Field distribution in a vacuum-nano diode

Jinpu Lin, and Peng ZhangDepartment of Nuclear Engineering and Radiological Science

University of Michigan, Ann Arbor

MIPSE Graduate Student Symposium

Ann Arbor, MI

October 7, 2015Work Supported by AFOSR Grant No.FA9550-14-0309

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Introduction• There is growing interest in miniaturization of anode-cathode (AK)

gap by using fine emission tips to realize a vacuum nano-diode, showing great potential for energy harvesting and conversion in solar cells

• A sharp anode tip sufficiently close to the flat graphene surface induces electron emissions from graphene surface

• Existing models usually assume that the emission tip is far away from the anode, whose effect on tip field enhancement is thus ignored.

• We study the effects of finite AK gap on the electric field distribution on and around an emission tip, using Schwarz-Christoffel transformation.

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[1]. F. Yesilkoy, et al, “A Mid-IR Antenna Integrated with a Geometrically Asymmetrical Metal-Insulator-Metal Rectifying Diode,” in Rectenna Solar Cells, G. Moddel and S. Grover, Eds. Springer New York, 2013, pp. 163–188.[2]. P. Zhang, and D. Hung, “An analytical model for ballistic diode based on asymmetric geometry”, J. Appl. Phys., 115, 204908 (2014).[3]. S. Santandrea, et al, “Field emission from single and few-layer graphene flakes,” Appl. Phys. Lett., 98, 163109 (2011).

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Schwarz-Christoffel Transformation

• Mapping from polygonal boundaries (z-plane) onto real axis of the w-plane.• We can then transform the points on emission tip into points on a straight

line.

Z w

[4]. F.B. Hilderbrand, Advanced Calculus for Applications (Printice-Hall, New Jersey, 1962)

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Schwarz-Christoffel Transformation for AK gap with a trapezoide tip

• a= half width of tip• h=depth of tip• d=distance of gap• = acute angle of tip

AB H

C D

𝐸1 𝐸2

F Gy

z

Z

ad

h α

W

uG’A’ B’ C’H’ D’ ’ ’ F’

v

- ∞- 1 𝑢3 𝑢3 ′ 𝑢4

𝜕𝑍𝜕𝑊

=𝑘𝑤−1 (𝑤−1 )− 𝛼𝜋 (𝑤−𝑢3 )

𝛼𝜋 (𝑤−𝑢3 ′   )

𝛼𝜋 (𝑤−𝑢4   )

− 𝛼𝜋

The transformation formula is given below:

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Schwarz-Christoffel Transformation for AK gap with a trapezoide tip

• Point B ()

• Point ()

• Point D (1, 0)

• Point ()

• Point ()

𝑧=𝑘∫𝑤 0

𝑤1𝑤

(𝑤−𝑢3   )𝛼𝜋

(𝑤−1 )𝛼𝜋

(𝑤−𝑢3 ′   )𝛼𝜋

(𝑤−𝑢4   )𝛼𝜋

𝑑𝑤+𝑧 0

For any given , we can calculate

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Example:

α=π2

𝑢3𝑢3 ′

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Example:

𝑢3

a/h=0.1

𝑢3 ′

E-field calculation

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As wAs w

Far away from the tip, uniform electric field is recovered.

Example: E vs. z/h

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𝑎h=0.1 ,𝛼=

𝜋2

| 𝐸𝐸0| | 𝐸𝐸0

| Along

• Electric field is enhanced near the tip when d/h is decreased.• Field enhancement is found at both cathode and anode.

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Current Emission and Field Enhancement Factor

• Consider Field Emission induced current:

[]

, ,

𝐸0=𝑉 𝑔

𝛼,

A1B1

𝛽= 𝑓𝑖𝑒𝑙𝑑 h𝑒𝑛 𝑎𝑛𝑐𝑒𝑚𝑒𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 ,

𝑊 ≡𝑤𝑜𝑟𝑘 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 [𝑒𝑣 ]

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Current Emission and Field Enhancement Factor

• Average current emitted from the tip is:

• Average current emitted from the flat surface is:

• Field Enhancement Factor is obtained from:

,

,

= local electric field at point s along the surface .

=DF, = local electric field at point s along the surface .

A1B1

Field enhancement factor vs. d/h

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𝛽1𝛽2

a/h=0.1,

• As the normalized gap space d/h decreases (<2), both • AS d/h>5, converges to constant values, corresponding the case of d/h

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Field enhancement factor vs. a/h

d/h=2,

𝛽1 𝛽2

• As a/h increases,

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Field enhancement factor vs. /h

a/h=0.1,

𝛽1𝛽2

• As increases,

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Field enhancement factor vs.

a/h=0.1

𝛽1𝛽2

• As increases, • is insensitive to .

[] []

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Summary & future work

• We calculated the local electric field inside the gap with a single trapezoid tip, by using the Schwarz-Christoffel Transformation.• We evaluated the

effective field enhancement factor for field emission, for both electrode with a tip and the flat electrode.

• Investigate tips of other shape, like semi-circle.• Investigate multiple tips

for electron emission.• Compare the results

with experiments.