(V(x) - E)1/2 dx )This question involves analyzing experimental measurements of electron tunneling through a capacitor (actually a MOS capacitor).
First, how good an insulator is SiO2? Well, we know that an electron requires an energy boostof about 3.2eV to enter the conduction band of SiO2 from theconduction band of silicon. This means that we can draw a band diagram for a Si--SiO2--Si structure. The barrier is fixed at about 3.2eV, so we'll get a leakage current through the oxide of Ileak = I0 exp(-3.2eV / kT), where I0 can be found approximately as follows:
We assume that at a barrier height of zero, we are allowed to dump all of the electrons in the vicinity of the barrier at full speed. So we need to know the size of the device, the concentration of the electrons, and the velocity of the electrons. Thus, I0 = A N v, where A is the area of our capacitor plate, N is the concentration of free electrons in the capacitor plate, and v is the average thermal velocity of the electrons in the capacitor plate.
Compute the room--temperature leakage current from silicon through silicon-dioxide, where
| A = 2 x 104 mm2 | 100 x 100 mm polysilicon plate leaking from both sides |
| N = 1019 electrons per cm3 | heavily doped n-type silicon |
| v = 106 cm per second | thermal velocity at 300K |
| kT = 0.026eV | thermal voltage at 300K |
The above estimates for values are very approximate, but it won't mattermuch. Be careful to use commensurate units everywhere. Don't hurt your calculator.
Quantum mechanics tells us that an electron's wave--function extends spatially somewhat beyond a barrier. If the barrier is thin enough, then, there is a reasonable probability that the electron will cross to the far side of the barrier. This process is called tunneling. A simple version of the theory for a square barrier of height DE and thickness d gives the probability of an electron reaching the far side of a barrier as
P = exp(-2d ( 2me DE ) 1/2 / (h/2p) )
For DE = 3.2eV, compute the distance required to give a probability of 0.5. The electron mass me is 5.7 x 10-12 eV s2/m2 and h/2p is 6.6 x 10-16 eV-s.
The gate oxide thickness in our fabrication process (2mm process) is about 43nm. This is much too far to have a significant tunneling probability, so how can we use tunneling at all? We have already seen that electrons can travel through SiO2; we need not tunnel all the way to the other side of the oxide layer if we can somehow arrange for the electrons to tunnel into the oxide conduction band. This process is called Fowler-Nordheim tunneling, and it happens when we apply a large enough electric field across the oxide. Draw a band diagram of this process. From the above band diagram, we see the electrons must tunnel through a triangular barrier. In Quantum Physics, the WKB approximation method is the tool to solve these kinds of problems. By the WKB approximation, the probability of an electron tunneling through the potential barrier defined by V(x) is
P = exp(
(-2 (2me)1/2 / (h/2p)
)
(V(x) - E)1/2 dx )
where E is the starting electron energy, a is the starting position of the barrier, and b is the final position of the barrier at the electron energy. Assuming that the electrons start at the conduction band, derive the tunneling probability using this approximation. In addition, plug in the above parameters for the tunneling probability.
The resulting electron current is a direct measure of the probabilty of the electron tunneling through this oxide. For the following measurements of electron tunneling (given below), curve fit this data to the analytical expression you derived above. The form of the equation should beI = exp( - Vo / V )
Experimental Measurements ( [ Applied_Voltage Measured_Current ] )