View Articles of the Electrical Properties of Cold Cathodes Mentioned Above Have Been Published Recently [1,2,3,4,9,10,11,12]

UNIVERSITY OF CINCINNATI

______, 20 _____

I,______, hereby submit this as part of the requirements for the degree of:

______in: ______It is entitled: ______

Approved by: ______ANALYSIS OF HIGH-FREQUENCY CHARACTERISTICS OF PLANAR COLD CATHODES

A thesis submitted to the

Division of Research and Advanced Studies of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE (M.S.)

in the Department of Electrical and Computer Engineering and Computer Science of the College of Engineering

2003

Rajesh Krishnan

B.E. (Electronics & Instrumentation) Annamalai University Tamilnadu, India, 2000

Committee Chair: Dr Marc Cahay

Abstract

The possibility of surface emitting cathode, which operates at room temperature, is attractive due to its compactness and reduced weight. Such cathodes, known as cold cathodes, can be used in a variety of electronic devices, including microwave vacuum transistors and tubes, pressure sensors, thin film displays, high temperature and radiation tolerant sensors, among others. Recently, M.Cahay and collaborators have proposed a new cold cathode emitter concept, making use of rare-earth sulphides to reach negative electron affinity at the surface.

In this thesis, we investigate two unchartered areas in the physical operation of these cathodes which entails a development of a small signal equivalent circuit of the cathode and the effects of noise (mostly due to shot noise of the injecting contact) on the anode current fluctuations. We find that the efficient direct modulation of the anode current with a small AC signal across the CdS layer is possible. For an InP/CdS/Las cold cathode, the degree of electron beam prebunching is dependent on the cathode to anode spacing but is found to be tunable up to frequency well within the K-band for cathode to anode spacing of a few microns. We also have used an Ensemble Monte-Carlo code to study effects of shot noise, in the anode current, in planar Metal/CdS/LaS cold cathodes. We have identified device parameters and biasing conditions for which regime of shot noise suppression but also shot noise enhancement in the anode current fluctuations can be observed for the same cathode.

Acknowledgements

First and foremost, I would like to thank my advisor Dr.Marc Cahay for his excellent guidance throughout the course of my thesis. This thesis would not be possible without his help, support and guidance. I would also like to thank the committee members,

Dr.Roenker and Dr.Boolchand for taking the time and effort to serve on my thesis defense committee. I would also like to thank the Air Force Research Laboratory, Wright

Patterson Air Force Base, Dayton, Ohio for their support.

I convey my heartfelt gratitude to my parents and my brothers, Suresh and Ramesh, for their love and encouragement. A special word of mention also goes to Uma, Harini and Vidya. A special word of gratitude goes to Shekhar, Ashita, Bijon, Mita, Karthick and Shobana for their sincere love and solid belief in me. A special word of thanks to

Aravind, who was a constant source of encouragement.

Thanks to all my friends Narain, Mukesh, Chandan, Raj, Texas Mohan, Pradeep, and all my undergrad friends who have helped me in many ways. I also would like to thank my roomies Prashanth, Thodur and Dom for their constant support. I also would like to thank my lab mates Yamini, Jugul, KP, Kalyan, Venkat and Philip. I have tried to include everyone, but unfortunately it is not possible to name all my friends in a single page.

Last but not the least; I thank God for showering His blessings upon me. Contents

1 Introduction and Signiﬁcance

of the Problem 1

1.1 Background: ...... 1

1.2 Paradigm Shifts ...... 5

1.3 Previous Analysis of Metal and InP/CdS/LaS

Cold Cathodes ...... 14

1.3.1 Self-Heating Eﬀects ...... 14

1.3.2 Current Crowding Eﬀects ...... 16

1.3.3 Space-Charge Eﬀects ...... 16

1.4 Outline of Thesis ...... 19

2 AC Current Crowding Eﬀects in Planar Cold Cathodes 26

2.1 Introduction ...... 26

2.2 Small AC Signal Equivalent Circuit ...... 29

2.3 Results ...... 33

2.4 Conclusions ...... 42

i 3 Sub-Poissonian and Super-Poissonian

Shot Noise 44

3.1 Introduction ...... 44

3.2 The Ensemble Monte-Carlo Approach ...... 48

3.3 The Eﬀects of Shot Noise ...... 52

3.4 Numerical Examples ...... 55

3.5 Shot Noise Power Spectrum ...... 57

3.6 Conclusions ...... 72

4 Conclusions and Future Work 78

4.1 Conclusions and Future Work ...... 78

4.2 Suggestions for future work ...... 80

4.3 List of Journal Publications, Conference Proceedings Papers and Conference

Presentations ...... 81

A Appendix 85

A.1 Ramo’s Current Expression ...... 85

ii List of Figures

1.1 NEA achieved through band bending at the surface of a semiconductor (p-

type doped). When the surface of the semiconductor is covered with a low

work function material (φ), the vacuum level outside the semiconductor can

end up below the minimum of the conduction band in the bulk. This leads to

an eﬀective negative electron aﬃnity, i.e., χeff < 0...... 4

1.2 Schematic representation of a Solid-State Field-Controlled Electron Emitter

(SSE). [13,14] ...... 6

1.3 Experimental Current Density J vs applied ﬁeld Fapp characteristics of a SSE.

[13,14] ...... 7

1.4 Schematic conduction band diagram of InGaN/GaN ﬁeld emitter. Electrons

travel ballistically across the InGaN layer and, thus, eﬀectively tunnel from

the maximum of the GaN conduction band at the GaN/InGaN interface. [15] 10

1.5 Bottom: I-V characteristics of InGaN/GaN (left) and GaN (right) ﬁeld emit-

ter array. [15] ...... 11

iii 1.6 Top: cross-section of the cold cathode proposed in refs [2, 1] between two

emitter ﬁngers. The substrate can easy be a metal or a heavily doped n-

type InP substrate. Trapping of electrons by the LaS semimetallic thin ﬁlm

leads to a lateral current ﬂow and current crowding in the structure. Bottom:

illustration of the partial reﬂection of the two-dimensional electron gas in the

LaS thin ﬁlm upon entering the three-dimensional contact regions where the

external bias is applied to Au contacts made to the thick LaS regions. The

emission window has a length L in the y direction...... 15

1.7 Illustration of the chopping action of the electrostatic potential energy proﬁle

in front of the LaS thin ﬁlm as a result of the space-charge eﬀects in the

cathode to anode region. The energy distribution h(E) of the current injected

into vacuum is shown under the approximation of ballistic transport(solid

line) and for the case of inelastic scattering in the CdS layer(dashed line). In

the current self-quenching regime, the potential energy proﬁle can oscillate

rapidly in front of the cathode(shown by vertical arrows) and the average

current collected at the anode is much smaller than the injected current. . . 18

2.1 Cross-section of a InP/CdS/LaS cold cathode with two emitter ﬁngers on

either side of the emission window. The total base current is the sum of Ib the

base current due to the shadowing action of the Au contacts and the current

0 Ib due to trapping of electrons in the LaS layer and lateral motion of carriers

along the semimetallic thin ﬁlm...... 27

iv 2.2 Small AC signal equivalent circuit of the cold cathode shown in Fig.1 including

a load resistance RL ...... 32

2.3 Plot of the anode current and base current as a function of the applied bias

across the CdS layer for the InP/CdS/LaS cold cathode with the parameters

listed in Tables 2.1 and 2.2...... 34

2.4 Plot of the transconductance gm and the inverse of the resistance rπ as a

function of the applied bias across the CdS layer for the InP/CdS/LaS cold

cathode with the parameters listed in Tables 2.1 and 2.2...... 36

2.5 Plot of bias dependence of the parameter z characterizing the eﬀects of DC

current crowding in the InP/CdS/LaS cold cathode with the parameters

listed in Table 2.1. DC current crowding eﬀects are negligible as long as the

parameter z stays below 0.3 [1]...... 38

2.6 Plot of the bias dependence of the temperature in the intrinsic portion of

the cold cathode. The back of the InP substrate is assumed to be at room

temperature (300K). The cold cathode parameters are listed in Tables 2.1 and

2.2...... 40

2.7 Plot of the voltage VL across the load resistance (RL = 1kΩ) for the InP/CdS/LaS

cold cathode with the parameters listed in Tables 2.1 and 2.2 for two diﬀerent

values of the cathode to anode spacing. The full and dashed curves correspond

to an anode to cathode spacing equal to 1 µm and 5 µm, respectively. . . . . 41

v 3.1 Top: cross-section of the cold cathode proposed in refs [3, 2] between two

emitter ﬁngers. The substrate can easy be a metal or a heavily doped n-

type InP substrate. Trapping of electrons by the LaS semimetallic thin ﬁlm

leads to a lateral current ﬂow and current crowding in the structure. Bottom:

illustration of the partial reﬂection of the two-dimensional electron gas in the

LaS thin ﬁlm upon entering the three-dimensional contact regions where the

external bias is applied to Au contacts made to the thick LaS regions. The

emission window has a length L in the y direction...... 46

3.2 Illustration of the chopping action of the electrostatic potential energy proﬁle

in front of the LaS thin-ﬁlm as a result of the space-charge eﬀects in the

cathode to anode region. The energy distribution h(E) of the current injected

into vacuum is shown under the approximation of ballistic transport (solid

line) and for the case of inelastic scattering in the CdS layer (dashed line). . 47

3.3 Frequency dependence of the spectral density SI (f) of the current density

ﬂuctuations for the Au/CdS/LaS cold cathode with the parameters listed in

Table.3.1. The curves labeled LINEAR and FULL POISSON correspond

to EMC simulations using a linear potential drop across the vacuum gap (i.e,

external electric ﬁeld) or a full self-consistent solution of Poisson’s equation,

respectively. The injected current density is equal to 153 A/cm2 and the in-

jected energy proﬁle across the Au/CdS/LaS cathode was calculated assuming

ballistic transport across the CdS and LaS layers [6]. The anode to cathode

spacing is equal to 15 µm and the bias across the gap Vgap = 5V...... 61

vi 3.4 Same as Fig.3.3 for a bias across the vacuum gap Vgap = 1V...... 62

3.5 Time dependence of the autocorrelation function CI (t) of the current density

ﬂuctuations for a Au/CdS/LaS cathode with the parameters in Table.3.1. The

cathode to anode separation is equal to 15 µm. The bias across the cathode to

anode spacing is equal to 5V.The curve of lower amplitude has been multiplied

by a factor 103 for clarity. It corresponds to the case of a linear potential drop

across the vacuum gap...... 64

3.6 Same as Fig.3.5 for a bias across the cathode to anode spacing equal to 1V.

The curve of lower amplitude corresponds to the case of a linear potential

drop across the vacuum gap...... 65

3.7 Time dependence of the Ramo current for a Au/CdS/LaS cathode with the

parameters listed in Table.3.1 for two diﬀerent biases across the vacuum gap.

The injected current density is equal to 153 A/cm2. The cathode to anode

separation is equal to 15 µm. The curve labeled Linear shows the time-

dependence of the current in the linear case, i.e, for a linear potential drop

across the gap. This curve has been shifted down by 80 A/cm2 from its

calculated value...... 66

vii 3.8 Time dependence of the potential minimum in front of the cathode for the

self-consistent EMC simulations for a Au/CdS/LaS cold cathode with the

parameters listed in Table.3.1. The cathode to anode separation is equal to

15 µm. The curves are labeled with the value of the bias across the gap. The

Vgap = 1V curve has been shifted down by -4 V for clarity. The amplitudes of

oscillation of the electrostatic potential minimum are much larger for Vgap =

5V...... 67

3.9 Time dependence of the location of the potential minimum in front of a

Au/CdS/LaS cold cathode in the case of a self-consistent EMC simulation.

The cold cathode has the parameters listed in Table.3.1. The cathode to an-

ode separation is equal to 15 µm. The curves are labeled with the value of

the bias across the gap. The Vgap = 1V curve has been shifted down by 2 µm

for clarity. The oscillations of the location of the potential minimum in the

vacuum gap are quite large for Vgap= 5V...... 69

3.10 Time dependence of the average value of the kinetic and potential energy of the

electrons within the vacuum gap. The potential across the gap is Vgap = 5V .

The cathode parameters are listed in Table.3.1. The injected current density

is equal to 153 A/cm2. Past the transient, plasma oscillations are clearly seen

with an exchange between the average kinetic and potential energy of the

carriers at a frequency equal to the the one observed in the oscillations of the

anode current (see Fig.3.7)...... 70

viii 3.11 Same as Fig.3.10 for Vgap = 1V . In this case, no clear plasma oscillation can

be seen...... 71

ix List of Tables

1.1 Materials Parameters of Some Sulﬁdes of Rare-Earth Metals (cubic form): a

(lattice constant in A˚), WF (work function at 300K), Tm (melting point in

oC), and ρ electrical resistivity (in µΩcm). [26] ...... 13

2.1 Material Parameters of the Cold Cathode ...... 35

2.2 Physical Parameters of the Cold Cathode ...... 37

2.3 Small Signal Parameters for the Cold Cathode with the parameters of Tables

2.1 and 2.2 for a bias of 2 V across the CdS layer ...... 39

3.1 Material Parameters of the Cold Cathode ...... 56

x Chapter 1

Introduction and Signiﬁcance of the Problem

1.1 Background:

Recently, there has been renewed interest into cold cathode emitters for applications to a variety of electronic devices, including microwave vacuum transistors and tubes, pressure sensors, thin panel displays, high temperature and radiation tolerant sensors, among others

[1,2,3]. The possibility of surface emitting cathode, that operate at room temperature is attractive since the introduction of such emitters would permit an unprecedented compactness and weight reduction in device and equipment design. Furthermore, low temperature operation in nonthermionic electron emitters is very desirable for keeping the statistical energy distribution of emitted electrons as narrow as possible, to minimize thermal drift of solid state device characteristics, and to avoid accelerated thermal aging or destruction by

1 internal mechanical stress and fatigue.

Many cold cathodes have been proposed based of the concept of field emission by placing a sufficiently high external electric field relative to the surface of the cathode to decrease the width of the potential barrier enough for reasonable tunneling currents to be produced.

Among the most widely investigated field emitters are the Spindt-type cathodes utilize field enhancement properties in a cone geometry to increase the electric field near the cone tip lo- cally to permit electron field emission [2,3]. Spindt-type cathodes have received considerable attention since their invention in the early 70’s. Several prototype field emission displays

(FEDs) based on metallic and semiconductor tips have been demonstrated at several dis- play industry events the last few years and are in the process of being commercialized [4].

However, these field emission cathodes still require too large a voltage for operation (several hundred volts). In practical field-emitter applications where many field-emitter tips are to be used simultaneously a critical requirement is that the tip arrays be highly uniform in order to maintain a uniformly high- emission current density. For arrays of Spindt cathodes, this uniformity in the distribution of the tips is a challenge to the most advanced lithographic technique and has led to malfunction of field arrays. Indeed, there are many experimental reports that malfunction of one tip out of millions can lead to the destruction of the entire array [2,3,4].

For that reason, several other approaches have been proposed to realize ﬁeld emission from various metallic and semiconducting materials (porous and polycrystalline silicon, ferroelectrics, and carbon nanotubes, among others). Alternatives to achieve cold cathode emission have been proposed based on the concept of low electron aﬃnity (LEA) or negative

2 electron aﬃnity (NEA) [5]. As illustrated in Fig.1.1, LEA or NEA can be achieved through bandgap engineering of semiconductor materials typically by treating a wide bandgap semiconductor with a low work function material such that the surface vacuum barrier is brought either slightly above (LEA) or preferably below (NEA) the bulk conduction band edge [5], respectively. This can be either occurring naturally like in the case of diamond and other wide bandgap materials such as for which there have been several reports of LEA or NEA whose value depends strongly on the crystallographic orientation of the substrate [6,7,8,9,

10,11].

It is well known that the amount of localized bending of the conduction band must occur near the surface of the material to match the Fermi level in the bulk with the surface state energy level. This requires a precise knowledge of the potential surface states for a given semiconductor surface, a process obviously depend on the exact preparation of the emitting surface (growth and condition, surface treatments, and ambient operation of the cold cathode). In order to reach NEA, the surface WF φ in relation to the semiconductor energy bandgap EG must obey one of the inequalities φ < 0.5EG or φ < EG if an intrinsic or p-type doped wide bandgap semiconductor is used, respectively. In that case, electrons excited into the conduction band have an energetically favorable chance of escaping the solid into the vacuum.

Several review articles of the electrical properties of cold cathodes mentioned above have been published recently [1,2,3,4,9,10,11,12]. More recently, another few unconventional approaches to realize cold cathodes have been proposed which could alleviate some of the diﬃculties linked with other approaches, especially in terms of reliability and reproducibility

3 Surface depletion region

E C χ eff VB χ

+ φ + E V − −

Figure 1.1: NEA achieved through band bending at the surface of a semiconductor (p- type doped). When the surface of the semiconductor is covered with a low work function material (φ), the vacuum level outside the semiconductor can end up below the minimum of the conduction band in the bulk. This leads to an eﬀective negative electron aﬃnity, i.e.,

χeff < 0.

4 of the electronic properties of the cathodes.

1.2 Paradigm Shifts

The Solid-State Field-Controlled Electron Emitter was recently proposed by Vu Thien

Binh et al. [13,14]. It consists of a ultra thin (a few nanometers thick) wide bandgap n-type semiconductor deposited on a metallic substrate (Fig.1.2). Explanation for the observed current density vs. applied external electric ﬁeld (Fig.1.3) has been given in terms of a two-step mechanism consisting of (a) injection of electrons across the metal-semiconductor

Schot junction followed by (b) emission of electrons from the wide bandgap semiconductor/vacuum interface. During this two-step process, the charge pile-up of electrons in the wide bandgap semiconductor eventually leads to low or even negative electron aﬃnity at the semiconductor/vacuum interface. One advantage of this cold cathode is that the threshold

field for substantial emitted current into vacuum was observed to be around 50 V/µm for a prototype P t/T iO2 cold cathode, which is about two orders of magnitude below the threshold electric field require to observe substantial emission from field emitter tips or diamond

ﬁlms.

InGaN/GaN Field Emitter Arrays Based on Piezoelectric Surface Barrier Low- ering [15].

To increase the emitted current density of cold cathodes, Mishra et al. have proposed

5 Figure 1.2: Schematic representation of a Solid-State Field-Controlled Electron Emitter

(SSE). [13,14]

6 Figure 1.3: Experimental Current Density J vs applied ﬁeld Fapp characteristics of a SSE.

[13,14]

7 to use wide bandgap semiconductors and lower the effective electron affinity at the surface by engineering the piezoelectric effect in these materials [15]. For instance, the piezoelectric effect in the thin InGaN surface layer grown on top of a GaN substrate is due to strain which, in certain crystal directions, can lead to a separation of core valence electrons. The latter generates an internal electric field in the InGaN thin layer as a result of the pseudo- morphic growth of this layer on the GaN substrate . Mishra et al. have shown that, for a structure grown in the (0001) direction, the internal electric field in the InGaN layer points towards the GaN substrate, leaving the positive end of the dipole in the InGaN layer at the InGaN/vacuum interface. This field leads to a potential drop across the InGaN layer and the effective electron affinity χeff at the surface of the cathode is reduced, as illustrated in Fig.1.4. As a result, the turn-on voltage of the field emitter is reduced, hence the emitted current density is increased. For field emitter arrays with similar cathode to anode spacing,

Mishra et al. observed a reduction of the turn-on voltage from 450V to 70V going from GaN to InGaN/GaN devices (Fig.1.5).

Rare-Earth Monosulﬁdes as A Mean to Achieve Negative Electron Aﬃnity

[16,17,2,1,3,6,5,4,24,5]

In the past, the most conventional approach to NEA has been the use of cesiated semiconductor surfaces. However, Cesium suﬀers from poor stability due to its willingness to release bonding electrons. Cesium melts at 28.5◦C and has a high vapor-pressure ( 10−3

Torr at 100 ◦ C). Cesium, when out into a vacuum structure, eventually migrates every-

8 where. It spreads to surfaces where electron emission is not desired and can make insulating surfaces undesirably conductive. It also spreads to interelectrode spaces where it will provide a relatively easy path for gaseous conduction and electrical arcs. The recent demonstrations of NEA from diamond and other wide bandgap materials are welcome alternatives to the cesiated surfaces but these investigations are still in their infancy and far from providing a technological breakthrough in the near future [9].

Typically, the operating lifetime of cathodes using cesiated surfaces is good as long as the tube is not subjected to excessive cathode currents from high-level usage. Otherwise, the emitted electrons from the cathode are accelerated and, depending on the accelerating voltage and cathode current density, there can be an electron scrubbing eﬀect of the anode resulting in positive ions being liberated and accelerated back onto the NEA surface and a continuous increase in the cathode WF. An accelerated cathode degradation which occurs under high current operating conditions is electron stimulated desorption of the Cs + 0 activation layer. In general, III-V NEA surfaces are not stable in ultra-high vacuum.

At 300 K, Cs will slowly desorb from the surface in UHV at a rate dependent, in part, on the partial pressure of Cs within the vacuum system. Additional Cs will restore a fraction of the original surface escape probability. A small quantum yield can still give rise to a large current if the excitation intensity is suﬃciently strong. Laser provide light intensity which are practically unlimited. However, excessive radiation may lead to the destruction of the activation layer by light induced desorption. This eﬀect must be counteracted by continuous cesiation during operation of the source.

Recently, M. Cahay and some of his former students have proposed to use sulﬁdes of

9 Figure 1.4: Schematic conduction band diagram of InGaN/GaN ﬁeld emitter. Electrons travel ballistically across the InGaN layer and, thus, eﬀectively tunnel from the maximum of the GaN conduction band at the GaN/InGaN interface. [15]

. 10 Figure 1.5: Bottom: I-V characteristics of InGaN/GaN (left) and GaN (right) ﬁeld emitter array. [15]

11 rare-earth elements as more stable alternatives to reach NEA at various III-V semiconductor surfaces [2,1,6,5]. These compounds do not suﬀer from all the limitations of cesiated surfaces.

A summary of some of the material properties of sulﬁdes of rare-earth elements in their cubic form are listed in Table 1.1. Of particular interest is the fact that the work function

(WF) at room temperature of these compounds, when extrapolated from high-temperature measurements [26], are quite small. It is therefore expected that these materials can be used to reach NEA when deposited on p-type doped semiconductors. For instance, LaS has a lattice constant (5.854A˚) very close to the lattice constant of InP (5.8688A˚) and NdS has a lattice constant (5.69A˚) very close to the lattice constant of GaAs (5.6533A˚). Since the room temperature WF of LaS (1.14eV ) and NdS (1.36eV ) are respectively below the band gap of InP (1.35 eV) and GaAs (1.41 eV), NEA can therefore be reached at InP/LaS and

GaAs/NdS interfaces using heavily p-type doped semiconductors. Recently, we conﬁrmed this result by means of a ﬁrst-principles electronic-structure method based on a local-density approximation to density-functional theory [16]. This analysis predicted a 0.9 eV and 1.1 eV WF for LaS and NdS, respectively, at low temperature.

Two other important features of the fcc cubic form of the rare-earth compounds listed in Table 1.1 are the fairly large melting temperature (about 2000o C) and their fairly good electrical resistivity. It is therefore expected that thin films of these compounds used to promote semiconductor surfaces to NEA should be stable and should not suffer from current crowding effects which would lead to non-uniformity in the emitted current into vacuum.

Based on the interesting properties of the rare-earth sulﬁdes described above, M. Cahay

12 ErS YS NdS GdS PrS CeS LaS EuS SmS

a 5.424 5.466 5.69 5.74 5.747 5.778 5.854 5.968 5.970

WF 1.36 1.26 1.05 1.14

Tm 2060 2200 2230 2450 2200 1870

ρ 242 240 170 25

Table 1.1: Materials Parameters of Some Sulﬁdes of Rare-Earth Metals (cubic form): a

o (lattice constant in A˚), WF (work function at 300K), Tm (melting point in C), and ρ electrical resistivity (in µΩcm). [26]

and collaborators have proposed new Metal(n++InP)/CdS/LaS cold cathode emitter concept making use of Lanthanum Sulfide (LaS) to reach NEA. The architecture of the proposed cold cathode is shown in Fig. 1.6. The main elements in the design and functioning of such an emitter are : (1) a wide bandgap semiconductor slab equipped on one side with a metal of heavily doped n-type InP substrate that supplies electrons at a sufficient rate into the conduction band and (2) on the opposite side, a thin semimetallic film (LaS) that facilitates the coherent transport (tunneling) of electrons from the semiconductor conduction band into vacuum. As shown in Fig 1.6, an array of Au fingers is defined on the surface of the

LaS thin ﬁlm to bias the structure. The bias is applied between the InP substrate and the metal grid with emission occuring from the exposed LaS surface. For the InP/CdS/LaS cold cathode, the choice of a LaS semimetallic thin ﬁlm grown on nominally undoped CdS is quite appropriate since the lattice constant of CdS (5.83 A˚) is very close to the lattice

13 constant of LaS (5.85 A˚) in its cubic crystalline form. Additionally, LaS is expected to have a quite low temperature work function (1.14 eV) [26], a feature when combined with the large energy gap (2.5 eV ) of CdS leads to NEA of the semiconductor material [5]. The electronic structure of CdS/LaS interface was recently investigated by means of a ﬁrst- principles electronic-structure method based on a local-density approximation to density- functional theory [16]. The extrapolated 1.14 eV low work function of LaS was reproduced by that theory. Furthermore, it was found that NaCl structured layers of LaS should grow in a well-behaved, epitaxial way on a CdS substrate with a ZnS structure [16].

1.3 Previous Analysis of Metal and InP/CdS/LaS

Cold Cathodes

1.3.1 Self-Heating Eﬀects

For an InP/LaS/CdS cold cathode, Malhotra et al. have analyzed the importance of self- heating effects as a result of various power dissipation mechanisms in the active portion of the device [5]. These include the power released as a result of (1) inelastic scattering in the CdS layer, (2) trapping of electrons in the LaS thin film, (2) Joule heating associated with the finite resistivity of the LaS thin film as the trapped electrons move to the side Au contacts of the LaS window (3) of Joule heating due to electron current moving from the

LaS thin ﬁlm region through the wider Au regions before reaching the top contacts, and (5)

Joule heating due to the blocking eﬀect of the wide Au ﬁngers needed to dc bias the cold

14 Figure 1.6: Top: cross-section of the cold cathode proposed in refs [2,1] between two emitter ﬁngers. The substrate can easy be a metal or a heavily doped n-type InP substrate.

Trapping of electrons by the LaS semimetallic thin film leads to a lateral current flow and current crowding in the structure. Bottom: illustration of the partial reflection of the two- dimensional electron gas in the LaS thin film upon entering the three-dimensional contact regions where the external bias is applied to Au contacts made to the thick LaS regions.

The emission window has a length L in the y direction.

15 cathode.

As shown in ref. [5], these various power dissipation mechanisms can lead to a substantial increases of the temperature of the active portion of the device. One of the most drastic eﬀect of this temperature rise is that the NEA at the surface could be lost since the WF of

LaS is known to increase with temperature at a rate of 2meV/K with temperature [27].

1.3.2 Current Crowding Eﬀects

For a Metal/LaS/CdS cold cathode, Mumford and Cahay have analyzed the importance of current crowding when a DC bias is applied across the CdS layer.[1] Because of the finite resistivity of the LaS thin films, there is a possibility for current crowding between any two fingers in the extraction grid configuration of the cold cathode shown in Fig 1.6. This current crowding is accompanied by a lateral electrostatic potential variation across the LaS emission window. In refs. [2,1], it was shown that current crowding effects can be minimized by keeping the width of the emission window (see Fig.1.6) under 50 µm while keeping the emission current density below 100 A/cm2.

1.3.3 Space-Charge Eﬀects

Recently Mumford and Cahay [3] and Modukuru et al. [6] have used an ensemble Monte-

Carlo approach to analyze the importance of space-charge eﬀects in the cathode to anode gap region of the Metal/CdS/LaS cold cathode and its inﬂuence on the energy distribution of the electrons collected at the anode. The analysis was based on a mean-free path approach to

16 include the eﬀects of inelastic scattering in the Cd layer following Fowler-Nordheim injection across the Metal/CdS interface [6].

As illustrated in Fig.1.7, one of the important feature of transport in the vacuum air gap between the cathode and the anode is that electrons are injected with a ﬁnite velocity much in excess of the thermal velocity of thermionic cathodes. At high injection current density, space-charge eﬀects lead to current self-quenching and the anode conduction current is much smaller than the Fowler-Nordheim injected current. However, the limiting value of the measured average anode current density is much larger than the Child-Langmuir limit [3,

6]. In this current self-quenching regime, they had already shown that the chopping action of the beam is sometimes accompanied by high frequency (several tens of GHz) ﬂuctuations in the minimum of the electrostatic potential in the air gap (see Fig.1.7). These oscillations leads to large oscillations in the anode current.

More recently, Modukuru and Cahay have revisited the issue of DC current crowding in InP/CdS/LaS cold cathodes since it can be by the trapping of electrons being reﬂected towards the cathode as a result of space-charge eﬀects in the vacuum gap discussed above [4].

They showed that a self-consistent solution of the interplay between current crowding and space-charge eﬀects can lead to a non-monotonic lateral distribution of the anode current density depending on the width of the emission window. This behavior is in sharp contrast with the monotonic decrease of the anode current density from the edges towards the center of the emission window when space-charge eﬀects in the vacuum region are neglected.

17 h(E)

E F

ANODE D

Figure 1.7: Illustration of the chopping action of the electrostatic potential energy profile in front of the LaS thin film as a result of the space-charge effects in the cathode to anode region. The energy distribution h(E) of the current injected into vacuum is shown under the approximation of ballistic transport(solid line) and for the case of inelastic scattering in the

CdS layer(dashed line). In the current self-quenching regime, the potential energy proﬁle can oscillate rapidly in front of the cathode(shown by vertical arrows) and the average current collected at the anode is much smaller than the injected current. 18 1.4 Outline of Thesis

A complete model of the Metal(InP)/CdS/LaS cold cathode described above should include the self-heating, current crowding, and space-charge effects in the cold cathode and the interplay between all these effects. The goal of this thesis is to investigate yet two uncharted areas in the physical operation of these cathodes which entails a development of a small signal equivalent circuit of the cathode and the effects of noise (mostly due to shot noise of the injecting contact) on the anode current fluctuations.

In chapter 2, we derive a small AC signal equivalent circuit of the planar InP/CdS/LaS cold cathode to investigate the possibility of electron prebunching of the electron beam injected into vacuum through the use of a small AC bias applied across the CdS layer. This possibility would greatly reduce the cost and weight of traveling wave tubes using this type of cathodes since it would remove the need for the extra circuitry typically used in current design to prebunch the electron beam after injection into vacuum [28,29,30]. In this work, the small signal equivalent circuit is developed under the approximation of negligible DC current crowding effects in the cathode and while neglecting the space-charge effects in the vacuum gap between the cathode and anode. The regime of negligible current-crowding effects is achievable by using a fairly narrow emission window (a few µm in width), as discussed in details in ref. [2]. A regime of operation with negligible space-charge effects can be obtained by putting a large enough bias across the vacuum gap [3,6]. However, the small signal AC model developed in this work takes into account self-heating effects in the intrinsic cathode but the latter are minimized in order to keep the NEA at the cathode/vacuum interface.

In chapter 3, we use the Ensemble Monte-Carlo code developed in ref. [3,6] to study

19 the effects of shot noise in planar Metal/CdS/LaS cold cathodes for which the emission into vacuum is characterized by an average injection energy in excess to the thermal energy typical of thermionic cathodes [3,6]. The cathode parameters are selected such that current- crowding and self-heating effects can be neglected. We identify device parameters and biasing conditions for which regime of shot noise suppression but also shot noise enhancement in the anode current fluctuations can be observed for the same cathode.

Finally, chapter 4 present our conclusions and suggestions for the continuation of this work. We also give a list of the papers submitted for publication and presented at conferences during the course of this Masters thesis.

20 Bibliography

[1] S. Iannazzo, ”A Survey of the Present Status of Vacuum Electronics”, Solid State Elec-

tronics, vol. 36, p. 301 (1993).

[2] I. Brodie and C. A. Spindt, ”Vacuum Microelectronics”, Adv. Electron. Electron Phys.,

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[3] I. Brodie and C. A. Spindt, ”Vacuum Microelectronic Devices”, Proceedings of the IEEE,

vol. 82, p.1006 (1994).

[4] D. Temple, ”Recent Progress in Field Emitter Array Development For High performance

applications”, Materials Science and Engineering Reports: A Review Journal, vol.24,

p.185 (1999).

[5] P. R. Bell, Negative Electron Aﬃnity Devices, Oxford: Claredon Press, 1973.

[6] F.J. Himpsel, J.A.Knapp, J.A. Van Vechten, D.E. Eastman, ”Quantum photoyield of

diamond (111) - A stable negative-aﬃnity emitter”, Phys. Rev. B 20, 624 (1979).

21 [7] J. van der Weide, Z. Zhang, P.K. Bauman, M.G. Wensell, J.Bernholc, and R.J. Ne-

manich, ”Negative-electron-aﬃnity eﬀects on the diamond (100) surface”, Phys. Rev B

50, 5803 (1994).

[8] B.B. Pate, ”The diamond surface: atomic and electronic structure”, Surf. Sci. 165, 83

(1986).

[9] J. E. Jaskie, ”Diamond Based Field Emission Displays”, Series on Directions in Con-

densed Matter Physics, vol.17, Insulating and Semiconducting Glasses, Editor: P. Boolc-

hand, World Scientiﬁc Singapore (2000).

[10] Proceedings of First International Symposium on Cold Cathodes, 198th Meeting of the

Electrochemical Society, Phoenix, AZ, 20-27 October 2000.

[11] J. Ristein, ”Electronic Properties of Diamond Surfaces - Blessing or Curse for Devices?”,

Diamond and Related Materials, vol.9, p.1129 (2000).

[12] G. Rosenman, D. Shur, Ya. E. Krasik and A. Dunaevsky, ”Electron Emission from

ferroelectrics”, Journal of Applied Physics vol.88, p.6110 (2000).

[13] Vu Thien Binh, ”Electron spectroscopy from solid-state ﬁeld-controlled emission cath-

odes”, Applied Physics Letters, vol.78, p.2799 (2001).

[14] Vu Thien Binh, V. Semet, J. P. Dupin and D. Guillot, ”Recent progress in the char-

acterization of electron emission from solid-state ﬁeld-controlled emitters”, J. Vac. Sci.

Technol. B, vol.19, p.1044 (2001).

22 [15] R. D. Underwood, P. Kozodoy, S. Keller, S. P. DenBaars, and U. K. Mishra, ”Piezoelec-

tric Surface Barrier Lowering Applied to InGaN/GaN Field Emitter Arrays”, Applied

Physics Letters, vol.73, p.405 (1998).

[16] O. Eriksson, J. Willis, P. D. Mumford, M. Cahay, and W. Friz, ”Electronic Structure of

the LaS Surface and LaS/CdS interface in a new Cold Cathode Conﬁguration”, Physical

Review B, vol.57, p. 4067 (1998).

[17] O. Eriksson, M. Cahay, J. Wills, ”Negative Electron Aﬃnity Material: LaS on InP”,

Physical Review B 65, 033304 (2002).

[18] P. D. Mumford and M. Cahay, ”Dynamic Work Function Shift in Cold Cathode Emitters

Using Current Carrying Thin Films” Journal of Applied Physics, vol.79, p.2176 (1996).

[19] P. D. Mumford and M. Cahay, ”Current Crowding Eﬀects in a CdS/LaS Cold Cathode”,

J. Appl. Physics 81, 3707 (1997).

[20] P. Mumford and M. Cahay, ”Space-Charge Eﬀects and Current Self-Quenching in a

Metal/CdS/LaS Cold Cathode”, Journal of Applied Physics, Vol.84, pp.2754-2767

(1998).

[21] Y. Modukuru, M. Cahay, H. Kolinsky, and P. D. Mumford, ”Onset of Current Self-

Quenching in a Metal/CdS/LaS Cold Cathode in the Presence of Inelastic Scattering

in the CdS Layer”, Journal of Applied Physics, Vol. 87, 3386 (2000).

23 [22] Y. Modukuru, M. Cahay, H. Kolinsky, and P. D. Mumford, ”Onset of Current Self-

Quenching in a Metal/CdS/LaS Cold Cathode in the Presence of Inelastic Scattering

in the CdS Layer”, Journal of Applied Physics, Vol. 87, 3386 (2000).

[23] Y. Modukuru and M. Cahay, ”Interplay of Current Crowding and Current Self-

Quenching Eﬀects in Planar Cold Cathodes” Journal of Vacuum Science and Technology

B 19, pp.2149-2154, Nov/Dec. 2001.

[24] Y. Modukuru, J. Thachery, H. Tang, A. Malhotra, M. Cahay and P. Boolchand,

”Growth and Characterization of Rare-Earth Monosulﬁdes for Cold Cathode Appli-

cations”, Journal of Vacuum Science and Technology B 19, pp.1958-1961 (2001).

[25] A. Malhotra, Y. Modukuru and M. Cahay, ”Self-Heating Eﬀects in a InP/CdS/LaS

Cold Cathode”, J. Vac. Sci. Technol. B, vol.16, p.3086 (1998).

[26] G. V. Samsonov, High Temperature Compounds of Rare-Earth Metals with Nonmetals,

Consultants Bureau Enterprises, Inc. (1965).

[27] The room temperature work function for LaS was calculated by extrapolating measured

work function values at high temperature as reported by S. Fomenko in Handbook of

Thermionic properties (Plenum, New York, 1966). Within the range of temperature

investigated by Fomenko, the LaS work function increases with temperature at a rate

of 2meV/K.

24 [28] L.H.Yu, ” Design parameters of the high gain harmonic generation experiment using

Cornell Undulator at the ATF”, Proceedings of the 1999 Particle Accelerator Confer-

ence, New York, 1999.

[29] Wurtele, J.S.Bekeﬁ, G.Chu, R.Xu, K, ”Prebunching in a collective Raman free electron

laser ampliﬁer”, 1989 JA Series, MIT Plasma Science and Fusion Center.

[30] W.D.Kimura, M.Babzien, I.Ben-Zevi, D.B.Cline, R.B.Fiorito, J.R.Fontana,

J.C.Gallardo, S.C.Gottschalk, P.He, K.P.Kusche, Y.Liu, R.H.Pantell, I.V.Pogorelsky,

D.C.Quimby, K.E.Robinson, D.W.Rule, J.Sandweiss, J.Skaritka, A.Van Steenbergen

and V.Yakimenko, ”Design and Model Simulations of Inverse Cerenkov Acceleration

Using Inverse Free Electron Laser Prebunching”, Proceedings of the 1997 Particle

Accelerator Conference, Vancouver, B.C., Canada, 1997.

25 Chapter 2

AC Current Crowding Eﬀects in

Planar Cold Cathodes

2.1 Introduction

The architecture of the cold cathode emitter model proposed in this thesis is shown in Fig.2.1.

The main elements of the cathode are: (1) a wide bandgap semiconductor (undoped CdS) sandwiched between a heavily doped n-type InP substrate, (2) a wide bandgap (intrinsic)

CdS layer, and (3) a thin semimetallic LaS ﬁlm that facilitates the coherent transport

(tunneling) of electrons from the semiconductor conduction band into vacuum. As argued in [2] the LaS thin film allows to reach NEA at the CdS surface. As shown in Fig. 2.1, metallic emitter fingers are defined on the surface of the LaS thin film to bias the structure.

Rectangular openings are deﬁned to expose the thin LaS ﬁlm which forms the active emission area of the cold cathode. If the applied voltage between the back of the substrate and the

26 Anode

R L

IA Ib Ib Vgap Au 2 2 Au LaS Vbias I’b I’b 2 2 i− CdS JFN n++ − InP

Figure 2.1: Cross-section of a InP/CdS/LaS cold cathode with two emitter ﬁngers on either side of the emission window. The total base current is the sum of Ib the base current due to

0 the shadowing action of the Au contacts and the current Ib due to trapping of electrons in the LaS layer and lateral motion of carriers along the semimetallic thin ﬁlm.

27 metallic grid is equal or larger than the semiconductor bandgap energy and the quotient of the applied voltage divided by the semiconductor thickness approaches 0.1eV/A˚, then electrons are tunnel injected into the conduction band and ascend during their travel across the semiconductor film to levels of increasing energy. The conduction band of the wide bandgap semiconductor provides the launching site for electrons where they are - through the semimetallic thin film - injected into vacuum. Negative electron affinity implies that the vacuum level would be located below the lower conduction band edge. In that case, electrons in the conduction band with momenta pointing toward the surface have a good chance to get emitted unless deflected by collision or trapped by impurities or defects. Electrons captured in the semimetallic thin film must flow laterally to the top contacts. Because of the finite resistivity of the LaS thin film, there is a possibility for current crowding in the cold cathode.

The choice of InP as a substrate is particularly attractive since the lattice constant of

InP (5.86 A)˚ closely matches the lattice constant of the zincblende cubic CdS (5.83 A).˚

Furthermore, there have been recent reports on the deposition of crystalline layers of CdS on InP by molecular beam epitaxy, chemical bath deposition and pulsed laser deposition .

The proposed cold cathode should therefore be realizable with present day technology.

Mumford and Dr.Cahay had already shown that the InP/CdS/LaS these cold cathodes are capable of low voltage (< 10 V) room temperature operation with emission current of several 100 A/cm2 [2]. For appropriate device parameters, they have discussed the conditions under which the eﬀects of DC current crowding and self-heating eﬀects due to the

ﬁnite thermal conductivity of the InP substrate can be neglected [1,5]. Hence, we use the analytical model derived in refs. [2,1,5] in which the eﬀects of current crowding and self

28 heating are minimized. We analyze the possibility to use InP/CdS/LaS cold cathode with a small AC signal on top of the DC bias applied across the CdS layer in order to prebunch the electron beam injected in vacuum. If this cathode has to be used for high power vacuum microlectronics applications, it is also important to address the frequency range over which the electron beam can be prebunched.

We develop a small AC signal equivalent circuit of the InP/CdS/LaS cold cathode structure depicted in Fig.2.1. A small signal equivalent circuit of the cathode is derived. We also present the SPICE simulations of the cathode as a function of the anode to cathode spacing to determine the frequency range over which eﬀective prebunching of the emitted current can be achieved.

2.2 Small AC Signal Equivalent Circuit

The cold cathode can be thought as a three-terminal device (similar to a bipolar or hetero- junction bipolar transistor) in which the role of the emitter is played by the heavily doped

InP substrate, the base consists of the LaS semimetallic thin ﬁlm with Au contacts, and the collector is the anode contact (typically placed several microns away from the cathode surface). In this analogy, the subcollector region is the vacuum region between the LaS thin

film and the anode. To derive the small AC signal equivalent circuit (similar to the hybrid-π model of transistors), we first realize that not all of the Fowler-Nordheim tunneling current emitted across the InP/CdS layer will be flowing through the opening between two adjacent emitter fingers. Some of the Fowler-Nordheim current, emitted under the Au lines used to bias the cathode, will not be able to make it into vacuum because of the shadowing action

29 of the top biasing fingers. This part of the Fowler-Nordheim tunneling current will be part of the base current since it will be forced to run through the Au lines. In addition, the base current also contains a component due that part of the Fowler-Nordheim current emitted under the emission window but which gets trapped in the LaS layer and will not escape into the vacuum gap. For simplicity, we neglect the effects of space-charge in the vacuum gap by assuming that the potential effect across the vacuum gap is large enough, otherwise this could lead to additional trapping of electrons into the LaS thin film.

Hereafter, we assume that the width WE of the Au ﬁngers on either sides of the emission window are equal to the width of the emission window itself (but the model can be generalized to any other geometrical conﬁguration). The amount of base current is then simply given by

I = J (2W L) + (J J )W L, (2.1) B F N E F N − A E where the first term comes from the shadowing action of the two emitters fingers and the second term is due to the current trapping of electrons in the LaS thin film. In Eq.(2.3), L is the length of the Au fingers.

In the simulations, we use the analytical model of the cold cathode described in detail in ref. [2,1,5] and ﬁt the base current to an expression of the form

αB Vbias IB = Io,Be , (2.2)

which is quite accurate for the small range of Vbias considered in the numerical simulations. our previous As a result, if a small AC signal is added to Vbias, the small signal resistance of the emitter (InP ) - base (LaS) junction is given by

dIB −1 −1 rπ = [ ] = (αIB) , (2.3) dVbias

30 which is a function of the DC bias across the CdS layer.

For simplicity, the capacitance between emitter and base is simply approximated at the parallel plate capacitor with a thickness equal to the width of the CdS layer and an area equal to 3 WE times the length L of the emission window. We use ² = 5.4²o for the dielectric constant of the CdS layer [6]. Since the DC bias and small AC signal are applied on a rectangular pad located at the end of the emitter ﬁngers of length L, we must therefore take into account of the ﬁnite resistance of the Au lines which have width WE, length L, and height H. This leads to a contact resistance RAu equal to

RAu = ρAuL/(WEH), (2.4)

where ρAu is the resistivity of the Au lines.

The small AC signal between emitter and base controls the anode current IA. The latter is also a function of the DC applied bias and is also parameterized as follows [2,1,5]

αAVbias IA = Io,Ae . (2.5)

This leads to a transconductance for the cathode given by

dIA gm = = αAIA. (2.6) dVbias

Finally, we include the eﬀects of the capacitance between the anode and the LaS thin ﬁlm and

Au lines by simply adding to the small signal equivalent circuit a parallel plate capacitance with a thickness equal to the separation between anode and cathode. The small signal equivalent circuit is then completed by adding the load resistance, schematically shown as

RL in Fig.2.1. The complete small AC signal equivalent circuit of the cathode is shown in

Fig.2.2.

31 C gap Rau

V g Vπ R be C Rπ m L Vπ

Figure 2.2: Small AC signal equivalent circuit of the cold cathode shown in Fig.1 including a load resistance RL

32 In the next section, we perform SPICE simulations of the circuit to determine the percentage of current modulation of the anode current, i.e, the ratio of the peak-to-peak value of the AC component of the anode current to the DC value of the anode current for a given DC bias and small AC signal across the CdS layer. Eﬃcient electron prebunching is achieved when this ratio can be made as close to unity as possible.

2.3 Results

Figure 2.3 is a plot of the anode and base current for the cold cathode with the parameters listed in Tables 2.1 and 2.2.

The total base current is larger than the anode current because of the shadowing action of the two emitter ﬁngers on either side of the emission window, as depicted in Fig. 2.1.

Figure 2.4 shows the corresponding bias dependence of the small signal parameters rπ and

−1 −1 gm calculated using Eq.(2.3) and (2.6), respectively. In our case, gm is larger than rπ in agreement with the fact that the base current is larger than the anode current.

The results plotted in Figures 2.3 and 2.4 where computed using the analytical treatment of the DC characteristics of the cathode following the model of of ref. [5] which includes the importance of DC current crowding and self-heating eﬀects in the cathode. As discussed in ref. [1], under DC biasing conditions, the importance of current crowding is characterized by a dimension parameter z deﬁned as follows

2 η(0) = JF N (0)/JF N (a) = cos z, (2.7)

where JF N (0)/JF N (a) is the ratio of the Fowler-Nordheim current density JF N emitted across

33 0.02

0.015

0.01 Current (A)

0.005 IA

0 1.8 1.85 1.9 1.95 2 2.05 2.1

Vbias (V) Figure 2.3: Plot of the anode current and base current as a function of the applied bias across the CdS layer for the InP/CdS/LaS cold cathode with the parameters listed in Tables

2.1 and 2.2.

34 Material InP i-CdS LaS

Lattice Constant (A˚) 5.8688 5.83 5.85

Workfunction (eV) 4.4 4.2 1.14

Bandgap (eV) 1.35 2.5

# of free electrons (1022cm−3) n++ — 1.99

Electron Mass (m0) 0.08 0.14 1.0

Electron Mobility (cm2V −1s−1) 400.0

Thermal conductivity @ 300K (W/cmK) 0.05..1 0.17

Electrical resistivity (273K) (µΩcm) 92.0

Melting temperature (K) 2500

Table 2.1: Material Parameters of the Cold Cathode

35 400

300

−1 gm 200 Ohms

100 rπ

0 1.8 1.85 1.9 1.95 2 2.05 2.1

Vbias (V)

Figure 2.4: Plot of the transconductance gm and the inverse of the resistance rπ as a function of the applied bias across the CdS layer for the InP/CdS/LaS cold cathode with the parameters listed in Tables 2.1 and 2.2.

36 Thickness of InP substrate 100 µm

Width of the CdS thin ﬁlm 150 A˚

Emission window length 100 µm

Emission window width 5 µm

Thickness of LaS thin ﬁlm 24.6 A˚

Thickness of emitter ﬁngers 5000 A˚

Cathode to anode spacing 1-5 µm

Electron mean free path in CdS and LaS thin ﬁlms 300 A˚

Table 2.2: Physical Parameters of the Cold Cathode

the InP/CdS interface calculated in the center (JF N (0)) and at the edge (JF N (a)) of the emission window. As shown in [1], DC current crowding eﬀects are negligible as long as z stays below 0.3. As seen in Fig.5, this requires the bias voltage for the cathode with the parameters in Tables 2.1 and 2.2 to be kept below 2.2 V.

Furthermore, self-heating eﬀects due to the ﬁnite resistivity of the substrate must be kept under controlled so that the temperature of the intrinsic portion of the device does not rise above 300K, otherwise the NEA would reduce to zero at the CdS/LaS interface.

Figure 2.6 shows the temperature of the intrinsic portion of the cathode as a function of the applied bias across the CdS layer. The temperature was calculated using the self-heating model model of ref. [5]. Figure 2.6 indicates that the bias across the CdS layer must be kept below 2.2 V for self-heating eﬀects to be negligible. Hence, the results of the previous

37 1.2

0.8

Z 0.6

0.4

0.2

0 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Vbias (V) Figure 2.5: Plot of bias dependence of the parameter z characterizing the eﬀects of DC current crowding in the InP/CdS/LaS cold cathode with the parameters listed in Table

2.1. DC current crowding eﬀects are negligible as long as the parameter z stays below 0.3

[1].

38 analysis therefore indicates that the bias across the CdS Layer should be kept below 2.2 V for both the eﬀects of DC current crowding and self-heating to be negligible in the cold cathode.

For these reasons, the results of the small AC signal analysis to be discussed hereafter were computed for Vbias = 2V across the CdS layer.

RAu C(pf) rpi (Ω) Cgap(pF ) gm(mS) RL(kΩ)

1 5 14 0.003/0.015 21 1

Table 2.3: Small Signal Parameters for the Cold Cathode with the parameters of Tables 2.1 and 2.2 for a bias of 2 V across the CdS layer

The value of Cgap in table 2.3 is listed for two diﬀerent values ( 1 and 5 µ m) of the distance between anode and cathode. When a small AC signal of 0.1 V peak-to-peak amplitude is applied across the CdS layer, a SPICE analysis of the high-frequency response of the cathode leads to the results shown in Fig.2.7 for the peak-to-peak value of the AC voltage across the load resistance.

At low frequency, the DC gain is about 20 as readily derived from the equivalent circuit shown in Fig.2.2 and the small AC signal parameters listed in Table 2.3. Furthermore, the frequency response has the expected low-pass characteristic with a 3dB frequency found equal to 9.5 (X-band) and 26 GHz (K-band) for an anode to cathode spacing equal to 1 and 5 µm, respectively. The 3dB frequency is lower for an anode spacing of 1 µm since the

39 540

510

480

450

420 T (K) 390

360

330

300 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Vbias (V) Figure 2.6: Plot of the bias dependence of the temperature in the intrinsic portion of the cold cathode. The back of the InP substrate is assumed to be at room temperature (300K).

The cold cathode parameters are listed in Tables 2.1 and 2.2.

40 2

1.8

1.6

1.4 3 dB line

1.2

(V) 1 L V 0.8

0.6

0.4

0.2

0 0 10 20 30 40 50 60 70 80 90 100 Frequency (GHz)

Figure 2.7: Plot of the voltage VL across the load resistance (RL = 1kΩ) for the InP/CdS/LaS cold cathode with the parameters listed in Tables 2.1 and 2.2 for two diﬀerent values of the cathode to anode spacing. The full and dashed curves correspond to an anode to cathode spacing equal to 1 µm and 5 µm, respectively.

41 impedance of the feedback capacitor is proportional to the anode to cathode spacing, leading to a smaller value of the output voltage across the load at a given frequency of the small

AC signal across the CdS layer. At the 3dB frequency, the ratio of the peak-to-peak value of the AC current ﬂowing through resistance to the DC anode current is found to be equal to 60 %. Therefore, the range of frequency at which electron prebunching is eﬀective can be tuned well into the K-band by varying (shortening) the anode to cathode spacing.

2.4 Conclusions

We have developed a first order small signal equivalent circuit of planar cold cathodes making use of semimetallic thin films at their surface to achieve negative electron affinity. The model was applied for the case of a specific InP/CdS/LaS cold cathode [2,1]. First, the DC biasing conditions of the cathode were selected so that both the effects of current crowding and current self-heating effects could be neglected. While neglecting the phase-delay linked to the finite time of propagation of the electron beam across the vacuum region, we have found that efficient direct modulation of the anode current (electron beam prebunching) with a small AC signal across the CdS layer is possible (with a 60% efficiency in our numerical examples). The degree of electron beam prebunching is dependent of the cathode to anode spacing but is found to be tunable up to frequency well within the K-band for cathode to anode spacing of a few microns.

42 Bibliography

[1] P. D. Mumford and M. Cahay, Journal of Applied Physics, Vol. 79(5), 2176 (1996).

[2] P. D. Mumford and M. Cahay, Journal of Applied Physics, Vol. 81(8), 3707 (1997).

[3] A. Malhotra, Y. Modukuru and M. Cahay, Journal of Vacuum Science and Technology

B 16, 3086 (1998).

[4] The room temperature work function for LaS was calculated by extrapolating measured

work function values at high temperature as reported by S. Fomenko in Handbook of

Thermionic properties (Plenum, New York, 1966). Within the range of temperature

investigated by Fomenko, the LaS work function increases with temperature at a rate

of 2meV/K.

[5] Y. Modukuru and M. Cahay, Journal of Vacuum Science and Technology B 19, 2149

(2001).

[6] S.M. Sze, Physics of Semiconductor Devices, 2nd Edition, Wiley, p. 849 (1981).

43 Chapter 3

Sub-Poissonian and Super-Poissonian

Shot Noise

3.1 Introduction

In thermionic cathodes, the onset of current self-quenching due to the space-charge built- up in the vacuum gap between cathode and anode leads to shot noise reduction in the anode current. This eﬀect has been observed experimentally and has been analyzed by several authors using analytical and computational techniques [1, 2, 3, 4, 5, 6]. In recently proposed cold cathode structures, the energy distribution of the injected electron beam into vacuum can be quite diﬀerent from the hemi-Maxwellian distribution typical of thermionic cathodes. We use an Ensemble Monte-Carlo (EMC) technique to study shot noise in planar cold cathodes in which the emission into vacuum is characterized by an average injection energy far in excess to the thermal energy typical of thermionic cathodes. Preliminary

44 Ensemble Monte-Carlo simulations of the space-charge eﬀects in the vacuum gap between anode and cathode of the structure shown in Fig.3.1 have been reported in refs [3,6].

The analysis, as mentioned in Chapter 1, was based on a mean-free path approach to include the effects of inelastic scattering in the CdS layer, following Fowler-Nordheim injection across the Metal/CdS interface. Space-charge effects in the vacuum gap can lead to the formation of a sizeable potential hump in front of the cathode which chops the injected beam and reduces the anode current, as shown in Fig.3.2 [3,6]. This is typically referred to as current self-quenching. In this regime, for the case of beam injection with a finite velocity much larger (two orders of magnitude) than the thermal velocity (107 cm/s) encountered in thermionic cathodes, the limiting value of the average anode current density is much larger than the Child-Langmuir limit [6,10,11,12,13]. In the current self-quenching regime,

Mumford and Cahay have shown in the past that the chopping action of the beam is sometime accompanied by a high-frequency (several tens of GHz) ﬂuctuations in the minimum of the electrostatic potential in the air gap [3]. These oscillations lead to large oscillations in the anode current density [3] which can strongly aﬀect the shot noise power spectrum will be studied in this chapter.

In ref. [6], it was also shown that, past the onset of current self-quenching, the energy spread of the electron beam collected at the anode is substantially narrower than for the beam injected from the cathode into vacuum due to the chopping of the beam by the oscillations of the electrostatic potential in front of the cathode. Under the approximation of ballistic transport through the CdS layer shown in Fig.3.1, the value of the cathode injection current at which the onset of current self-quenching occurs was found to be in very good agreement

45 Figure 3.1: Top: cross-section of the cold cathode proposed in refs [3,2] between two emitter ﬁngers. The substrate can easy be a metal or a heavily doped n-type InP substrate.

The emission window has a length L in the y direction.

h(E)

¡ ¡ ¡

¢¡¢¡¢ ¡ ¡ ¡

¢¡¢¡¢ E (LaS) ¡ ¡ ¡

¢¡¢¡¢ F

¡ ¡ ¡

¢¡¢¡¢

¡ ¡ ¡

¢¡¢¡¢

¡ ¡ ¡

¢¡¢¡¢

¡ ¡ ¡

¢¡¢¡¢

¡ ¡ ¡

¢¡¢¡¢

¡ ¡ ¡

¢¡¢¡¢ ¡ ¡ ¡

¢¡¢¡¢ D

ANODE

Figure 3.2: Illustration of the chopping action of the electrostatic potential energy proﬁle

in front of the LaS thin-ﬁlm as a result of the space-charge eﬀects in the cathode to anode

region. The energy distribution h(E) of the current injected into vacuum is shown under

the approximation of ballistic transport (solid line) and for the case of inelastic scattering in

the CdS layer (dashed line).

47 with the results of a theoretical hydrodynamic description of electron transport across the gap under the assumption of electron injection from the cathode at a ﬁnite velocity [12,13].

In the current self-quenching regime, the EMC simulations also predict that, when the eﬀects of inelastic scattering through the CdS layer are included, the transition to the regime of current self-quenching (i.e, drop in the anode current as a function of the injection current) is more gradual compared to the case of ballistic transport through the CdS layer. Hereafter, we extend the EMC simulations of [3,6] to study the power spectrum of the shot noise level in the anode current for the cold cathode depicted in Fig 3.1.

3.2 The Ensemble Monte-Carlo Approach

To begin a simulation of electron flow in the free space region of our emitter structure, the energy distribution of the injected carriers must be determined. In our case, we use a flux method to calculate this energy spectrum of the injected electron flux into the CdS following Fowler-Nordheim tunneling injection at the metal/CdS interface. More specifically, we calculate the energy dependence of the flux function h(E) defined as the total energy distribution for emitted electrons or the differential current density divided by the electronic charge, i.e,

+∞ JF N = e h(E)dE, (3.1) Z0 where JF N is the Fowler-Nordheim emission current and E is the total energy of the electron.

Since we are only considering ballistic transport through the CdS and LaS regions along with the assumption of prefect transmission (equal to unity for all electrons) at the CdS/LaS

48 and LaS/vacuum interfaces, the ﬂux h(E) can then provide an electron energy distribution.

This distribution is used to provide the initial electron energy to simulate injection into and through the space charge region between the LaS thin ﬁlm and the anode.

For the case of Fowler-Nordheim injection assumed here, JF N is given by [14,15]

4πme +∞ E JF N = 3 f(E)D(Ex)dEx, (3.2) h Z0 Z0 where f(E) is the Fermi distribution function and D(Ex) is the transmission coeﬃcient for an electron with an energy Ex normal to the direction of motion.

Following the derivation of Good and Muller [15], including the classical lowering of the barrier by the image charge potential (Schottky eﬀect) at the metal/CdS interface, the transmission coeﬃcient D(Ex) can be calculated exactly in the WKB approximation as

E E D(E ) = exp( c + x − F ). (3.3) x − d

This expression is valid for electrons with energies near the Fermi energy EF in the metallic contact. As shown explicitly by Good and Muller, the parameters c and d are both dependent on the electric ﬁeld strength F outside the metallic contact (in the CdS layer in our case) and are given by 2π√2m∆3 c = v(y ), (3.4) 3heF 0 heF 1 d = 2 . (3.5) 4π√2m∆ t (y0)

Here ∆ is the barrier height at the metal/CdS interface before including the Schottky eﬀect.

The functions v(y) and t2(y) have been tabulated by Good and Muller [15]. The quantity yo is the value of the variable eF 1 y0 = , (3.6) s4π² V E 0 o − x 49 calculated for Ex equal to the Fermi energy EF in the metallic contact; Vo is the sum of the barrier height ∆ and EF in eV.

The energy distribution ﬂux h(E) can be found after substituting Eq.(3.3) into Eq. (3.2) which leads to 4πmd E E h(E) = exp( c + x − F )f(E) (3.7) h3 − d

While deriving Eq.(3.3), Good and Muller have shown that this expression is only valid if the following conditions are satisﬁed;

y < 1, (3.8) and

d > kBT. (3.9)

A more accurate calculation of the energy distribution should include the variation in the electron eﬀective mass as they move across the interface from one material to another, but we expect the resulting energy distribution for the emitted electrons to be qualitatively similar to Eq.(3.7).

To determine the limits of applicability of the distribution function h(E) in Eq.(3.7), we use the two constraints (3.8) and (3.9) while replacing y by y0 (i.e, setting Ex = EF in Eq.

(3.7)). The latter approximation is reasonable since, at room temperature, the majority of the electrons emitted from the metallic contact will have a total energy near the Fermi level.

Setting Ex = EF in Eq.(3.6), Eq.(3.8) can be rewritten as

eF < ∆, (3.10) s4π²0

50 or equivalently,

3.7954 10−4√F /∆ < 1, (3.11) × with F and ∆ expressed in V/cm and eV, respectively.

Since t(y) is a slowly varying function ranging from 1 to 1.1 as y increases from 0 to 1

[15], Eq.(3.9) can be approximated as follows

−9 F 9.3055 10 > kBT, (3.12) × √∆ with F and ∆ expressed in V/cm and eV, respectively.

The energy distribution function h(E) is only valid for a range of applied electric ﬁeld

[Fmin, Fmax] whose lower and upper values are determined using Eqs.(3.11) and (3.12), respectively. Constructing a plot using the conditions (3.11) and (3.12) provides a graphical visualization of the region in which h(E) is a valid energy distribution function. The range of electric ﬁeld over which Eq.(3.7) is applicable increases for a barrier height at the metal/CdS interface much larger than ∆∗. In the numerical examples below, we will consider cold cathodes with Au/CdS interface for which ∆ is equal to .78 eV [16].

Once the injected energy distribution h(E) has been found as a function of the applied bias across the CdS layer, we use an EMC description of electron transport between the

LaS thin film and the anode. The EMC approach was chosen to be able to account for dynamical effects linked to space-charge flow as well as random fluctuations linked to the effects of shot noise in the metallic contact [17]. Our approach here is similar to the one developed over 40 years ago by Tien and Moshman to study the minimum noise figure for microwave beam-type amplifiers [19].

51 The simulation is conducted as follows. First, the emitted current density is determined as a function of applied bias across the CdS layer for a specific set of cathode structural parameters using semi-analytical treatment. Second, the bias dependence of the parameters yo and d are calculated to assure the range over which conditions (3.11) and (3.12) are satisfied for a valid use of the flux energy distribution h(E). Third, the energy spectrum of the injected carriers is determined and normalized using Eq. (3.7) to provide an initial injection energy for the electrons. Finally, the EMC simulation of the transport through the free space interelectrode region between the LaS thin film and the anode is performed.

The EMC simulator evaluates the movement of carriers during a small interval of time

(selected to be 50 fs in our simulations) after which the conduction band energy proﬁle between the LaS emitting surface and the anode thin ﬁlm is calculated through a solution of the Poisson equation. The boundary conditions used to solve the Poisson equation are the predetermined, and assumed static, values of the electrostatic potential applied at the

LaS thin film and anode contacts. Carriers which are reflected towards the cathode (LaS thin film) as a result of the conduction band maximum in front of the cathode and cross the LaS surface with a negative velocity are automatically removed from the simulation.

Similarly, electrons crossing the plane of the anode are also assumed to be absorbed and therefore removed from the simulation.

3.3 The Eﬀects of Shot Noise

Shot noise in electronic devices is due to the discrete nature of carriers. The aggregate of emitted electrons contain current ﬂuctuations, due to the random emission times, and

52 velocity (and therefore energy) ﬂuctuations, since the number of electrons in any small range of energies within the distribution also ﬂuctuates as a consequence of the random emission times. By random emission, we mean simply that the emission of every electron is completely independent of every other electron emission. We describe such a random process by assuming that the number of electrons emitted per unit time is given by a Poisson distribution (Schottky model [1,2,20,21,22,23]) and their corresponding emission energies is given by the energy distribution density h(E) described. Therefore, set of random numbers must be generated to simulate the cathode emission and which possess all the statistical features of shot noise. The average number of electrons emitted in a time interval ∆T is nav. This number is computed for a given emission current and emission window area.

The probability that s electrons will be emitted in this time interval follows the Poisson distribution e−nns f(s) = , (3.13) s! where s = nav∆T . The cumulative sum of this distribution, F (s) = Σf(s), is the probability that s or fewer electrons are emitted in a time interval ∆T . Typically, nav = 500 is the numerical simulations described below.

To determine the number of electrons emitted in a time interval [ t1 , t1 + ∆T ], we generate a random number α1 and compare it with the values of cumulative distribution function F (s). If α1 is such that

F (m 1) < α < F (m), (3.14) − 1 the number of electrons emitted is m.

53 Next, we generate s additional pseudo-random numbers. If the random numbers are

β1, β2, β3, ... , the exact times at which electrons are emitted are t1 + β1∆T , t1 + β2∆T , t1 +β3∆T, ..., respectively. In the actual computation, instead of taking the electrons emitted at times t1 + βi∆T (i = 1, 2, 3, ...) as described above, we assume for simplicity that all were emitted at the same time but a varying small distance in front of the cathode. Speciﬁcally,

th the i electron is assumed to be emitted at time t1 but at a distance xi = βi∆T vi where vi is the initial velocity of the ith electron. The error involved in this approximation is negligible if

∆T is chosen suﬃciently small. The initial velocity is determined using a rejection technique starting with the energy distribution density h(E) given above. For simplicity, we assume zero velocity components in the direction perpendicular to the direction of current ﬂow. This is a reasonable approximation since the average kinetic energy of carriers in the plane parallel to the cathode surface is around the thermal energy kBT which is much smaller than the longitudinal kinetic energy of injected carriers in the numerical simulations described below.

Under a ﬁxed applied voltage Vgap, the anode current density can be calculated using

Ramo’s theorem(see appendix B)

q N(t) I(t) = v (t), (3.15) L i Xi=1 th where vi(t) is the instantaneous velocity component along the ﬁeld direction of the i particle and N(t) is the total number of particles in the vacuum gap per unit area. This expression contains the contributions of both the particles current and displacement current [1,2,20].

Since this number can be quite large in actual devices, it is typically reduced to more reasonable size ( < 50, 000 ) by attributing an eﬀective charge to each electron much larger than the electronic charge. This is equivalent to replacing q by qQeff in Eq.(3.15). The

54 number N(t) then represents the number of superparticles being simulated in the simulations domain. This approach is physically sounds as long as physical quantities of interest are independent of the number of superparticles used in the simulations. This technique has been widely used in the simulations of space-charge eﬀects of thermionic cathodes and more recently in the simulations of submicron semiconducting structures [24,25,26].

Space-charge eﬀects in the vacuum gap can lead to the formation of a sizeable potential hump in front of the cathode which chops the injected beam and reduces the anode current

(see Fig.3.2). This is typically referred to as current self-quenching. In this regime, for the case of beam injection with a finite velocity much larger than the thermal velocity encountered in thermionic cathodes, the limiting value of the average anode current density is then much larger than the Child-Langmuir limit [6,12,13]. In the current self-quenching regime, we have shown in the past that the chopping action of the beam is sometime accompanied by a high-frequency (several tens of GHz) fluctuations in the minimum of the electrostatic potential in the air gap [3]. These oscillations lead to large oscillations in the anode current density [3] which can strongly affect the shot noise power spectrum, as will be shown next.

3.4 Numerical Examples

Recently, we used the EMC described above to study current self-quenching eﬀects in a

Metal/CdS/ LaS cathode while including the eﬀects of inelastic scattering in the CdS layer

[6]. Hereafter, we investigate the eﬀects of shot noise in a speciﬁc Au/CdS/LaS cold cathode with the parameters listed in the table below.

55 In all simulations, the spacing between the LaS thin ﬁlm and anode is set equal to 15

Material Au i CdS LaS − Lattice Thickness (A)˚ optional 75.0 24.6

Lattice Constant (A˚) 4.04 5.83 5.85

Work function (eV) 4.3 4.2 1.14

Bandgap (eV) 2.5

# of free electrons (1022cm−3) (5.86) — 1.99

Electron Mass (m0) 1.0 0.14 1.0

Table 3.1: Material Parameters of the Cold Cathode

µm. We consider a cold cathode with a Au/CdS interface for which ∆ is equal to 0.78 eV [2].

For this cathode, the energy distribution of the injected electron proﬁle h(E) was calculated explicitly in ref. [6]. The simulation domain is divided into 300 bins of 500 A˚ each. This is the length traveled by an electron with a velocity of 108cm/s (a typical velocity of electrons in the injected beam [6]) over a period of 50 femtoseconds, the time interval at which we upgrade the Poisson solver. To make the EMC simulator more eﬃcient, the bin size was selected so that electrons could not cross several bins between solutions of the Poisson solver.

To calculate the steady-state value of the anode current, the EMC simulations are run for

20,000 time steps (1 ns). When current self-quenching occurs, the anode current can contain an AC component of large frequency (several tens of GHz) equal to the chopping frequency of the electrostatic potential maximum in front of the cathode [3], the anode current density

56 must be calculated while averaging over many oscillations of the AC signal.

3.5 Shot Noise Power Spectrum

The expression for the current density across the vacuum gap given in Eq.(3.15) is ﬁrst rewritten as follows qQ I(t) = eff N(t)v(t), (3.16) L where the average velocity (over superparticles) is given by

1 N(t) v(t) = v (t). (3.17) N(t) i Xi=1 The current density ﬂuctuations can then be calculated as

δI(t) = I(t) < I >= (qQ /L)[N(t)v(t) < N(t)v(t) >], (3.18) − eff − where the brackets denote a time average taken past the transient regime following the application of the external potential Vgap. The autocorrelation function of the current-density

ﬂuctuations is then calculated as follows

0 0 CI (t) =< δI(t ) δI(t + t) >, (3.19)

0 where the brackets stand for a time average taken over t . The spectral density of the current-density ﬂuctuations are then calculated as the Fourier transform of the autocorrelation function

+∞ SI (f) = 2 exp(2iπft)CI (t)dt. (3.20) −∞ Z It is well known from a study of shot noise power spectrum in thermionic cathodes that, when the ﬂux of carriers through the gap are uncorrelated, the shot noise spectral density

57 at low frequency (i.e, small compared to the inverse transit time through the vacuum gap) is given by [1,2]

SI (0) = 2qIdc. (3.21)

This is the Schottky (or full shot noise) result or full shot noise result corresponding to a

Poisson statistics description of the injected carriers [1]. For the case of space-charge eﬀects in thermionic cathodes, it is also well-known that, in the self-quenching regime, the eﬀects of space-charge leads to some degree of correlation among carriers in the vacuum gap which translates in a reduction of the shot noise value at low frequency below the Schottky result.

For thermionic cathodes, this shot noise reduction has been analyzed using various analytical treatments [1,2,3,4,5,6] and Monte-Carlo simulations [19,21,26,27]. The shot-noise reduction phenomenon is often characterized by the addition of a prefactor γ on the righthand side of Eq.(3.21) (smaller than unity) called the suppression factor. This shot noise suppression has also been demonstrated more recently in mesoscopic devices [28]. In this case, sources of correlation among carriers leading to shot-noise reduction include (1) statistical correlations due to the Pauli-exclusion principle, (2) short-range Coulomb interactions (electron-electron scattering), and (3) long-range Coulomb interaction due to a self-consistent treatment of the electrostatic potential.

Hereafter, we study the effects of space-charge on the power spectral density of shot noise for the Au/CdS/LaS considered above. It should be stressed that the calculation of current-density autocorrelation function are often described as noisy [24]. This means that the time average in Eq.(3.19) must be taken over a sufficiently long time using small time steps to include the effects of space-charge and any relevant dynamical effects resulting

58 from a self-consistent solution of Poisson’s equation. The reliability of the calculation of the autocorrelation function must be tested by increasing the overall time of the EMC simulation but also by checking for its independence on the time step used in the self-consistent solution of the Poisson solver.

In order to investigate the importance of space-charge effects on the shot noise spectral density, the latter was calculated including a fully self-consistent solution of Poisson’s equation but also assuming a linear potential drop across the gap. The latter correspond to a non self-consistent calculation where the electric field in the vacuum gap is assumed to be equal to the externally applied electric field. Hereafter the two types of simulations are referred to as self-consistent and linear case, respectively. In this second case, the spectral density SI (f) should have all the characteristics of a white noise process up to a frequency corresponding to the inverse transit time of carriers across the vacuum gap, above which

SI (f) should decrease rapidly [29]. At low frequency (large enough for the effects of Flicker noise to be negligible), the importance of space-charge effects can then be assessed by taking the ratio of the shot noise spectral density with and without the effect of the Poisson solver included. The calculation of the autocorrelation function of the anode current density was performed starting with Eq.(3.21). All simulations discussed hereafter were carried for an injection current density of 153 A/cm2 at the cathode/vacuum interface, assuming an electron mean free path of 300 A˚ in the CdS layer. The bias across the gap was set equal to 1V or 5V. For the structure with the parameters listed above, it was shown in ref. [6] that for an injected current density of 153 A/cm2 the cathode is in the regime of current self-quenching when both a potential of 1V and 5V is applied across the gap. However, the shot noise power

59 spectrum of the anode current density is quite diﬀerent for these two applied biases across the gap, as will be shown below.

0 To calculate the time average value (sum over t in Eq.(3.21)), we used the numerical results of the EMC simulations for the time interval extending from 8000 (0.4 ns) to 40000

(2ns) timesteps. The average over time was only carried out for time in excess to 8000 time steps (0.4 ns) to remove the transient regime following the application of the bias across the gap. The resulting shot noise power spectra SI (f) for the linear and self-consistent case are displayed in Figures 3.3 and 3.4. These results were obtained while averaging over 5 simulations corresponding to ﬁve diﬀerent choices of initial seeds in the EMC simulations.

This averaging was needed to reproduce the expected white noise characteristics of SI (f) at low frequency in agreement with a Schottky (Poissonian) injection of carriers in the vacuum gap. This averaging would correspond to the average over different time intervals for a specific cathode or over different cathodes which would need to be performed experimentally to confirm the theoretical results. The average was only carried out over five different initial seeds because of the large amount of data generated in the EMC simulations.

From Figures 3.3 and 3.4, we note that the shot noise spectra calculated in the linear case are fairly smooth function of frequency, being independent of frequency for low frequency and then suddenly dropping at frequency above 70 GHz. The latter corresponds to the calculated inverse of the average transit time of carriers across the gap for the assumed injected distribution of carriers. For the self-consistent case, there is a large peak in the shot-noise spectra which occurs around 72 GHz and 62 GHz for the case of a voltage across the vacuum gap equal to 1V and 5V, respectively. Simultaneously, there are also smaller

60 6 10 V = 5V GAP

4 10 FULL POISSON ) −4 s cm 2

A 2 10 LINEAR POISSON −14 (f) (5*10 I

S 0 10

−2 10 8 9 10 11 12 13 10 10 10 10 10 10 Frequency (Hz)

Figure 3.3: Frequency dependence of the spectral density SI (f) of the current density fluctuations for the Au/CdS/LaS cold cathode with the parameters listed in Table.3.1. The curves labeled LINEAR and FULL POISSON correspond to EMC simulations using a linear potential drop across the vacuum gap (i.e, external electric field) or a full self-consistent solution of Poisson’s equation, respectively. The injected current density is equal to 153 A/cm2 and the injected energy profile across the Au/CdS/LaS cathode was calculated assuming ballistic transport across the CdS and LaS layers [6]. The anode to cathode spacing is equal to 15 µm and the bias across the gap Vgap = 5V.

61 3 10 V = 1V GAP

2 10 LINEAR POISSON ) −4

1 10 s cm 2

A FULL POISSON −14

0 10 (f) (5*10 I S

−1 10

−2 10 8 9 10 11 12 13 10 10 10 10 10 10 Frequency (Hz)

Figure 3.4: Same as Fig.3.3 for a bias across the vacuum gap Vgap = 1V.

62 peaks in the spectra at frequencies equal to multiples of the ﬁrst peak. These harmonics are due to the non-sinusoidal waveform of the autocorrelation functions. The latter are displayed in Figures 3.5 and 3.6, respectively.

The consistency of the numerical simulations can be addressed through the power spectrum relation [1,2] ∞ 2 < δI >= SI (f)df. (3.22) Z0 which can be easily derived by inverting Eq.(94) and setting t=0.

Referring to Figures 3.3 and 3.4, the integral of SI (f) is indeed larger in the linear case than in the self-consistent case for Vgap = 1V. This is in agreement with the fact that the time average of the current density ﬂuctuations < δI 2 > is larger in the linear than in the self-consistent case (See Fig.3.7). Hence, shot-noise reduction is observed for these biasing conditions. The shot noise reduction factor is found to be about 0.1. For Vgap = 5V, shot- noise enhancement is observed because < δI 2 > is much larger for the self-consistent case than for the linear case (See Fig.3.7), in agreement with the larger area under the curve for SI (f) for the former. The shot noise enhancement factor is found to be around 100 in this case. To understand the origin of the large value of < δI 2 > in the case of shot noise enhancement, the amplitude of oscillations of the minimum in electrostatic potential in the vacuum gap is plotted in Fig.3.8. This ﬁgure clearly shows the high-frequency oscillations in the electrostatic potential minimum responsible for the peak in the spectral densities

SI (f). However, the amplitude of the oscillations of the potential minimum are quite large for the case where Vgap = 5V, compared to the case Vgap = 1V. More importantly, Fig.3.9 indicates that the location of the minimum oscillates over a range of about one micron for

63 80 VGAP = 5V 60

) 40 −4 cm 2 20

(t) (A 0 I −20 C

−40

−60 1 1.2 1.4 1.6 1.8 2.8 4 timesteps(*10 )

Figure 3.5: Time dependence of the autocorrelation function CI (t) of the current density

ﬂuctuations for a Au/CdS/LaS cathode with the parameters in Table.3.1. The cathode to anode separation is equal to 15 µm. The bias across the cathode to anode spacing is equal to 5V.The curve of lower amplitude has been multiplied by a factor 103 for clarity. It corresponds to the case of a linear potential drop across the vacuum gap.

64 0.1

VGAP = 1V 0.05 ) −4 cm 2 0 (t)(A I C −0.05

−0.1 1 1.2 1.4 1.6 1.8 2.0 4 timesteps(*10 ) Figure 3.6: Same as Fig.3.5 for a bias across the cathode to anode spacing equal to 1V.

The curve of lower amplitude corresponds to the case of a linear potential drop across the vacuum gap.

65 80

LINEAR ) 2 60

VGAP = 5V

20 Ramo Current (A/cm

VGAP = 1V 0 1 1.2 1.4 1.6 1.8 2.0 4 timesteps(*10 )

Figure 3.7: Time dependence of the Ramo current for a Au/CdS/LaS cathode with the parameters listed in Table.3.1 for two diﬀerent biases across the vacuum gap. The injected current density is equal to 153 A/cm2. The cathode to anode separation is equal to 15 µm.

The curve labeled Linear shows the time-dependence of the current in the linear case, i.e, for a linear potential drop across the gap. This curve has been shifted down by 80 A/cm2 from its calculated value.

66 0

−2

V = 5V (V) −4 GAP min V

−6

VGAP = 1V −8 0 0.4 0.8 1.2 1.6 2.0 4 timesteps(*10 ) Figure 3.8: Time dependence of the potential minimum in front of the cathode for the self- consistent EMC simulations for a Au/CdS/LaS cold cathode with the parameters listed in

Table.3.1. The cathode to anode separation is equal to 15 µm. The curves are labeled with the value of the bias across the gap. The Vgap = 1V curve has been shifted down by -4 V for clarity. The amplitudes of oscillation of the electrostatic potential minimum are much larger for Vgap = 5V.

67 Vgap = 5V whereas the location of the minimum is pretty much constant for Vgap = 1V.

For the latter, the small glitches seen in the location of the potential minimum close to steady-state correspond to a ﬂuctuation of the minimum location over a bin size (500 A˚).

The large oscillation in the location of the minimum of the electrostatic potential for Vgap

= 5V is ultimately responsible for a large oscillation in the number of superparticles in the simulation domain, leading the shot noise enhancement. To further illustrate the diﬀerence between the Vgap = 1 and 5V, we plot in Figures 3.10 and 3.11 the time dependence of the average (over the superparticles) of the electron kinetic energy and potential energy in the vacuum gap.

Figure 3.10 shows clearly that, past the transient, a steady-state is reached where the average (over the particles) of the total energy of the carriers stay constant but there is a swapping between kinetic and potential energy of the electrons in the vacuum gap. The average potential and kinetic energies are out of phase and vary with the same frequency as the anode current. This is a clear evidence of a plasma oscillation in front of the cathode.

Past the transient, the average value of the kinetic energy of the carriers is equal to 1.4 eV and the peak-to-peak amplitude of the average kinetic energy is about 2 eV, or 40 times the thermal energy kBT (T = 300 K). For Vgap = 1 V, the amplitudes of the oscillations in the average kinetic and potential energies are much smaller. In that case, the time average of the average kinetic energy of the particles in the vacuum gap is 0.21 eV and the maximum peak-to-peak amplitude of the average kinetic energy is only 0.045 eV which is less than 2 kBT (T = 300 K). The enhanced shot-noise in the power spectrum of the anode current is therefore characteristic of sustained plasma oscillations in the vacuum gap with substantial

68 4

V = 5 V 3 GAP

m) 2 µ (

min 1 X

V = 1 V 0 GAP

−1 0 0.4 0.8 1.2 1.6 2.0 4 timesteps(*10 ) Figure 3.9: Time dependence of the location of the potential minimum in front of a

Au/CdS/LaS cold cathode in the case of a self-consistent EMC simulation. The cold cathode has the parameters listed in Table.3.1. The cathode to anode separation is equal to 15

µm. The curves are labeled with the value of the bias across the gap. The Vgap = 1V curve has been shifted down by 2 µm for clarity. The oscillations of the location of the potential minimum in the vacuum gap are quite large for Vgap= 5V.

69 20

VGAP = 5 V 15

Potential Energy 10 Kinetic Energy Energy (eV) 5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 4 Timesteps (*10 ) Figure 3.10: Time dependence of the average value of the kinetic and potential energy of the electrons within the vacuum gap. The potential across the gap is Vgap = 5V . The cathode parameters are listed in Table.3.1. The injected current density is equal to 153 A/cm2. Past the transient, plasma oscillations are clearly seen with an exchange between the average kinetic and potential energy of the carriers at a frequency equal to the the one observed in the oscillations of the anode current (see Fig.3.7).

70 6

VGAP = 1 V 4

Kinetic Energy 2

Energy (eV) 0

Potential Energy −2

0 0.2 0.4 0.6 0.8 1.0 4 Timesteps (*10 )

Figure 3.11: Same as Fig.3.10 for Vgap = 1V . In this case, no clear plasma oscillation can be seen.

71 amplitude (along the energy and spatial axes) leading to an appreciable oscillation in the number of particles in the vacuum gap. The latter is ultimately responsible for the observed shot noise enhancement.

3.6 Conclusions

Our EMC simulations have shown that the highly non-linear nature of the feedback mechanism between current transport and electrostatics in the vacuum gap of a Metal/CdS/LaS planar cold cathode (shown in Fig.3.2) can lead to a large amplitude in the energy potential hump and its location in front of the cathode. More speciﬁcally, it was shown that there exist biasing conditions for which shot noise enhancement compared to the Schottky value can be observed. In practical applications, this regime of shot noise enhancement should be avoided since it would be detrimental to the high-frequency performance of the cold cathode.

The phenomenon of shot noise enhancement for non-cathodic shot noise in a spiraling electron beam of, e.g., a gyroklystron, was the subject of an intensive analysis by Anton- sen, et al. [30], who found that noise reduction by shielding of the electron charge is in competition with ampliﬁcation of unstable collective modes for an electron beam in an inho- mogeous magnetic ﬁeld. Elsewhere, experimental evidence and theoretical analysis of shot noise enhancement in nanoscale structures such as resonant tunneling devices [31] and Al-

GaAs/GaAs semiconductor heterostructures [32] have also been reported recently. Various physical mechanisms leading to shot noise enhancement in mesoscopic systems operating in the ballistic regime have been summarized recently by Blanter and Buttik¨ er [28].

We have found that both shot-noise reduction and enhancement can be observed in

72 the same device depending on the biasing conditions. This possibility of transition from sub-Poissonian to super-Poissonian shot noise in a speciﬁc device was reported recently by

Blanter and Buttik¨ er for the case of double barrier resonant tunneling structures [33].

The shot noise enhancement found here for a speciﬁc cathode could be a more common phenomenon in cold cathode design characterized by an energy distribution of the injected beam far from the thermal equilibrium conditions typical of thermionic cathodes.

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77 Chapter 4

Conclusions and Future Work

4.1 Conclusions and Future Work

In this thesis, we have analyzed the possibility of using the InP/CdS/LaS cold cathode proposed in [1] for prebunching the electron beam if the cathode is to be used for high power high frequency applications. This electron prebunching would be accomplished through the use of a small AC applied voltage across the thin CdS layer in addition to the DC bias.

In this thesis, we have developed a first order small signal equivalent circuit of planar cold cathodes making use of semimetallic thin films at their surface to achieve negative electron affinity. While neglecting the phase-delay linked to the finite time of propagation of the electron beam across the vacuum region, we have derived an analytical expression for the unity current frequency of the cathode as a function of the load resistance. The model was applied for the case of a InP/CdS/LaS cold cathode we have proposed recently [1,2]. For a cathode with the parameters listed in Table II, the effects of DC current crowding are found

78 to be negligible. The effects of AC current crowding are negligible for a cathode operated at sufficiently low bias (1.55 V) across the CdS layer. AC current crowding is more important for a bias around 1.7V . This could have a deleterious effect on the uniformity of the emitted electron beam and counteract the advantage of using a prebunched electron beam when the cathode is operated at too high a frequency. More specifically, if the InP/CdS/LaS cold cathode analyzed here is to be used for effective prebunching of an electron beam to be used in a klystron operating in the X-band (10 GHz), our analysis indicate that the emission window width would need to be reduced below 5µm in order to avoid the deleterious effects of AC current crowd

In chapter 3, we have used Ensemble Monte Carlo simulations to study the importance of space-charge effects on the shot noise power spectrum of the anode current fluctuations for a planar Metal/CdS/LaS cold cathode [1, 3] in which the energy distribution of the injected electron beam at the cathode is very different from the hemi-Maxwellian distribution characteristic of thermionic cathodes [3]. we find that the injected electron beam has an average velocity of the order of 108cm/s which is one order of magnitude larger than the thermal velocity of carriers at room temperature.

For a ﬁxed bias across the vacuum gap, the shot noise power spectrum was calculated as a function of the injected current density from the cathode. When evaluated at a frequency much larger than the range at which the eﬀects of Flicker noise cannot be neglected but much smaller than the inverse of the average transit time of carriers across the space-charge region, the shot noise power spectrum is found to agree with the Schottky result for Jem below the threshold for the onset of current self-quenching. For values of Jem within the

79 transition regime between negligible space-charge and saturation mode of operation of the cathode, the shot noise power is below the Schotty value (Shot noise reduction). Far into the saturation mode, i.e when self-quenching eﬀects due to space-charge are dominant, the shot noise power spectrum is found to be far in excess to the Schottky result (Shot noise enhancement). The onset of shot noise enhancement is related to large amplitude (plasma) oscillations of the electrostatic potential energy proﬁle in front of the cathode not only along the energy but also along the spatial axis.

The shot noise enhancement found here for a speciﬁc planar cold cathode operated at large injection current densities well within the regime of current self-quenching could be a more common phenomenon for cold cathodes characterized by an energy distribution of the injected beam far from the thermal equilibrium conditions typical of thermionic cathodes.

This shot noise enhancement could have a detrimental eﬀect on the practical use of cold cathodes for high-power high-frequency microwave sources.

4.2 Suggestions for future work

As outlined in the introduction, for the planar cold cathodes using rare-earth monosulfides to achieve negative electron affinity at the vacuum interface, operating conditions must be carefully selected to avoid DC current crowding [1,4] and self-heating effects [5] in the cathode. In the past [3,4,6], we also have shown that space-charge effects in the vacuum gap can have a drastic influence on the average (DC) anode current. Except for the analysis of ref.

[4] in which the interplay of current crowding and space-charge eﬀects were analyzed, there is a need to build a comprehensive model of the cathode to include all the eﬀects mentioned

80 above self-consistently since they weill occur simultaneously under speciﬁc biasing conditions.

For instance, a thorough investigation of the eﬀects of shot noise would require an extension of the one-dimensionla Monte Carlo analysis of Chapter 3 to at least two dimensions to be able to include the eﬀects of DC current crowding.

Some additional work could exame the theoretical possibility to integrate the rare earth cold cathode materials with hetero-junction bipolar transistors. For high current beam generation, the cold cathode structure must be operated at a high current density (in excess of 103 A/cm2). To achieve this current density, the cold cathode structure must be provided with a high current input since the structure does not produce gain on it own. Heterojunction bipolar transistors are attractive for this application beacuse of their high current drive and high frequency capabilities. Future work can be done to achieve a monolothic integration of the rare-earth cold cathode materials with hetero-junction bipolar transistors for possible use as wideband microwave ampliﬁers in vacuum microelectronics.

4.3 List of Journal Publications, Conference Proceed-

ings Papers and Conference Presentations

The results presented in chapters 2 and 3 have been recently submitted for publication. One manuscript was published in the proceedings of a symposium on cold cathodes held as part of the 201st Meeting of The Electrochemical Society in Philadelphia in May 2002. Some of the results of this thesis were also presented at three conferences.

1. R. Krishnan and M. Cahay, ”Transition from Sub-Poissonian to Super-Poissonian Shot

81 Noise in Planar Cold Cathodes”, submitted to Journal of Vacuum Science and Tech-

nology B, October 2002.

2. R. Krishnan and M. Cahay, ”Electron Beam Prebunching in Planar Cold Cathodes

With Surface Current Currying Thin Films”, in preparation, to be sumitted to Journal

of Vacuum Science and Technology B.

3. R. Krishnan, M. Cahay, and K.L. Jensen, ”Sub-Poissonian and Super-Poissonian Shot

Noise in Planar Cold Cathodes” Proceedings of the Second International Symposium on

Cold Cathodes, 201st Meeting of The Electrochemical Society, Vol. 2002-18, Philadel-

phia, May 12-17, 2002.

4. R. Krishnan and M. Cahay, ”Transition from Sub-Poissonian to Super-Poissonian Shot

Noise in Planar Cold Cathodes”, The Ninth Van der Ziel Symposium on quantum 1/f

ﬂuctuations and low frequency noise, School of Engineering, Virginia Commonwealth

University, Richmond, VA August 2-3, 2002.

5. R.Krishnan, M.Cahay and K.L.Jensen, ”Sub-Poissonian and Super-Poissonian Shot

Noise in Planar Cold Cathodes”, 201st meeting of the Electrochemical Society, Philadel-

phia, PA, May 17-22, 2002.

6. R.Krishnan and M.Cahay, ”Shot-Noise Eﬀects in Planar Cold Cathodes”, Annual

American Physical Society Meeting, March 18-22, 2002.

82 Bibliography

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84 Appendix A

Appendix

A.1 Ramo’s Current Expression

In Chapter 3, we use Ramo’s theorem (Eq.(3.15)) to calculate the anode current density under the assumption of a constant bias across the vacuum gap between anode and cathode.

Hereafter, we derive Eq.(3.15) starting with the general expression for the current density

δEx(x, t) Jx = e vxf(v, x, t)d~v + ε0 , (A.1) Z δt where the first term is the conduction current and the second term represents the displacement current due to the fluctuations in the electric field. In our case, these fluctuations are due to the self-consistent solution of Poisson equations and Newton’s equation to describe the motion of carriers using the Ensemble Monte-Carlo simulations. As we have seen in chapter

3, this self-consistent solution can lead to substantial plasma oscillations in the vacuum gap, hence non negligible contributions from the displacement current. In Eq.(A1), f(x,~v,t) is the

85 distribution of carriers which obey Boltzmann equation. In our case, that distribution is determined using EMC simulations. Since we neglect electron-electron interaction, are description of the motion of carriers is purely one-dimensional, i.e, the electric ﬁeld is assumed to have a non-zero component only in the direction x of current ﬂow from cathode to anode, we get

q D δ J(t) = dx d~vvxf(x,~v, t) + ε0 Vgap, (A.2) D Z0 Z δt where Vgap, in our case, is the constant applied bias between cathode and anode, and q =

1.6 10−19 Coulombs. Hence, the anode current density is given by, ∗

1 N(t) J(t) = q v (t). (A.3) L i Xi=1

Sice the number N(t) can be quite large for practical cathodes, each electron must be replaced by a superparticle with an eﬀective charge to keep the number of superparticles N(t) in the

EMC simulations reasonably low (a few tens of thousands). Ofcourse, physical quantities of interest, like the shot noise power spectrum of the anode current, should not depend on the selection of the eﬀective charge of carriers, a feature which must be checked numerically to test the accuracy of the EMC simulations.