Modelling of electron emission from sandwich cathodes

Vacuum/volume Bl/number 7/pages 297 to 30111981 Printed in Great Britain

Modelling cathodes

0042-207X/81/070297-05$02.00/0 Pergamon Press Ltd

of electron

emission

from

sandwich

R Hrach, Department of Electronics and Vacuum Physics, Faculty of Mathematics and Physics, Charles University, Prague, Povltavskri 1, 780 00 Praha 8, Czechoslovakia received

6 February

1981;

in revised

form

24 March

1981

The electron emission from M/M systems is studied by means of a computer simulation. A model of the sandwich structure with a scattering by traps and phonons in the dielectric layer is proposed. The distributions of emitted electrons (both angular and energy) for various combinations of electron mean free paths have been obtained by means of the Monte Carlo method.

1. Introduction

Sandwichcathodescan beusedfor the study of a chargetransport in thin metalanddielectriclayers.The analysisof temperatureand voltage dependences of the leakageand emissioncurrents and their distributions provides information about scattering processes, particularly in dielectrics1-3.When a voltageexceedingthe work function of the upper metalelectrodeis applied(Figure l), electrons tunnel through the potential barrier between the negative electrode and the dielectric, after which they are acceleratedby the appliedelectric field in the dielectriclayer and finally someof them can passthrough the positive electrodeinto vacuum and create the emissioncurrent. METAL

1

DIELECTRIC

METAL

2

VACUUM

_------=--7-------I, eU

I

-5.5 Figure

1.

N IE) NIE,) N,iSI

]

PI t-

Energydiagramof the sandwich cathode.

2. Experimental

results

The measurements werecarried out on sandwichcathodesof the AI-AI,O,-Au structure. Dielectric films were prepared by the anodic oxidation of the basic metal electrode. The dielectric thicknessvaried from 5 to 20 nm the thicknessof the top metal electrode being about 15 nm. The electron emission was distributed uniformly over the wholesurfaceof the cathode.The I-V characteristicof the emissioncurrent had an approximately

exponentially increasingform which correspondsto the proposed tunnel mechanism. For further evaluations we usedthe experimentally derived distributions of emitted electrons-angular distribution N,(9), energy distributions N(E) and N(E,) and the energy-angular distribution N(E,9). Typical resultsareshownin Figure2. In these experimentsthe sandwichcathodewith dielectricthicknesss= 16 nm and metal thicknesst = 14 nm was used.The characteristics were measuredat T=300 K and the applied voltage U = 11 V. The measurementsof angular distribution and total energy distribution were performed with a hemisphericalcollector divided into concentric zones.The cathode was placed in the centre of the collector and wasenvelopedby a positively biased hemisphericalgrid. The energy distributions were obtained by meansof the retarding potential technique.The original distribution function is the energy-angulardistribution N(E,3) while both the distributions N(E) and N,(9) can be derived by integration over one of variables’. The angular information is derived with someinaccuracy(of order of 3-5 degrees)causedby averagingover collector zones.The normal energy distribution N(E,) was measured in experimental system with planar geometry. All distributions of emitted electrons are temperature dependent but the most significant changecan be seenon the angular distribution. With a decreasingtemperaturethe cosine part (background)of the distribution islesspronouncedand at 80 K the whole angular distribution usually consistsonly of the central peak. On the contrary, the central peak is nearly temperatureindependent. A further studiedcharacteristicwasthe transferratio a, i.e. the ratio of the emissioncurrent to the total current flowing through the dielectric. This ratio was usually very small depending exponentiallyon the thicknessof the upperelectrodet. Evenin the caseof the extrapolation to t = 0 the calculatedvalue txOusually did not exceed 10e3. From all our results it follows that a substantialrole in the chargetransport through the cathodeis played by the scatteringprocesses in the dielectric. 297

R Hrach:

Modelling

of electron

emission

from

k 0

3.

2. Experimentally

Proposed

As the

cathodes

N(E.1

N,(8)

Figure

sandwich

derived

characteristics

2

E,E.[eVl

6 '

of emission

current.

model

are rather by means of a computer simulation. When modelling the action of the sandwich cathode, we followed two purposes: (1) To find out whether the proposed mechanisms can be utilized for the explanation of the measured characteristics. (2) To estimate the intensity of the scattering processes by means of a comparison with experimental results. Our model of a sandwich cathode is based on the following principles. It is assumed that the transport of electrons into the dielectric is due to a tunnel transition. In the dielectric layer we assume four processes: (i) trapping followed by the thermal reexcitation (the intensity of the process is described by means of the mean free path I.,,); (ii) trapping followed by the liberation by the PooleFrenkel effect (A,); (iii) generation of optical phonons (L,,); (iv) elastic scattering or generation of accoustical phonons (,I,). According to the model, the first two processes result in the loss of the whole electron energy. The only difference is that after the re-excitation into the conduction band of the dielectric by the thermal process the electron can gain an energy E with the probability proportional to exp( - E/kT). The generation of phonons is modelled by the isotropical scattering which will be preceded by a loss of energy AE for optical phonons. The energy of optical phonons in Al,O, is about 0.1 eV. The transition of electrons through the upper metal electrode is considered as ballistic-after the interaction with conduction electrons of the metal the hot electrons lose such a great part of their energy that they cannot overcome the barrier between the metal and vacuum. Therefore, the transition of hot electrons through the positive electrode will not disturb the distribution of emitted electrons but only reduces the total emission current and changes the value of the transfer ratio CL.The transition into vacuum is treated with a quantum-mechanical assumption about the possibility of an overbarrier reflection. inaccurate,

4 Results

usual analytical methods of computations an attempt was made to solve the problem

of computer

experiment

The evaluations have been carried out by means of the Monte Carlo method, i.e. by modelling the transport of a great number of electrons through the sandwich cathode. The electrons which will 298

4

E [eVl

7

get into vacuum will be added into the distributions N,(9), N(E), N(E,) and N(E,9), respectively. The program was written in the FORTRAN language and solved on the ICL 4-72 computer. All calculations were performed with the following cathode parameters: dielectric thickness s= 10 nm, work function metaldielectric 0 =2 eV, work function metal-vacuum cp=4.7 eV, Fermi energy of metal electrode E,= 5 eV, energy loss at generation of optical phonon AE=O.l eV and applied voltage U = IO V. A typical number ofemitted electrons was about IO 000, the necessary computation time was of the order of IO’ to lo3 s (depending on the values of mean free paths). No scattering process. The distributions of electrons emitted (in our model) without any scattering in the dielectric film differ strongly from the experimental data. All electrons are emitted with energies corresponding to the Fermi level of the negative electrode and their velocities are normal to the cathode surface.

41.

with the thermal excitation. Some results obtained for the single scattering process in the dielectric layer described by the mean free path I,, are shown in Figure 3 for room temperature. The decrease of the temperature narrows the angular distribution and removes the high energy electrons from the energy distributions.

42. Trapping

with the field excitation. When the only scattering mechanism in the dielectric layer is supposed to be the capture by traps and the re-excitation by the Poole-Frenkel effect, electrons are emitted perpendicularly to the cathode surface. The angular distribution of these electrons has the &form and all energy distributions N(E), N(E,) and N(E, 9) coincide. The energyangular distribution is depicted in Figure 4. All electron distributions are practically independent of the temperature.

4.3. Trapping

Generation of optical and acoustical phonons. The angular distributions corresponding to both mechanisms are approximately cosine. The energy-angular distributions are plotted in Figures 5 and 6 for the same mean free paths. From these figures the form of energy distributions can be deduced as well. For the scattering mechanism with optical phonons the distribution N(E) is wide and the energy of its maximum decreases with decreasing mean free path APO, while for the elastic scattering all electrons are

4.4

R Hrach:

Modelling

of electron

emiaaion

from

sandwich

h

N,l41

cathodes

NfE,) N (E,$)

\ 30’

\ ‘603 -go-Figure

3. Computed

distributions

Figure

4. Computed

energy-angular

of emitted

electrons

distribution

for I,, = 0.5 nm and

T=300

K

for I,, = 0.5 nm.

4 -,,; ;::,;:~~

go

Figure with

““‘:’

4 Figure optical

5. Computed phonons with

EIeVl

energy-angular the mean

6

free

distribution path A,,=O.5

for the nm.

generation

of

emitted with the same energy E corresponding to their original energy near the Fermi level in the metal. The distributions N(E,) derived from both processes are similar and resemble the distribution N(E) for the scattering by optical phonons. Combination of several processes. A comparison of results derived with the use of a single scattering mechanism in the dielectric (Figures 3-6) with the corresponding experimental data shows that though some features of distributions are similar, the single mechanism cannot explain the behaviour of emitted electrons completely. The best way seems to be a combination of several processes. In order to enable the choice of the proper mechanisms we can

4.5.

E [eV1

6. Computed the mean free

’

energy-angular path I,,=O.5

distribution

for the elastic

scattering

nm.

use the dependences of the leakage and emission currents or the transfer ratio. From the computer calculations we can derive the transfer ratio extrapolated to the zero thickness of the top electrode, aO. The dependence of a,, on the intensity of the scattering process for all four mechanisms considered is plotted in Figure 7. From this figure it follows that if it is necessary to reduce a,, to values corresponding to experimental data, at least one of the real scattering processes must be connected with the trapping. Taking into account the small temperature dependence of the central peak of the angular distribution and the high value of the electric field in the dielectric, the scattering by traps followed by the liberation with the help of the Poole-Frenkel effect can be preferred.

5. Discussion

It was found (e.g.4) that a typical sandwich cathode is macroscopitally and microscopically uneven and that this unevenness can disturb the distributions of emitted electrons. Particularly, the central part of the angular distribution will be wider-the change will be at least several degrees. When this process is taken into account then probably the most 299

R Hrach:

Modelling

of electron

emission

from

sandwich

cathodes

of the coefficientCLand to suppressall electronsflowing directly from the negative electrode without any interaction in the cathode. As the mostsensitivecharacteristicof emitted electronsseems to be the angulardistribution, one can comparethe influenceof various interactions upon it (Figure 8). Figure 8(a) gives the experimentalcharacteristic.In Figure 8(b) isshownthe influence of trapping followed by the thermal excitation for two temperatures-dashedline was derived at 300 K, dotted-anddashedline at 100K. Figure 8(c) showsthe distributions of electrons after interaction with phonons, both optical and acoustical.(Dashedlinewascalculatedat agreaterintensityof the scatteringprocess(1=0.5 nm), dotted-and-dashed lineat 1= 1 nm, when someelectronspassthrough the cathode without interaction andcreatea very narrow peak.) Finally, Figure 8(d) shows the resultsof combinationsof two processes in unevencathodes.In thesegraphsweworked with the followingparameters:&PO=1 nm, I.,/= 1 nm (dashedline); &,= 1 nm, &=2 nm (dotted line); I,, = 1 nm, ,I,, = 5 nm (dotted-and-dashedline). In further evaluationswe tried another combinationof cathodeparameters,too. Our aim wasto find sucha combination of scatteringprocesses which producesthe distributions of emitted electronssimilar to experimental results. From Figure 8 it can be seenthat the resemblance betweenthe dotted line andthe experimentalcurve is rather good.Similarly, the proposedprocesses provide a sufficient agreementin further characteristicsof emitted electrons.

7. Dependenceof extrapolated transfer ratio II,, on the meanfree path of single scattering mechanism 1.

Figure

significant scattering mechanisms in sandwich cathodes are: (a) capture by traps in the dielectric with the field re-excitation; (b) generation of optical phonons in the dielectric (process in aluminium oxide more probable than the generation of acoustical phonons); and (c) interactions with conduction electrons in the top metal electrode. Furthermore, the intensity of the scattering must be great enough to obtain sufficiently small value

a

‘. 0 I -IT

/ //60” //11 *_-’

\

i i i i i i i i i i !! !! 11

,,/

30'

\’30

C

30'

ii ii

,i 1..

\ -60*\ 4 -90'

1

-90=

!

!: !ICl

/

Figure 8 Angulardistribution of emittedelectrons derived:(a)experimentally; (b) modelled with A,,=O.S nm, R,,=L,,= 1 and 0.5 nm; (d) modelled with simultaneous action of three processes (i) ‘trapping+ field excitation’, unevenness’. 300

T=300 and 100 K; (c) modelled (ii) ‘optical

phonons’

with and (iii) ‘surface

R Hrach:

Modelling

of electron emission from sandwich cathodes

6. Conclusion

The comparisonof resultsof computer simulationwith experimentalcharacteristicsenablesthe estimationof the mostprobable scatteringprocesses in a sandwichcathode. From the analysisof the distributionsofemitted electronsit followsthat the interaction with optical phononsisabout 2 or 3 timesmoreprobablethan the interaction with traps.Thesedistributionsgive usonly therelative proportions of correspondingmeanfree paths.From the analysis of the transferratio a the value of about 0.5 nm for the meanfree path for interaction with traps can be obtained. The model describedin the present paper was only onedimensional,thereforethe derived meanfreepaths representthe projection of real multidimensionalpaths onto the normal to the cathodeplane.

.____

where

L,‘,‘,’ l-exp[ -As. ( 4, 4, I, ‘OC4-exp(

-t)-exp(

-E)-exp(

-$)-exp(

-2)’

(5) When the electron was scattered, its energies E and E, are changed in following way6 : (i) trapping with thermal excitation E=-kT.Inc, E,=E.(l-2.5,)‘;

References

‘R M Handy, J appl Phys, 37,462O (1966). ‘L Eckertovl,J Vat Sci Technol, 6, 509 (1969).

(ii) trapping with field excitation

sV S Kortov and P P ?Zdlnikdv, Ph&ico &us solidi (a), 31, 331 (1975). ‘1 M Sobol’,Num&zol Methods Monte Carlo. Izd Nauka,Moscow (1973) (in Russian). ‘L D Landau and E M Lifschitz, Quantum Mechanics. Izd Nauka. Moscow (1%3) (in Russian). sR Hrach, Thin Solid Films, 15, 65 (1973).

(iii) generation of optical phonons

3RHrachandM Sobotka,Csfasfyr, A29, 58(1979)(in Czech). 4R Hrach.Czech J Phvs. B22. 490 (1972).

Appendix.

>I

A,,

Basic features

of computer

experiment

(1) Electrons tunnel from the negative electrode into the dielectric. Trajectories start from the point 0 (Figure 1) on the bottom of the dielectric conduction band. (2) The electron state is described by one-dimensional position x measured from the point 0, total kinetic energy E and part of kinetic energy corresponding to the momentum in the normal direction E,. Initial values of both energies at point 0 are equal to zero. (3) Electron trajectory in the dielectric is divided into small steps As (about 0.1 nm)“. Alter each step electron energies E and E, increase by an increment of e . U/s . As. This approach requires less computer time than the standard computation based on free paths. (4) Alter each step the possibilities of various scattering events are tested by comparing the random number &(O, 1) with probabilities

h=h.[l-erp(-:)]

E=E,=O;

E=max(O, E- AE) (l-2. &)‘;

E,=E.

(iv) generation of acoustical phonons E unchanged E,=E.(l-2.<,)‘. Here 5, and t2 are random numbers from the interval (0, 1). (6) When electrons leave the dielectric their energies change. As a new zero energy the level of vacuum is chosen instead of the bottom of the dielectric conduction band:

.rw=EOM+O-cp E-=Ex“M+O-~. x

(7) The transport through the top metal electrode is supposed to be ballistic, therefore it needs not to be taken into account for the distributions of emitted electrons. (8) The probability of electron transfer into vacuum is described by the expression’ 2. [E,. (E,+E,+rp)]“* P(E’)=[E~.(Ex+E,+(p)]L’2+E~+(E,+cp)/2 =o

E,>O

E,
Therefore, electrons with E,< 0 are excluded and then the possibility of the overbarrier reflection is tested by means of comparison of P(E,) with random number. (9) Energy and angular distributions are formed from the emitted electrons.

301