modeling
Numerical Calculation Results and Discussion
- Square Barrier
- Spherical Schottky Barrier
Conclusion
Source : http://web.tiscali.it/decartes/phd html/node3.html
Schottky Contact
Poisson Equation
• Potential Energy Barrier
Schrodinger Equation
• Transmission Coefficient
Fermi-Dirac Function
• Current Density
Depletion approximation𝜌 = 𝑒𝑁𝑑
Multiply 𝑟2 and integrate
Divide by 𝑟2 and integrate
At the end of the depletion layer (𝑟𝑑), E=0
The ground of the potential is set to be zero at the end of the depletion layer, 𝑟𝑑
𝑟𝑑 can be calculated from the condition
𝑉𝑠 : intrinsic potential drop at the contact 𝑉𝑏𝑖 : bias voltage
Calculation of 𝑟𝑑 for various bias voltages. 𝑉𝑠 is set to be -1 V.
Potential barrier shapes for various bias voltages. 𝑉𝑠 is set to be -1 V.
To be solved :
,where
The solution for the Schrodinger equation at the constant potential = plane wave
Plug in the initial condition at x=a (𝐴𝑇 is set to be 1)
and integrate the Schrodinger equation numerically.
So we get , and
From the form of the solution,
Get 𝐴𝐼, and 𝐴𝑇
And the transmission probability is
Normalized solution on the left/right side
Periodic boundary condition
with
The current density for this quantum state is
The total current density that flows from left to right is
: the probability that the quantum state on the left side is occupied
: the probability that the quantum state on the right side is empty
If we change the expression into the integral form
Change the integration variable into the energy,
Similarly, the current density for the left moving electron is
The total current density is the difference of the two
𝑈0= 2 eV
𝜉𝑟= 2 eV
𝜉𝑙= 2 eV
𝑇=300 K
𝑈0= 2 eV
𝜉𝑟= 2 eV
𝜉𝑙= 2 eV
𝑎= 1 nm
𝜉𝑟= 2 eV
𝜉𝑙= 2 eV
𝑎= 1 nm
𝑇= 300K
𝑈0= 2 eV
𝜉𝑙= 2 eV
𝑎= 1 nm
𝑇=300 K
𝑈0= 6 eV
𝜉𝑙= 2 eV
𝜉𝑟= 2 eV
𝑇=300 K
𝑈0= 6 eV
𝜉𝑙= 2 eV
𝜉𝑟= 2 eV
𝑎=1 nm
Source : http://web.tiscali.it/decartes/phd html/node3.html
𝜉𝑙= 2 eV
𝜉𝑟= 2 eV
𝑎=1 nm
𝑇=300 K
𝜉𝑙= 2 eV
𝜉𝑟= 2 eV
𝑎=1 nm
𝑇=300 K
𝑈0= 6 eV
𝜉𝑙= 2 eV
𝑎= 1 nm
𝑇=300 K
*Thermoionic Emission Dominates
Conventional Schottky diode behavior
Small 𝑈0Large 𝜉Large 𝑎High 𝑇
*Tunneling Dominates
Reverse rectifying behavior
Large 𝑈0Small 𝜉Small 𝑎Low 𝑇