Lab 8
Midterm Exam
In this exam you will perform a Fowler-Nordheim analysis of the analog current and voltage data you collected.
C. F. Eyring, S. S. Mackeown, and R. A. Millikan. Phys. Rev. 31, 900-909 (1928).
Field Currents from Points.pdf.
I. Introduction
A series of experiments were performed between 1918 and 1928 that would change the way in which physicists interpreted natural phenomena. In 1928 Eyring, Mackeown, and Millikan (a graduate student, later of oil-drop fame) placed a needle-like metal cathode called a tip in front of a flat metal anode in high vacuum, creating a vacuum diode. See Figure 1.
Figure 1. A Vacuum Diode
When the dc voltage, V, was increased to several thousand volts an electron current, I, was detected and was related to the voltage by an equation of the form:
where B and C are constants.
Surprisingly, the current increased as the radius of curvature of the tip apex, R, was decreased (at a fixed voltage). This observation could not be explained in terms of classical physics. According to classical physics an electron current could only flow across the vacuum gap if: (1) the cathode was heated to a very high temperature causing thermionic emission of electrons, or (2) the cathode was exposed to energetic photons leading to the emission of electrons by the Photoelectric Effect.
Later in 1928, Fowler and Nordheim explained the experimental observations using newly introduced principles of quantum mechanics. According to the Fowler-Nordheim theory, an electric field at the the tip apex would reduce the width of the energy barrier that confined electrons to the interior of the cathode. When the barrier width approached ≈10 Å electrons would tunnel with high probability through the barrier, emerging as free particles in the vacuum gap where they would accelerate to the anode producing the measured current. Electron tunneling is a phenomenon of quantum mechanics. It has no counterpart in classical physics.
According to the Fowler-Nordheim theory the electric field at the cathode surface, E, is proportional to the applied voltage, V, where:
KR is the field-voltage proportionality factor where, K~5, is a dimensionless constant that depends on the electrode geometry and, R, is the average radius of curvature of the tip apex (for an isolated sphere in space K=1). This lead to an expression for the current as a function of applied voltage that predicted the experimental results:
where the constants are defined by:
Here, A, is the emission area (in cm2); α is the dimensionless Nordheim image correction factor; µ is the Fermi energy of the cathode (in eV) and, φ, is the average work function of the cathode tip apex (in eV). A plot of ln (I /V2) versus (1 /V) is called the Fowler-Nordheim equation and has the form:
Fowler-Nordheim theory has been successfully applied to a variety of electron tunneling phenomena including tunneling in Metal-Oxide-Semiconductor structures. The success of the Fowler-Nordheim theory was one of the earliest confirmations of quantum mechanics.
II. Fowler-Nordheim Analysis
A Fowler-Nordheim analysis typically consists of three parts:
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1. Demonstrate Fowler-Nordheim tunneling (ln (I /V2) versus (1 /V) is linear).
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2. A calculation of the field-voltage proportionality factor, KR.
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3. A calculation of the electron emission area, A.
How to Calculate KR
KR (in centimeters) can be calculated from the slope of the Fowler-Nordheim equation using the following procedure described by Robert Gomer in 1961 (see References).
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1.Let α = 1 and KR = 1.
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2.Find a new value of KR from the slope of the Fowler-Nordheim equation where:
and φ = 4.55 eV (the experimental value for clean tungsten).
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3.Find the mean value of the field strength, Em, from the mean value of the voltage, V:
where Vm = Σ V / N and, N, is the number of voltage readings, V.
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4.Find a new value for the Nordheim image correction factor from the approximation:
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5.Repeat steps 2 through 4 until the absolute value:
How to Calculate the Tip Radius
The average radius of the tip apex, R, can be calculated from KR using K= 5. This will result in a radius accurate to about 20%. A more accurate radius can be obtained by direct measurement from an electron microscope image of the tip apex.
How to Calculate the Emission Area
The emission area, A (in square centimeters), is calculated from the slope and the intercept of the Fowler-Nordheim equation. Solving for the emission area gives:
where φ = 4.55 eV (the experimental value for clean tungsten) and µ = 10.46 eV (the theoretical value of the Fermi energy for tungsten). The value for the Nordheim image correction factor is found from the calculation of KR, above.
III. Tabulate the Experimental Data (10 points)
Use the experimental data you collected in Lab 8 to prepare a table with the appropriate headings. Note the units. See Figure 2.
Figure 2. Table Heading
Show the calibration parameters used to calculate the current from the op-amp output. Use data from Section VII to complete the last column.
IV. Plot the Experimental Data (10 points)
Plot the current, I, as a function of the corresponding voltage, V .
V. Demonstrate Fowler-Nordheim Tunneling (30 Points)
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(a) Plot ln (I /V2) versus (1/V) from your experimental data.
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(b) Perform a Least Squares analysis on the Fowler-Nordheim data.
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(c) Add the Least Squares Line to your plot.
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(d) Does your plot demonstrate Fowler-Nordheim tunneling? Explain.
Recall the calculation for the slope, m, and the intercept, b, of the Least Squares line:
m = [ N ∑ xi yi - ( ∑ xi ) ( ∑ yi ) ] / [ N ∑ xi2 - ( ∑ xi )2 ]
b = [ ( ∑ yi ) ( ∑ xi2 ) - ( ∑ xi yi ) ( ∑ xi ) ] / [ N ∑ xi2 - ( ∑ xi )2 ]
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(e) Show your calculations for the slope and the intercept of the Least Squares line.
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(f) Summarize your results in a table with the heading shown in Figure 3.
Figure 3. Table Heading
VI. Find KR (15 points)
Find the field-voltage proportionality factor, KR, using the iterative procedure outlined in Section II. Show your calculations for each step of the iteration cycle in a table with the heading shown in Figure 4. The iteration process should take about 10 cycles to complete.
Figure 4. Table Heading
VII. Find the Tip Radius (5 points)
Calculate the tip radius from the value of KR using the procedure outined in Section II.
VIII. Find the Electric Field (15 points)
Use the equation for the electric field, E, in Section 1 to calculate the field for each value of voltage that was tabulated in Section III.
IX. Find the Emission Area (15 points)
Find the emission area, A, using the equation given from Section II.
R. H. Fowler and L. Nordheim. Proc. R. Soc. (London) A119, 173-181 (1928).
Electron Emission in Intense Electric Fields.pdf
E. Stern, B. S. Gossling, and R. H. Fowler. Proc R. Soc. (London) A124 699-723 (1929).
Further Studies in the Emission of Electrons from Cold Metals.pdf
R. Gomer. Field Emission and Field Ionization (Harvard University Press, 1961) p. 47. Reprinted in 1993 (American Vacuum Society Classic Series, Number 1-56396-124-5).
Field Emission and Field Ionization p46.pdf.
Kleint. Prog. Surf. Sci. 42, 101-115 (1993).
On the Early History of Field Emission Including Attempts of Tunneling Spectroscopy.pdf
