ELECTRON EMISSION DUE TO THE SCHOTTKY EFFECT

3. The Design of the X-ray Tube and the MgO Cold Cathode

3.2.3. ELECTRON EMISSION DUE TO THE SCHOTTKY EFFECT

If the phenomenon of electron emission is due to the Schottky [3] effect, then, according to Shroff [2], the Richardson-Dushman equation (equation 3.2.1) can be used, so that the thermoelectronic functioning of the cathode at temperature T under conditions of saturation is taken into account.

Under such circumstances it follows, that once the cathode is exposed to an exterior electric field E, the emission current then becomes proportional to the temperature of the emitter [2]. Thermal electron emission from all types of cathodes can be adequately described by the Richardson- Dushman equation:

(3.2.1)

The value of the work function ct> varies approximately linearly with temperature according to the relation

(3.2.2)

where <1> _o

=

work function at T

=

0 oK and a

=

d<1> / dT , the temperature coefficient of the work function. According to W H. Kohl [19], equations 3.2.l and 3.2.2 above can be combined into one expreSSIon:

(3.2.3)

In equation 3.2.3 the term Ao' exp[ - : ] is also referred to as the Richardson or effilSSlon constant of a material. While its value lies typically between 30x 104

and 70x 104 A m-2

deg"2 for pure metal surfaces [19], for oxide cathodes and thin films the emission constant assumes a value close to 1x104 A m-2

deg"2 and even less according to W H. Kohl [19]. For the following calculation it will therefore be assumed, that the MgO film of the cold cathode will have an emission constant of 1x104 A m-2

deg"2. Hence by substituting Ao . exp[ - : ] ⁼ 1x104 A m-2 deg-2, T

=

293 OK and <1>0

=

3.1 eV, the work function of MgO [1], into equation 3.2.3 the following value for the saturated thermal emission current density j s is obtained:

( 0)2 ( 4 -2 -2) [-3.1eV x 1.602 x 10-¹⁹ C

1

j s

=

^{293 K . 1 x lOA m deg . exp 23 0 1 0}

1.38x10- J K- ·293 K

~ 1.336x 10⁴⁷ A m-2

Using a disk shaped cathode which has an outside diameter of 1.4 cm and an inner diameter of 0.6 cm and hence a total surface area of A

=

1.256 cm^2, the thermal saturation current is can then be calculated by multiplying the emission area A of the cathode with is:

(3.2.4)

1.678 x 10-⁵¹ A

=

1.678 x 10-⁴⁸ rnA

Clearly this current is practically negligible and it will be shown in the following sections that autoelectronic emission from the MgO cathode is due to effects other than the Schottky effect.

Table 3.2.2 presents further values for is at still higher temperatures. These values were computed using the same approach as above for a MgO cold cathode with an emission surface area of 1.256 cm2

. Even at temperatures as high as 1100 OK the emission current is only 9.41lxlO-5 rnA.

So far the above calculations were performed with respect to a single MgO layer without taking into account the entire cathode structure i.e. the Ni-NiO-MgO structure. With the work function of Ni equal to 5 eV and MgO equal to 3.1 eV, it is obvious that the work function of the Ni-NiO- MgO structure will have a value between 3.l eV and 5 eV. By SUbstituting values for the work

function greater than 3.1 eV into equation 3.2.3 the values for

is

and hence is will be even lower than those obtained in Table 3.2.2 at the corresponding temperatures. The quantities obtained in Table 3.2.2 may therefore be considered the maximum currents, that a MgO cathode may emit provided its substrate, which is nickel in our case, has a work function ~ 3.1 e V.

Table 3.2.2: Absolute currents (rnA) at different temperatures emitted by a MgO film of 1.256 cm² surface area assuming the entire cathode structure has a minimum work function of 3.1 eV.

TfK Is/rnA

293 l.678xl0-⁵¹

300 9.053x 1 0-48

500 l. 734 X 10-²⁶

700 2.898xlO-¹⁸

900 4.385xlO-¹²

1100 9.411xlO-⁵

3.2.4. FOWLER-NORDHEIM ELECTRON EMISSION FROM A COLD CATHODE

The density of the emitted current dependent on the electric field can also be described by the Fowler-Nordheim relation [2][21][26]:

where

. . E Eo

( )

2

]=]0 Eo

exp[-E]

4&

^{~ ~}

Eo

= _ _

^(/J2

=

6.383 X 10-⁸ (/J2 ,

31]e

J.

=

2em (/J2

=

6 037 x 10-⁸⁵ (/J2

o 9 3 2 ' , 1] ;rr

E =V

d '

(3.2.5)

where 1] = elm, V is the voltage applied across the dielectric medium and d refers to the thickness of the dielectric medium.

It can be seen that the above expression is independent of temperature, which is a characteristic of the field effect. According to Shroff [2] the above expression can finally be presented as shown in equation 3.2.6:

dlog--4 J

_---.--:V_ 1

=

-0 3d(Jj2 . d-

V

(3.2.6)

Using the above expression Shroff [2] determined the thickness d of the NiO layer, that is fonned between the MgO layer and the nickel substrate. If one considers the graph in Figure 3.2.2 as an

1 example one can see that the logarithm of current density j is plotted as a function of - for

V various nickel based MgO samples, which Shroff [2] used in his investigations of MgO cold cathodes.

J..

^(Ay-2)

• ^{- r}

¹

-'

^{_ i}

,

.y 2

1. ,

,

.-

,

1

--,

\ I 1.

\ ' ^\

\

\.

\

10-7 \ _'\ \

\ \ _{\ ~ --\}

^{j i}

"

^{,,'\. 1}

,

^{' ''- II. \ n. '\}

,

'\

,

^-~

,

1. \. \

'\

, "

_Y ^\.' \1 ^\. \ ^1. \ \. ^\ \

\ \ \ , \ i \ \

10-8

\ \ ^{- \ - -- \}

^~

~ __

~-

'f\=-\

^{c-f-- -}

' ^\-\~.

^{\-- 1}

³

'\.

" " "

^{-y \. .. ,}

0

--- X - _{l\~\ \} ^\~

_\ ^{Z395 , -}

^{_ . ..,} _I

10.9

\~\~~ ~. \1~Z39R

^IZ432

016 08 "\ ^t-Z387 .

Z385

~- I ₁₀³ (V-1~

--.~

--- -.-

I ^V

2 3 4 5 6 1 8

Fig.3.2.2: The characteristic log

~ ^{= .{} ~)

^[2].

The gradients of these curves, namely 1 ' enabled Shroff [2] to evaluate d, the dielectric

d -

V

layer thickness in question, provided its work function <l> is known. By considering the work

function <l> ofMgO to have a value 3.1eV, he then found d to be equal to 780

A

[2]. He deduced that this value represented the thickness of a nickel oxide interface between the metal and the magnesium oxide layer, and approximated it to 1000

A.

Furthermore he verified the existence of such a layer by a simple X-ray examination of the cathode and by an evaluation of the Debye- Scherrer diagram obtained from such an analysis. Even though the value for the thickness of the NiO layer has thus been established by applying the Fowler-Nordheim relation, the author disagrees with Shroff's statement, that the electron emission from the MgO cold cathode is one of the Fowler-Nordheim type. However, before elaborating further on this point, it is necessary to first calculate the different voltages and electric fields that develop not only across the Ni-NiO- MgO structure of the cold cathode, but also across the cold cathode-vacuum-anode structure of the entire X-ray tube. In Figure 3.2.3 a working model and a band structure model for Ni-NiO system is displayed.

NiO Ni

1---0+

Fig. 3.2.3.a: Working model of a Ni-NiO-MgO structure [2).

Fermi level

Ni=5eV 1)=2.8eV

1

0.4< X < 1.6 eV

I) = 2.J8eV

- - - -

^~---

Fig. 3.2.3.h: Band structure model for a Ni-NiO system (2).

By taking into account the work function ofNi (<1>

=

5 eV) and the electron affinity ofNiO (X

=

1.6 eV) and thus establishing an overall work function for the Ni-NiO system of 3.4 eV, Shroff[2]

showed that the Schottky effect was practically negligible even for fields of the order of lOs V cm-^I (of the order of 10-³³ A cm-² at T

=

300 OK as compared to orders of lO+^sA cm-² for the field effect at the same temperature). The fact, that the Schottky effect does not take place in field dependent electron emission has also been verified in section 3.2.3 earlier. This means, that the electron emission from a cold MgO cathode subjected to a high electric field is affected by temperature to a negligible extent, and so it can be stated that autoelectronic emission can be treated as a field dependent phenomenon only. Taking that into account the following emitter-collector structure representative of a cold cathode X-ray tube must be considered:

Ni-NiO-MgO-vacuum-anode.

Here one can see the presence of three dielectric layers, namely the nickel oxide, the magnesium oxide and the vacuum layers, whereby the substrate (i.e. Ni) and the anode are functioning as the electrodes [2]. The following material thicknesses, dielectric constants and electric fields are taken into consideration:

1. d}, &}, EI for NiO;

2. d2, e2, E2 for MgO;

3. d3, &3, E3 for the vacuum;

Similarly one must consider the corresponding potential differences V}, V₂ and V3 which add up to a total accelerating voltage between the end electrodes of the X-ray tube V= 85 kV. With

one obtains the following expressions for VI, V₂ and V3:

One can now find the following values for VI, V₂ and V3:

VI

=

1.279x 1 0-² V for d_l

=

I x 1 0-⁷ m,

V3

=

84980.79 V for d3

=

0.056 m.

where

EI = EoKI = 1.05x 1 0-¹⁰ F m-^I and KI, the relative dielectric constant for NiO, = 11.9 [20],

E2= EoK2= 8.54xlO-^1I F m-^I and K2, the relative dielectric constant for MgO, = 9.65 [20],

E3= EoK3= 8.85xlO-¹²F m-^I and K3, the relative dielectric constant for vacuum, = I, dj = the thickness of the NiO layer (m),

d₂ = the thickness of the deposited MgO layer on the nickel substrate (m),

d₃ = the anode-cathode spacing of the X-ray tube (m), which has been derived in Section 3.l.3, where it is denoted as z.

Using the above values for VI, V₂ and V3 and dI , d2 and d3 the corresponding electric fields EI, E2

and E3 can be found:

Cobine [1] states that electrostatic fields of the order of 10⁵ V m-^I are sufficient to cause autoelectronic emission. From the above values for the electric fields it can therefore be seen that these values are sufficient for cold cathode electron emission to take place. Another aspect of this type of emission, which has to be mentioned, is the existence of superficial charges at the NiO- MgO interface. According to Shrojf[2] the electric field can also be expressed as the ratio between the surface charge density (J s (C m-²) and the dielectric constant & of a given material, i.e.

E

=

(Js [14]. By taking the earlier obtained values ofE₂

=

l.573x10⁵ V m-^I and &7

=

8.54xlO-¹¹

E -

F m-^I, (J s is obtained as

which, when divided by the charge value for the elementary charge e

=

1.602x J 0-¹⁹ C, is equivalent to approximately 8.385xl0\3 charges cm-² If one takes into account, that the electrons are actually drawn from the nickel base of the cathode through the NiO and MgO layers during avalanching, it becomes apparent that the number of free electrons must exceed the minimum charge density of 8.385xl013 charges cm-² in order for electron emission from the cathode under the presence of high electric fields to take place. Figure 3.2.4 illustrates the equilibrium potentials with and without such charges. It can be shown that nickel contains 9.129xl022 atoms cm-³, which means that the number of available electrons is clearly even higher than both that value and the required minimum

of available charges of 8.385x1013 charges cm-^2. Therefore a nickel substrate will be sufficient to supply electrons into the NiO and MgO layers, so that the required MgO surface potentials can develop for autoelectronic emission to take place. In Figure 3.2.4 the initial equilibrium potentials across the NiO, MgO and vacuum layers, i.e. the equilibrium potentials in the absence of charges, are illustrated. Once a steady flow of charges is established across these layers, so that practically a short connection is established across the vacuum, the entire voltage applied to the tube develops across the NiO and the MgO layers (see curve denoted 'equilibrium state with charges' in Figure 3.2.4). As can be seen below most of that potential drop then develops across the NiO, thus making it the layer most prominent in accelerating the electrons from the nickel substrate.

It can therefore be said, that by having theoretically established the thickness of a NiO layer that forms in a nickel based MgO cold cathode it is possible to establish the equilibrium potentials and electric fields within the different dielectric layers (i.e. NiO, MgO, vacuum). The calculations for these fields have shown, that the application of an accelerating potential of 85 kV in the aforementioned X-ray tube structure will indeed produce field magnitudes within the different dielectric layers high enough to produce electron emission. Also it can be seen that the electron emission from a cold MgO cathode would in this case be one arising from the presence of high electric fields within the different dielectric layers of the cathode rather than from the temperature dependent Schottky type. Having established a value for the electric field E

=

E2 across the MgO layer and keeping in mind that the work function <l> of MgO is 3.1 eV, the value for the emitted cold cathode current density j can be determined by substituting the above values into equation 3.2.5. In this way one finds, that

and

; .

o

=

^2em J 2 · ^(/)2

=

1489 X 10-¹²¹ A m ^-j , 9'7 Jr

and so, with E in equation 3.2.5 equal to E2,

With the current density j in equation 3.2.5 attaining a value approaching zero it can therefore be stated, that even though the electron emission from the cold cathode is entirely dependent on an applied electric field, the Fowler-Nordheim concept of electron emission from a MgO cold cathode does not offer a satisfactory description for this phenomenon at all. Note should be taken of the fact, that the value assumed for the NiO layer thickness of 1 x 1 0-⁷ m is merely an approximation made by Shroff [2], which he attempted to explain via the Fowler-Nordheim equation (equation 3.2.5), yet only verified by X-ray examination of the cathode.

S5000~----~---=~~---~~

>

"'-

Q) C)

-

(0

Q

>

MgO dz

84080.79 V

Vacuum

Thickness

Equilibrium state without charges

Fig. 3.2.4: Equilibrium potentials with and without surface charges.

3.2.5. COLD CATHODE ELECTRON EMISSION DUE TO CHANGING ELECTRIC FIELDS