The Quantum Limit of Etching

NANOSYSTEMS PRINCIPLES

Kristian Dolghier

Professor Mangilal Agarwal

Submitted 8 December 2022

Abstract

An electron has a measurable percent chance to tunnel through any gap no matter the size. As the distance between gates in modern CPUs continues to decrease, we see the percent chance of quantum tunneling increase. This transmittance percentage will be simulated with a small gate to gate gap to find the theoretical limit before a significant amount of quantum tunneling takes place. The gate to gate gap length as well as the depth of the gates will be iteratively simulated for a variety of sizes.

Introduction

As the scale of transistors within computation devices continues to become smaller there is a larger possibility of quantum interaction [5]. These quantum interactions include quantum tunneling that will flip certain bits and cause a degradation of the computation abilities of all processors. [6] This is because in standard computation, one cannot tell what the but inputs were from the output. This is in contrast to a quantum computer where all logic gates are reversible so there is no error if a qubit is flipped [7].

To explain how quantum tunneling affects standard computation, one must understand how computation occurs. This topic, as well as the nature of Quantum Computation, is further discussed in my paper coined 'Quantum Computation' [1]. Standard computation relies upon logic gates which is a logical operation based on the input of certain bits. A logic gate-in modern systems-is based upon a n and p channel which underlies CMOS logic. There are several types of logic gates such as 'NOT' and 'AND' gates that either change the state of a bit (can only be a 0 or a 1) or keep the bit the same number [8].

To more easily understand this concept, one can look at Figure 1 which displays various inputs and what their outputs will be.

Figure 1: A truncated graph from [1] that demonstrates given a certain bit input from A and B and what the output will be given various operators

As one can see in Figure 1, if you are given an input from A that is a 1 and an input from B that is a 1, then given a NOR operator (which depends upon the input of two bits) you will get a bit output of 0. In fact, the only time a NOR operator will give a bit output of 1 is when both A and B bits are 0.

Now one must understand the nature of how these bits and operators work in the real world. To succinctly state it, using a combination of lithography and various methods of etching, one can produce patterns of gold and other substances [9]. These patterns are on a nanoscale and with the combination of different types of wells and metals, one will eventually form a transistor. The easiest way to symbolize it is the CPU will have a series of wells and bridges that act as transistors. Trillions of these transistors, via certain algorithms, form computation devices [10]. These devices are abbreviated as a CPU and take various inputs the user sends it, operators on it, and then gives an output.

As we develop newer and newer forms of lithography and etching, we can make these transistors smaller and smaller. We transferred from a gate size of hundreds of nanometers in the 2000s to five nanometers in the 2020s [11]. As the scale of these transistors decrease with new technology, there is a higher chance of quantum interaction between gates. For brevity, this paper will assume we are only factoring in a two gate system (also called a well) that perfectly represents the smallest dimension within a CPU.

Using a program in Matlab we can visualize each well and plot the wavefunction of a single electron passing through the barrier between the two gates. We will compare the percent chance of quantum tunneling using various dimensions before quantum tunneling will cause too much decoherence in the computations. From this, we can gather the smallest critical dimension before quantum tunneling needs to be seriously accounted for.

Methodology

The code that is to be used is based upon a prior experiment that involved finding the wave function as well as the various energy states of a single electron passing through a Dirac Comb[14]. In addition to this, one of the wells was doped (the depth was increased) and the same calculations were performed. This program was rewritten in Matlab by David Buchanan for a prior project and then additionally modified for this project. [2]

The nature of the code and calculations within are as follows. Initially, you state the total length of the plane which was stated at 600 atomic units. This is a 2 dimensional plane with a X and a Y axis. To visualize it, one could say that the electron masses from the left to the right at 0 on the Y axis over the length defined. Within this length defined, a dip is stated as a well with perfectly straight sides. The sides of the well are in the negative y direction so when the electron passes through the dip, it must quantum tunnel to surpass the potential between the two wells; there is a small percentage that passes through. The percentage that will pass through can be found by measuring the distance between the center of the two wavefunctions and comparing that to the max height. See Figure 2 for a visualization.

Figure 2: A depiction of the wells we simulate and the wavefunction generated

To actually code this, we must define the mass to be 1 for an electron, as well as hbar (a constant)[13]. Following this we state the parameters of the wells including the distance between wells, the width of each well, and the potential of each well. Finally, we must state that elsewhere, the potential is 0. Then a matrix labeled 'M' is created that represents the finite differences method. 'M' is then dialogized to normalize the results. We then store the normalized square of the eigenvectors of matrix 'M' to eventually find the normalized wave vectors for each state. Following this we plot the wave vectors that we found to show the wave vector.

A few things to note regarding the nature of the values selected. An electron is chosen because a proton or other heavier particle will have significantly less quantum tunneling, and an electron can flip a bit [12]. Additionally, one must define the whole plane to be significantly larger than the distance and width of the wells. If the plane is too small, then the wave function will not approach 0 fast enough, and the value for the percent of quantum tunneling will be significantly off. Ideally the plane is several times larger than the relevant distance of the wells. See the Supplementary Info section for the full code.

Finally, we modify the code to iteratively change either the depth of the wells, the width of the barrier, or the width of the wells. We do this by defining a matrix that starts at a value then either increases or decreases by a small decimal point to a desired value. The other two parameters remain constant. Since a for loop must run over whole numbers, we use the 'numel' function to count the number of values in the matrix previously defined. Then we create a new variable that is from 1 to the number of values defined. Now, all of our for loops will run over the full dimension defined.

As mentioned, we use the finite differences method to obtain the wave functions for each iterative dimension simulated. From here we use the 'findpeaks' function to find the local minima and maxima to obtain the quantum tunneling percentage. Eventually we will either graph the wells as well as the wavefunction of the percent transmittance for either parameter simulated.

Lastly, this is all simulated in atomic units of length which is defined as the bohr radius[15]. 1 au is approximately 5.291*10^-11 meters of length or around .053 nanometers. This was done as it is easier to know how close we are to approaching the size of an atom.

Results

For the calculations, two different simulations were performed. The first one that was performed simulated the various heights of the barrier between two separate wells. WIth each well having a width of 10 atomic units and the distance between the wells remaining a constant 10 atomic units. The height of the barrier was changed from a height of '0'-which made the barrier a height of one atomic unit-to a height of -1 which made the height of the barrier even with the depth of the wells. This was performed with a step size of .01 which approximately visualized in Figure 3.

Figure 3: This is the first 3 iterations from a height of 1 to a height of 0.7 and their corresponding wave functions.

The antecedent graph was formed as a result of this experiment. The percent of the electrons that quantum tunneled was calculated by comparing the local minimum to the local maximum of the wavefunction. The local minimum is a representation of the electron that quantum tunneled. The full scale of each iteration is shown in Figure 4 that shows each wave function and depth is amalgamated into a single graph

Figure 4: All barrier height iterations from a height of 1 to a height of 0 and their corresponding wave functions.

As the height of the barrier decreased, we saw an increase of the magnitude of the local minima with respect to the local maxima. As a matter of fact, as we decreased the barrier height to a point, we stopped getting local minima and just got a maxima. The percent transmittance of these electrons were found by comparing the height of the wavefunction when the barrier height is at zero.

From this, we found that, as expected, when the barrier height is decreased, we get a significantly higher percentage of electrons quantum tunneling or just having enough energy (potential) to pass above the barrier. This is shown in Figure 5. It should be 100% transmittance at <-0.91 atomic units but due to the barrier being almost 0, we get 100% transmittance and actually little to no tunneling.

Figure 5: Here is the percent transmittance as the height of barrier was reduced from a height of 0 to -1

A similar experiment was performed that varied the width of the barrier. Due to constraints of the matlab code, the smallest width that could be achieved was a width of one atomic unit. Since, we are attempting to find a practical smallest dimension of etching, it was judged that one atomic unit is small enough to where beyond that is too small and impractical.

Thus, we experiment by changing the distance from each well (or the width of the barrier) from one to 30 atomic units. The first three iterations are shown in Figure 6.

Figure 6: A representation of the first 3 barrier widths from 1 to 3 au.

The amalgamation of all 30 widths are shown in Figure 7. As we can see, the increase in barrier width causes the second well moving to the right more than the left well moves to the right due to how the code is written.

Figure 7: A representation of the full simulation of 30 various barrier widths.

From these simulations, we again found the wavefunction in between the two wells and found the percent transmittance from this as shown in Figure 8.

Figure 8: A graph of the percent transmittance when the barrier width is increased from 1 atomic unit to 30.

It was found that beyond 30 atomic units, the height of the wavefunction at the barrier was 0 and thus, irrelevant. We found some percent transmittance until a width of 18 atomic units-past where there was no quantum tunneling found.

A similar experiment was therefore run to find the optimal depth of the two well system. As demonstrated by Figure 9.

Figure 9: A representation of the three different depths of wells and their ensuing wavefunctions. The difference between the local max and minimum representing the percent transmittance. This was subsequently rerun with an iterative depth until 10 atomic units

This was then extended to a depth of ten atomic units which produced Figure 10.

Figure 10: The full simulation with an iterative depth until 5 atomic units

Furthermore, the percent transmittance was found in Figure 11 dependent on the depth of the two wells.

Figure 11: The full simulation with an iterative depth until 10 atomic units

Now we have some preliminary data on the rate of quantum tunneling when we vary the width between two different wells, the depth of the two wells, as well as the height of the barrier between the two wells. From these we can simulate a realistically sized gate within a CPU. In order to simulate a proper gate, we must know the approximate proportion of the height of each well, the distance between wells and the width of each well. Using a TEM image of Intel's 14 nm process node in Figure 12, we can find the approximate proportion of width to height to distance.

Figure 12: TEM image from Intel's 14nm transistor [3].

From the image provided, it is found that the width of each 'well' is approximately 10 nm with a peak to peak width of 30 nm and a height of 50 nm. This correlated to a proportion of 1:3:5. Combined with the prior simulations already performed, we found that on the highest end, we found a non-negligible amount of quantum tunneling at approximately 18 atomic units on the variable width of barrier simulation. Using that as our benchmark, we can state that the simulation should have the proportions of 6:18:30 atomic units. Using these values, we get Figure 13 that shows the wave functions of such a simulation.

Figure 13: Two well simulations with a proportion of 6:18:30 atomic units

As shown, there is a higher probability of an electron remaining within the second well. This was already seen in prior simulations within [1] and [2]. When the two wells are close enough, we see the wavefunction being the same above each well, but as the distance increases between wells, we will see the electrons favoring one of the wells. This is mainly because at a barrier width of 18 atomic units, we already almost stopped seeing any quantum tunneling, therefore, we will only see tunneling when the depth of each well is very shallow. Following this train of logic, we get Figure 14 that echoes this sentiment.

Figure 14: Here is the graph of the percent transmittance. Note that at the depth of 30, matlab gives a percent transmittance of 0, but this is due to rounding, as the tunneling is nonzero.

It is time consuming to simulate every possible simulation but we can easily simulate some proportions as shown in Figure 15.

Figure 15: The percent transmittance at each multiple of the base ratio of 1:3:5. Note that due to Matlab requiring whole numbers for some of the code, it was needed to round up or down.

Discussion

To begin with, let's discuss the limitations of the code that was written within Matlab. To begin with, there is a major concern of over what distance we should run this two well simulation for. This length was designated as 600 au and should be several orders of magnitude larger than the distance at which the wavefunction goes to zero. If the length is defined too shortly, one can observe the whole wave function not being to propagate fully and cause the percent transmittance to be artificially high. This length was written in the original Fortran code as 600 au, but it was not analyzed to see if it is long enough.

Another possible concern is the number of points over which this simulation takes place. As written in the code, this number is 2048, which is relatively small when the total length of the simulation is 600 au. This is likely the reason that the sides of each well do not instantly go downwards, but instead quickly slope downwards. However, it was deemed that this sharp slope was significantly more realistic that a straight up and down line and thus was kept. But the wave function also depends upon this number and as such, their resolution could certainly be improved upon.

With regards to how each well dimension is defined there are no concerns. This is due to the fact that I can iteratively simulate every dimension to whatever small degree that I so desire. The only dimension that this does not apply to is the width of the barrier which can only be changed iteratively with whole numbers. This would not change the percent transmittance, but could improve the precision as the atomic units approach 0.

The simulations of each of the three parameters is robust and has been compared to each other to make sure changing the code did not change the transmittance. The final simulations with each proportion given in atomic units is very intriguing however. Given that 19 au is approximately 1 nm, we do see that at the last ratio of 6:18:30, it is almost shocking that the percent transmittance approaches 0. Looking at the smallest dimension of 6 au, this is 15x smaller than the current smallest production length of 5nm (94.48 au)[4]. From this, we can understand that the current smallest dimension of commercial CPU's has a significant way to go until quantum tunneling becomes a major concern.

Conclusion

This paper set out to find the percent transmittance through CPUs as the distance of gates continues to decrease. We successfully simulated the percent transmittance of a variety of different configurations. From these simulations, we got an approximate understanding of at what dimensional limitation one could state that there is an insignificant amount of quantum interference.

Following this, we used an existing 14 nm CPU and found the proportion of the gate width, the distance between gates and the height of the gate. With these proportions and having prior simulations, we found that the quantum interference at a ratio of 6:18:30 atomic units approaches 0 and could not be measured using our simulation. Additionally, various other ratios were simulated and a measurable amount of quantum tunneling was observed. Considering that there are trillions of transistors in a CPU and trillions of calculations performed, there is a serious concern that any measurable amount of quantum tunneling would cause crashes and instability. However, the nature of error correction within algorithms may be enough to account for some of these flipped bits [1].

One can state that at a ratio of 6:18:30 atomic units, there is an insignificant amount of quantum tunneling and with the current proportions of gates remaining the same, this will be the critical dimension that we should strive for in the future. From a quantum tunneling perspective, this is the critical dimension that more engineering will be needed to be involved to surpass this limit.

Acknowledgments

I would like to thank Dr. Maxim Sukharev who wrote the base code in Fortran 90 that I used in my effects of doping on finite wells experiment.

I would like to thank David Buchanan who ported and heavily modified the code from Fortran 90 to matlab.

I would like to thank Dr. Mangilal Agarwal who gave me the push and credit for a project that I had long since wanted to run. Also, I appreciate the additional days to turn this in.

References

[1] K. Dolghier, "Quantum Computation," Kdolghier, 04-May-2021. [Online]. Available: https://kdolghier.webnode.page/quantum-computation/. [Accessed: 08-Sep-2022].

[2] David Buchanan, Kristian Dolghier, and Richard Gerst "Doping Simulations," Kdolghier, 16-Dec-2021. [Online]. Available: https://kdolghier.webnode.page/doping-simulations/. [Accessed: 11-Dec-2022].

[3] TechInsights. TechInsights - The Much Anticipated Intel 14 Nm Is Finally Here! 30 June 2018, www.prnewswire.com/news-releases/techinsights---the-much-anticipated-intel-14-nm-is-finally-here-281598701.html.

[4] Iriarte, Mariana. "Marvell and TSMC Collaborate to Deliver Data Infrastructure Portfolio on 5nm Technology." HPCwire, www.hpcwire.com/off-the-wire/marvell-and-tsmc-collaborate-to-deliver-data-infrastructure-portfolio-on-5nm-technology/.

[5] Sperling, Ed. "Quantum Effects at 7/5nm and Beyond." Semiconductor Engineering, 13 Feb. 2019, semiengineering.com/quantum-effects-at-7-5nm.

[6] Beau, M.; Kiukas, J.; Egusquiza, I. L.; del Campo, A. (2017). "Nonexponential quantum decay under environmental decoherence". Phys. Rev. Lett. 119 (13): 130401. arXiv:1706.06943. Bibcode:2017PhRvL.119m0401B. doi:10.1103/PhysRevLett.119.130401. PMID 29341721. S2CID 206299205.

[7] C. H. Bennett, "Logical reversibility of computation", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525-532, 1973

[8] Jaeger, Microelectronic Circuit Design, McGraw-Hill 1997, ISBN 0-07-032482-4, pp. 226-233

[9] *, Name. "CMOS Fabrication Using N-Well and P-Well Technology." ElProCus, 18 Apr. 2019, www.elprocus.com/the-fabrication-process-of-cmos-transistor/.

[10] Townsend, Whitney J.; Swartzlander, Jr., Earl E.; Abraham, Jacob A. (December 2003) [2003-08-06]. "A Comparison of Dadda and Wallace Multiplier Delays" (PDF). SPIE Advanced Signal Processing Algorithms, Architectures, and Implementations XIII. The International Society. doi:10.1117/12.507012.

[11] International Roadmap for Devices and Systems: 2021 Update: More Moore, IEEE, 2021, p. 7, archived from the original on 7 August 2022

[12] Feroz, Farhan, and Dr. A. B. M. Alim Al Islam. "Scaling up Bit-Flip Quantum Error Correction." 7th International Conference on Networking, Systems and Security, ACM, Dec. 2020, https://doi.org/10.1145/3428363.3428372.

[13] Sornborger, Andrew T. "Quantum Simulation of Tunneling in Small Systems." ScientificReports, vol. 2, no. 1, Springer Science and Business Media LLC, Aug. 2012, https://doi.org/10.1038/srep00597

[14] K. Dolghier "Effects of Doping on Finite Wells ::" kdolghier.webnode.page/effects-of-doping-on-finite-wells.

[15] CODATA Value: Atomic Unit of Length. physics.nist.gov/cgi-bin/cuu/Value?Abohrrada0.

Supplementary info

Matlab Code for variable depth of well sim:

%program adapts code written by Kris Dolghier to find eigenvectors and eigenvalues of an electron in a

%1-d comb of doped potential wells; original code was in Fortran 90 and used discrete value representation

%method. Adapted code uses finite differences method. Original version used positively and

%negatively numbered wells, which did not affect%output, while this version uses only positively

%numbered wells. This%version adds an array to denote which wells are doped

%Created by David Buchanan November 2021

%remodified by Kristian Dolghier October 2022 for quantum interaction at

%nanoscale approximation

close all;clc;clear;

L=600; %length

num = 2048; %number of points; fortran code has 2049

dx = L/(num-1); %distance between points

%express physical constants in atomic units

hbar = 1;

m = 1;

a = -hbar*hbar/(2*m*dx*dx);

%define parameters of wells

distance=10; %distance between wells (only works in certain numbners divisible by 2)

width=10 ; %width of well; fortran code has as float type

V0= -1.0; %undoped well

V1= -1.0; %doped well

allV2=0.0:-0.01:-1.0;

numV2=numel(allV2);

minvalue=zeros(1,numV2);

maxvalue=zeros(1,numV2);

for kk=1:numV2; %height of barrier btwn 2 wells

V = 0; %placeholder for now, will be used in matrix to set doping

V2=allV2(kk);

%create matrix to represent potential

U= zeros(1,num);

number_of_wells = 2;

offset=floor(num/2-number_of_wells*width/2-(number_of_wells-1)*distance/2);

doped_wells =zeros(1,number_of_wells);%matrix defines which wells are doped

%wells can be selectively doped by setting corresponding matrix element

%to any non-zero value

doped_wells(1) = 0;

for well = 1:number_of_wells

if doped_wells(well) == 0

V = V0;

else V = V1;

end

for i =0:width-1

U(offset + distance*(well-1/2) + (well-1)*width +i) = V;

%U(offset - distance*(well-1/2) - (well-1)*width -i) = V0;

%fortan code has negatively numbered wells

end

end

for i =0:width-1

U(offset + distance*(well-3/2) + (well-1)*width +i) = V2; %defining the height of barrier

end

%create matrix M to represent finite differences method

M = zeros(num,num);

M(1,1) = -2*a + U(1);

M(1,2) = a;

for i=2:num-1

M(i,i-1) = a;

M(i,i) = -2*a + U(i);

M(i,i+1) = a;

end

M(num,num-1)= a;

M(num,num) = -2*a + U(num);

%diagonalize M and normalize results

[evectors,evalues] = eig(M);

N=zeros(1,num); %store normalized square of eigenvectors of matrix M

start = 1; %this and next line allow choice of range of eigenvectors to

%include in matrix of normalized eigenvectors

max0 = 1;

for j = start:max0

for k=1:num

N(k,j) = evectors(k,j)*conj(evectors(k,j))/dx;

%normalized square of wavevector for l-th state

end

end

%plot(evalues) %used to examine eigenvalues, not pertinent to paper

Uscaled = zeros(1,num);

scale = 1;

for i = 1:num

Uscaled(i) = U(i)/scale;

end

plot(Uscaled);

N=N(:,1);

% hold on

% plot(N)

% title("2 well")

% axis ([1000 1060 -1.5 1.5])

N=N;

minimum=findpeaks(-N);

min2=-min(minimum);

min3(kk)=min2;

maximum=findpeaks(N);

max2=max(maximum);

max3(kk)=max2;

minmax=[min3;max3]';

end

minmax2=[minmax,allV2'];

percent=abs(minmax2(:,1)-minmax2(:,2));

percent1=100-100*(percent./minmax2(:,2));

percent2=[percent1(:,1),allV2']

plot(percent2(:,2),percent2(:,1))

set(0,'defaultfigurecolor',[1 1 1])

title("2 Well Sim")

xlabel('Height of Barrier (au)')

ylabel('Percent Transmittance (%)')