Why can not calculate the ElectronDensity of Bulkconfiguration using ATK-SE?

[Anders Blom]

It's a nanoscale device, so yes the numbers are small.

[esp]

Thank you .. i will post the final script, could be helpful to others after all this discussion ... This is a 2 probe GNR TFET device, 16nm long, 1.29nm wide

Code

	diffDensity1 = nlread(fName, ElectronDifferenceDensity, object_id="diffDensity[vg-0.25000][vds0.50000]")[0]
	diffDensity2 = nlread(fName, ElectronDifferenceDensity, object_id="diffDensity[vg-0.20000][vds0.50000]")[0]
	diffDensity3 = nlread(fName, ElectronDifferenceDensity, object_id="diffDensity[vg0.00000][vds0.50000]")[0]

	s1 = diffDensity1[:, :, :].sum()
	s2 = diffDensity2[:, :, :].sum()
	s3 = diffDensity3[:, :, :].sum()
	print s1, s2, s3, s2-s1, s3-s2

	# Find the volume elements.
	dX1, dY1, dZ1 = diffDensity1.volumeElement()
	dX2, dY2, dZ2 = diffDensity2.volumeElement()
	dX3, dY3, dZ3 = diffDensity3.volumeElement()
	print "unit lengths", dX1,dY1,dZ1
	print "unit lengths", dX2,dY2,dZ2
	length_unit1 = dX1.unit()
	length_unit2 = dX2.unit()
	length_unit3 = dX3.unit()
	# Calculate the volume of the volume element.
	dV1 = numpy.dot(dX1, numpy.cross(dY1,dZ1)) * length_unit1**3
	dV2 = numpy.dot(dX2, numpy.cross(dY2,dZ2)) * length_unit2**3
	dV3 = numpy.dot(dX3, numpy.cross(dY3,dZ3)) * length_unit3**3
	print "volumes", dV1, dV2, dV3
	numE1 = s1*dV1
	numE2 = s2*dV2
	numE3 = s3*dV3
	print "Total # electrons: ", numE1, numE2, numE3

	# let's use s2-s1 as difference in # electrons in channel due to voltage
	numElecDiff1 = (numE2-numE1)
	numElecDiff2 = (numE3-numE2)
	print "numElecDiff", numElecDiff1, numElecDiff2

	# convert to coulombs
	chargeDiff1 = numElecDiff1*1.6e-19
	chargeDiff2 = numElecDiff2*1.6e-19
	voltDiff1 = 0.05
	voltDiff2 = 0.25
	capacitance1 = chargeDiff1/voltDiff1
	capacitance2 = chargeDiff2/voltDiff2
	print "capacitance", (capacitance1), (capacitance2)

	print "capacitance in fF", (capacitance1)*1e15, (capacitance2)*1e15

my results for reference:

Code

2.49328980937 1/Bohr**3 2.62952644165 1/Bohr**3 3.18669569532 1/Bohr**3 0.136236632277 1/Bohr**3 0.557169253673 1/Bohr**3
unit lengths [ 0.69774531  0.          0.        ] Bohr [ 0.          0.69769726  0.        ] Bohr [ 0.          0.          0.70126857] Bohr
unit lengths [ 0.69774531  0.          0.        ] Bohr [ 0.          0.69769726  0.        ] Bohr [ 0.          0.          0.70126857] Bohr
volumes 0.341388054851 Bohr**3 0.341388054851 Bohr**3 0.341388054851 Bohr**3
Total # electrons:  0.851179358199  0.897688917092  1.08789984483 
numElecDiff 0.0465095588926  0.190210927734 
capacitance 1.48830588456e-19  1.2173499375e-19 
capacitance in fF 0.000148830588456  0.00012173499375 

[esp]

since my device channel is 16nm x 1.29nm, i can say this capacitance is in units of:  fF / (16*1.29 sq nm) correct? 

Then I can convert to um^2 by multiplying by 16*1.29, and then by 1000*1000 ... this gives:
capacitance in fF per um sq 3071.86334574  2512.61027099

Now if all that is right, how can I convert to ITRS units of fF per um device width? .... multiply by the length in um? That gives:
capacitance in fF per um width 49.1498135318  40.2017643359

Does that make sense?

[Anders Blom]

The last point seems to make sense.

I was looking at this: http://arxiv.org/abs/0705.3068 - it gives numbers in pF/cm.

So if you have C of the order 1e-5 fF over a "gate length" of 1 nm it is 1 pF/cm.

[esp]

addition to script:

Code

	# convert to coulombs
	chargeDiff1 = numElecDiff1*1.6e-19
	chargeDiff2 = numElecDiff2*1.6e-19
	voltDiff1 = 0.05
	voltDiff2 = 0.25
	capacitance1 = chargeDiff1/voltDiff1
	capacitance2 = chargeDiff2/voltDiff2
	print "capacitance", (capacitance1), (capacitance2)

	capacitance1 /= 1e-15
	capacitance2 /= 1e-15
	print "capacitance in fF", (capacitance1), (capacitance2)

	# (a in 1/nm**2)*(b in nm**2)=cap
	# (a in 1/nm**2) = cap/(b in nm**2)
	# above is fF over area of device, which is 16nm x 1.29nm = 20.64nm**2
	deviceArea = 20.64 # in nm^2
	capacitance1 /= deviceArea
	capacitance2 /= deviceArea
	print "capacitance in fF per nm^2", (capacitance1), (capacitance2)
	capacitance1 = capacitance1*1e6
	capacitance2 = capacitance2*1e6
	print "capacitance in fF per um^2", (capacitance1), (capacitance2)

	# multiply by device length in um
	deviceLength = 0.016 # in um
	capacitance1 = capacitance1*deviceLength
	capacitance2 = capacitance2*deviceLength
	print "capacitance in fF per um width", (capacitance1), (capacitance2)

Results:

Code

numElecDiff 0.0465095588926  0.190210927734 
capacitance 1.48830588456e-19  1.2173499375e-19 
capacitance in fF 0.000148830588456  0.00012173499375 
capacitance in fF per nm^2 7.21078432443e-06  5.89801326307e-06 
capacitance in fF per um^2 7.21078432443  5.89801326307 
capacitance in fF per um width 0.115372549191  0.094368212209 

[esp]

one thing is bothering me here still ...

you said >> "a decent fraction of an electron (0.05-0.2) per 0.1 V or so (I assume) sounds about right"

So thinking of the definition of an electronvolt, yes that seems to make sense ... now, my numbers say that with .25V voltage difference on the gate, there are 0.19 more electrons in the channel ... now I know from transmission calculations that the whole range of gate voltage for this device, 0.5V, gives a current increase of current of about 100000x, or 10^5 ... so then it seems to me that the extra electrons in channel are really not helping the current much at all ... is that right?  i suppose since it is a tfet, and tunneling is happening only at one side (source side for n tfet) .. and it does not depend on their being any mosfet-like channel bridging source-drain for operation, then it all makes sense? 

I suppose as in the attached figure from "Tunneling Field-Effect Transistor: Capacitance Components and Modeling", my numbers are in line (my 0.1 fF/um versus theirs about 1.5 fF/um for a not so dissimilar device) with published results .. but then if that is the case .. then i think their picture of the inversion layer near the drain is misleading. .. if there are only a few extra electrons .. i would draw it extremeley thin .. do I have the right understanding here?

[esp]

actually this figure is more appropriate .. they are using Si while i am using Graphene, but this figure shows a 25nm length device, mine is 16nm, so almost similar, and there they show Cgd (which makes up most of Cg), as between 0.2 and 1 fF/um .. so very similar numbers ..

[Anders Blom]

i think their picture of the inversion layer near the drain is misleading

Their picture of the charge distribution is most likely based on a purely classical simulation (or actually not even a simulation, it's just a conceptual view), so there is no account for quantum effects. I think what we will increasingly see in the near future, as people start to study true nano-scale devices from an atomistic perspective, is that things are not so simple as the continuous models predict when you get down to these "extreme" length scales. Well established concepts will break down and a new understanding of how electrons behave in very confined dimensions will be necessary. Well, how the electrons themselves behave is rather well understood, but now we are adding electrostatic control, via the gates, and it's another level of complexity immediately.

So a lot of new simulations need to be done even for simple toy systems just to establish the new paradigm. I'm doing some calculations right now on a very, very thin Si nanowire set up as a p-i-n junction with a cylindrical wrap gate. The voltage drop is not symmetric!