quantum mechanical theory of electronic photon-stimulated field emission by transfer matrices and...

Quantum Mechanical Theory of ElectronicPhoton-Stimulated Field Emission byTransfer Matrices and Green’s Functions

ALEXANDRE MAYER,1 MARK J. HAGMANN,2 JEAN-POL VIGNERON1

1Laboratoire de Physique du Solide, Facultés Universitaires Notre-Dame de la Paix,Rue de Bruxelles 61, B-5000 Namur, Belgium2BSD Medical Corporation, 2188 West 2200 South, Salt Lake City, UT 84119-1326

Received 26 February 2000; revised 5 April 2000; accepted 6 April 2000

ABSTRACT: For the purpose of simulating photon-stimulated field emission by takingaccount of three-dimensional aspects, the transfer matrix and Green’s functionsformalisms, used in previous works for the simulation of the Fresnel projectionmicroscope, was combined with a Floquet expansion of the wave function in order tostudy quanta exchanges between the electrons and the external radiation. In this newformulation, the wave function is expanded along basic states that describe its spatialvariation for the various discrete energy levels associated with absorption/emission ofphotons. The transmission of these basic states between the two electrodes providing thefield emission voltage is described within the transfer matrix formalism, where specifictechniques are used to preserve stability. The transmission of these basic states from thecathode to a distant screen is achieved within the Green’s functions formalism. Thetechnique takes advantage of n-fold symmetries and provides control of computationaccuracy. The theory is applied to the stimulation of field emission from a 1-nm-longtungsten nanotip. The application specifically focuses on the current increase due toelectromagnetic radiation. Total energy distributions (TED) and current-relative-increasecurves are provided for radiation wavelengths ranging from 0.1 to 10 µm and power fluxdensities ranging from 1010 to 1012 W/m2. The results point out a resonance in the currentenhancement at a radiation wavelength of approximately 0.3 µm. c© 2000 John Wiley &Sons, Inc. Int J Quantum Chem 80: 816–823, 2000

Key words: photon-stimulated field emission; transfer matrix; Green’s functions;Floquet expansion

Correspondence to: A. Mayer; e-mail: [email protected].

Contract grant sponsors: Belgian National Fund for ScientificResearch (FNRS); Interuniversity Research Project (PAI).

International Journal of Quantum Chemistry, Vol. 80, 816–823 (2000)c© 2000 John Wiley & Sons, Inc.

ELECTRONIC PHOTON-STIMULATED FIELD EMISSION

Introduction

A t present GaAs photomixers are usable at fre-quencies up to 1 THz [1], but there are no

wideband tunable sources for use at higher fre-quencies. Microfabricated triode field emitter arrays(FEA) offer promise as microwave amplifiers [2].However, thus far no FEA has been shown to havea gain exceeding unity above 1.3 GHz [3], and thislimit is attributed to shunting by capacitance of thegate structure. Previous simulations show that pho-tomixing in resonant photon-assisted field emissioncould be used to generate signals from direct current(DC) to 100 THz [4]. However, these signals occurat the apex of the tip, and so there is the practicalproblem of how to efficiently couple the signals toan external load.

By means of one-dimensional numerical sim-ulations [5, 6] confirmed by preliminary experi-ments [7], a resonance was found in the interac-tion of tunneling electrons with a radiation field inwhich the tunneling current is markedly increased.The mechanism for resonance is reinforcement ofthe wave function by reflections at the classical turn-ing points. Thus, for the special case of square barri-ers, resonance occurs when electrons are promotedabove the barrier by absorbing quanta from the ra-diation, and the length of the barrier is an integermultiple of one-half of the de Broglie wavelength.The extremely large bandwidth is made possible bygating the current with a radiation field in place ofthe grid, thus avoiding shunting by the gate capaci-tance.

For the purpose of studying three-dimensionalaspects, the transfer matrix and Green’s functionsformalism [8 – 12], used in previous works for thesimulation [13 – 15] of the Fresnel projection micro-scope [16], was combined with a Floquet expan-sion [17] of the wave function in order to considerquanta exchanges between the electrons and the ex-ternal radiation. The details of this new techniqueare given in the following section.

The theory is applied to the simulation of fieldemission from a 1-nm-long tungsten nanotip. Theapplication specifically focuses on the current in-crease due to electromagnetic radiation. The sim-ulation of the (time-dependent) potential-energydistribution around the field emission structure isdescribed in the third section. The results of the sim-ulations are presented in the fourth section, wheretotal energy distributions (TED) and curves of thecurrent relative increase are provided, for radiation

wavelengths ranging from 0.1 to 10 µm and powerflux densities ranging from 1010 to 1012 W/m2. Theresults point out a resonance in the current enhance-ment at a wavelength of approximately 0.3 µm.

Theory of Photon-StimulatedField Emission

BASIC FORMULATION

Let us consider the following three-dimensionalpotential-energy distribution:

V(r, t) = Vstat(r)+ Vosc(r) cos(�t), (1)

where the time-dependent radiation is representedby the second term [18].

Floquet’s theorem [17] is used to expand thewave function according to:

9(r, t) =+∞∑

k= −∞9k(r)e−i(E+kh̄�)t/h̄. (2)

By inserting this expression in the time-depend-ent Schrödinger equation[

− h̄2

2m1+ V(r, t)

]9(r, t) = ih̄

∂

∂t9(r, t),

the components 9k(r) of the wave function expan-sion (2) turn out to verify the exact set of coupledequations:

− h̄2

2m19k(r)+ Vstat(r)9k(r)+ 1

2Vosc(r)

[9k−1(r)

+9k+1(r)] = (E+ kh̄�)9k(r). (3)

This equation reveals that the time-dependentpotential Vosc(r) cos(�t) is responsible for photon-assisted coupling to occur exclusively betweenwave function components whose associated en-ergy values are separated by a single quantum h̄�.

CONSIDERATION OF n-FOLD SYMMETRY

The wave function components 9k(r) of expres-sion (2) can be expanded in terms of basis functionsthat account for the φ and ρ dependences. Their setis forced to be enumerable by specifying that thescattering electron remains localized inside a cylin-der with radius R [19] in regions I and II. One canthus write

9k(r) =∑m,j

8(m,j),k(z)φ(m,j)(ρ,φ). (4)

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 817

MAYER, HAGMANN, AND VIGNERON

In this last expression, the two integer subscriptsm and j enumerate the basis functions 9(m,j)(ρ,φ),given by:

φ(m,j)(ρ,φ) = Jm(km,jρ)eimφ√2π∫ R

0 ρ[Jm(km,jρ)]2 dρ(5)

and characterized by a radial wave vector km,j solu-tion of Jm(km,jR) = 0. The z dependence of the wavefunction is contained in the coefficients 8(m,j),k(z) ofthe expansion.

By writing Vstat(r) and Vosc(r) like

Vstat(ρ,φ, z) = V0stat(z)+

+∞∑q=−∞

Vqstat(ρ, z)eiqnφ , (6)

Vosc(ρ,φ, z) = V0osc(z)+

+∞∑q=−∞

Vqosc(ρ, z)eiqnφ , (7)

the propagation equation (3) turns out to become

− h̄2

2md28(m,j),k(z)

dz2 + V0stat(z)8(m,j),k(z)

+ 12

V0osc(z)

[8(m,j),k+1(z)+8(m,j),k−1(z)

]+∑q,j′

Sq,j′m,j(z)8(m−qn,j′),k(z)

+ 12

∑q,j′

Oq,j′m,j(z)

[8(m−qn,j′),k+1(z)+8(m−qn,j′),k−1(z)

]=(

E+ kh̄�− h̄2

2mk2

m,j

)8(m,j),k(z), (8)

where the coupling coefficients Sq,j′m,j(z) and Oq,j′

m,j(z)are defined by the expressions:

Sq,j′m,j(z) =

∫ R

0ρV

qstat(ρ, z)Jm(km,jρ)Jm−qn(km−qn,j′ρ) dρ

×(∫ R

0ρ[Jm(km,jρ)

]2 dρ)−1/2

×(∫ R

0ρ[Jm−qn(km−qn,j′ρ)

]2 dρ)−1/2

, (9)

Oq,j′m,j(z) =

∫ R

0ρV

qosc(ρ, z)Jm(km,jρ)Jm−qn(km−qn,j′ρ) dρ

×(∫ R

0ρ[Jm(km,jρ)

]2 dρ)−1/2

×(∫ R

0ρ[Jm−qn(km−qn,j′ρ)

]2 dρ)−1/2

. (10)

FIGURE 1. Geometry of the situation considered, i.e.,scattering from region I (z ≤ 0) to region III (z ≥ D),through an intermediate region II (0 ≤ z ≤ D).The electrons are confined in regions I and II to enforcea quantification of the basic states to consider for thewave function expansion.

The propagation equation (8) can be imple-mented in a transfer matrix procedure [20] in situa-tions that involve scattering from (metallic) region I(z ≤ 0) to region III (z ≥ D), through an intermediateregion II (0 ≤ z ≤ D) (see Fig. 1). The propagationof solutions through region III to the screen will beachieved within the Green’s functions formalism.

IMPLEMENTATION OF LOCAL SCATTERING BYTRANSFER MATRICES

Boundary States for z ≤ 0 (Region I) and z = D(Extraction Grid)

Let us assume the potential energy V(r) to take aconstant zero value in region III (z ≥ D). For z = D,the propagation equation (8) takes then the specificform:

− h̄2

2md28(m,j),k(z)

dz2 =(

E+ kh̄�− h̄2

2mk2

m,j

)×8(m,j),k(z), (11)

and the corresponding boundary states are givenby:

8D,±(m,j),k(z) = e

±i√

(2m/h̄2)(E+kh̄�)−k2m,jz. (12)

The ± sign will refer to the propagation directionrelative to the z axis, which is oriented from region Ito region III.

In the metallic region I (z ≤ 0), the componentsVstat(r) and Vosc(r) take the constant values Vstat =eV−W−EF and Vosc, where V is the extraction bias,W and EF are, respectively, the work function andFermi energy of the metallic region I. The propaga-

818 VOL. 80, NO. 4 / 5


tion equation (8) takes here the specific form:

− h̄2

2md28(m,j),k(z)

dz2 + Vstat8(m,j),k(z)

+ 12

Vosc[8(m,j),k+1(z)+8(m,j),k−1(z)

]=(

E+ kh̄�− h̄2

2mk2

m,j

)8(m,j),k(z). (13)

There is no coupling between wave functioncomponents with different (m, j) subscripts. If onerestricts the k subscripts to the range [−N,+N], theboundary states 8I,±

(m,j),k(z) are found by solving thematricial equation:

− h̄2

2md2

dz28(z)+[(

Vstat − E+h̄2k2

m,j

2m

)I

+M]8(z) = 0, (14)

where the 8(z) vector is defined by:

8(z) =

8I,±(m,j),N+1(z)

8I,±(m,j),N(z)

...8

I,±(m,j),−N−1(z)

8I,±(m,j),−N(z)

, (15)

I is the identity matrix and M is defined by:

M =−(N+ 1)h̄� 0

Vosc/2 −Nh̄� Vosc/2. . . . . . . . .

Vosc/2 Nh̄� Vosc/20 (N + 1)h̄�

.

(16)

The two zeros in the off-diagonal bands of the Mmatrix aim at enforcing the eigenvalues of (Vstat −E + h̄2k2

m,j/2m)I +M to take the required values ofVstat − (E + kh̄� − h̄2k2

m,j/2m). The boundary states

8I,±(m,j),k(z) are obtained after expressing this matrix

as V3V−1, where the two matrices 3 and V con-tain, respectively, the eigenvalues and eigenvectorsof (Vstat − E+ h̄2k2

m,j/2m)I+M.

Transfer Matrix Computation

Once the 9I,±(m,j),k(r) = 8

I,±(m,j),k(z)φ(m,j)(ρ,φ) and

9D,±(m,j),k(r) = 8

D,±(m,j),k(z)φ(m,j)(ρ,φ) boundary states are

defined, one needs to compute the solutions corre-sponding to single incident boundary states withenergy E in region I. These solutions are describedwithin the transfer matrix formalism as:

9+(m,j),0(r)=9 I,+(m,j),0(r)+

∑k

∑(m′ ,i)

t−+[(m′ ,i),k],[(m,j),0]

×9 I,−(m′ ,i),k(r), for z ≤ 0,

=∑

k

∑(m′ ,i)

t++[(m′ ,i),k],[(m,j),0]9D,+(m′ ,i),k(r),

for z = D.

(17)

The two transfer matrices t++ and t−+ contain theamplitudes of the transmitted and reflected bound-ary states corresponding to a given incident state9

I,+(m,j),0 with energy E and subscripts (m, j) in region I.

These solutions result from the linear combinationof solutions obtained by numerical propagation ofthe transmitted states 9

D,+(m,j),k(r) through interme-

diate region II [8 – 10, 12]. The three-dimensionalpotential-energy distribution given in Eq. (1) is con-sidered in the propagation step.

Stability Considerations

In order to take account of all propagative states(in the final region III) corresponding to the variousenergy levels E + kh̄� considered, the (m, j) sub-scripts are restricted by the condition:

h̄2

2mk2

m,j ≤ E+Nh̄�. (18)

The recommended number of layers nlayer to con-sider for the computation of the transfer matrices(see Ref. [10]), is given in this context by:

nlayer = 4D

nbit ln 2

√2m

h̄2 (V + 2Nh̄�), (19)

where nbit is the number of binary digits used for therepresentation of the fractional part of real numbers(53 in double precision).

The intermediate states used between two adja-cent layers inside region II must be propagative inorder to maintain the stability of the computation.This is achieved by considering a reference poten-tial energy of −2Nh̄� between two adjacent layers.Since the width of the separation between two adja-cent layers is zero, this choice is of no conceptualsignificance. For the same stability reasons, onlypropagative states must be considered for z ≥ D.



IMPLEMENTATION OF FAR PROPAGATION BYGREEN’S FUNCTIONS

The solutions given in Eq. (17) can be propagatedfrom the plane z = D to a distant screen within theGreen’s functions formalism [11] by an expressionof the form:

9+(m,j),0(r, θ ,φ) =∑

k

eikE,kr

r

∑(m′ ,i)

t++[(m′ ,i),k],[(m,j),0]

× σ (θ , m′, i, k, E)eim′φ , for r� 0, (20)

with kE,k =√

2m(E+ kh̄�)/h̄2.Within the Neumann boundary condition (no z

component of the current in the z = D plane forρ > R), the σ (θ , m, j, k, E) coefficients are given by:

σ (θ , m, j, k, E) = −Nm,j,k,E

√k2

E,k − k2m,j i1−m

× ei(√

k2E,k−k2

m,j−kE,k cos θ)

D√2π∫ 2π

0 [Jm′(km,jρ)]2ρ dρ

×∫ R

0dρ ρJm(km,jρ)Jm(kE,k sin θρ).

(21)

The normalization coefficients Nm,j,k,E ensure ex-act current conservation, i.e., enforce the transmit-ted states 9D,+

(m,j),k to be propagated according to thecondition:∫ 2π

0dφ∫ R

0dρ ρJz(ρ,φ, z = D)

=∫ 2π

0dφ∫ π/2

0dθ sin θr2Jr(r, θ ,φ). (22)

To obtain complete current-density distributions,the solutions 9+(m,j),0 have to be computed for all sig-nificant values of m, j, and E. The current densitiescorresponding to these solutions are then added, bytaking account of the associated density of states inthe metal z ≤ 0.

Description of the Field Emission Tip

The theory of the previous section enables thecomputation of current density distributions result-ing from a field emission process, when photonstimulation is present. The extraction bias V is es-tablished between the metallic region I (z ≤ 0) andthe plane z = D. The field emission tip is con-sidered as an extension of the supporting metallicregion I. In this study, the tip and its supporting

region I are made of tungsten and described as aSommerfeld continuous medium, characterized bya Fermi energy EF of 19.1 eV and a work functionW of 4.5 eV. The tip is an axially symmetric ellipticalemitter with a height and basis radius of 1 nm.

The static potential-energy distribution Vstat(r) inregion II (0 ≤ z ≤ D) is computed by relaxationtechniques (see Ref. [9]). The amplitude Vosc(r) ofthe time-dependent potential-energy distribution iscomputed in the same way, by considering a di-electric constant εr of 3.4 + i2.7 (typical value fora radiation with 0.5 µm wavelength λ) in the tip.The potential-energy Vosc(r) is given a zero valuefor z = D and a constant value Vosc = eDE forz ≤ 0, the amplitude of the effective electric fieldE being related to the radiation power flux densityS by S = cε0E2/2. This way of computing Vosc(r) isjustified by the metal-grid distance D being around1000 times smaller than the radiation wavelength.

Results

The real part of the static and time-dependentcomponents of the potential energy correspondingto an extraction bias V of 8 V, an electrode sepa-ration D of 3 nm, and a power flux density S of1012 W/m2 are illustrated in Figure 2. There is animportant penetration of the time-dependent partof the electric field in the tip, the penetration depthtaking a value of λ/(2π2.7) = 28.9 nm.

The current density obtained on the extractiongrid z = D and on a 10-cm distant screen when pho-ton stimulation is not considered is illustrated at thetop of Figure 3. The bottom of this figure shows thetotal energy distribution (TED) of the field-emittedelectrons.

It is interesting to compute the relative increase ofthe emission current that one obtains by consideringphotonic stimulation, with radiation wavelengthsranging from 0.1 to 10 µm. This interval corre-sponds to photon energies ranging, respectively,from 12.397 to 0.12397 eV. The results are illustratedin the left part of Figure 4. The right part of this fig-ure presents the results obtained when there is noemission tip on the flat metallic support z ≤ 0. Thepower flux densities are respectively 1010, 1011 and1012 W/m2 (upwards).

The relative increase in the current due to pho-tonic stimulation is more pronounced for the flatemitting structure. This is due to the potential bar-rier to tunnel through being larger in the case of

820 VOL. 80, NO. 4 / 5


FIGURE 2. Real part of the static (left) and time-dependent (right) components of the potential-energy distribution inthe XZ plane. The extraction bias and the distance between the tip holder and the conducting grid are, respectively, 8 Vand 3 nm. The radiation power flux density is 1012 W/m2.

FIGURE 3. (Top) Total current density (in A/cm2) on the conducting grid (left) and on a 10-cm distant screen (right).(Bottom) total energy distribution (TED) of the field-emitted electrons. The extraction bias and the distance between thetip holder and the conducting grid are, respectively, 8 V and 3 nm.



FIGURE 4. Relative increase of the emission current as a function of the radiation wavelength. The power fluxdensities are, respectively, 1010, 1011, and 1012 W/m2 (upwards). The results are obtained with a 1-nm-long fieldemission tip (left) and the flat metallic support (right).

a flat emitter. The transmission probability beingintrinsically lower, the sensitivity of the emissioncurrent to photon stimulation is more pronounced.There is typically one order of magnitude betweenthe curves associated with the different values of thepower flux density, which is consistent with thesevalues being also magnified by a factor of 10.

There is a substantial increase in the emission cur-rent when the photon energy enables the electronsat the Fermi level to reach the top of the potentialbarrier. This happens at a wavelength of 0.569 µm(corresponding to a photon energy of 2.18 eV) whenthe tip is present and at a wavelength of 0.456 µm(corresponding to a photon energy of 2.72 eV) inthe other case. The maximal value of the currentincrease is, however, encountered at a shorter wave-length (around 0.3 µm), which corresponds to a

photon energy around 4.1 eV. Such a resonance wasalready observed in one-dimensional simulations[5, 6]. It occurs when the line integral of the momen-tum of the photon-stimulated electrons around theclosed contour between the classical turning pointsof the potential barrier is an integral multiple ofPlanck’s constant.

The resonance is illustrated in Figure 5 by to-tal energy distributions corresponding to a photonenergy h̄� of 2.05, 4.1, and 8.2 eV. These energyvalues correspond to radiation wavelengths of 0.60,0.30, and 0.15 µm. The resonance in the photon-stimulated electrons is clearly visible. The expo-nentially decreasing shape of the photon-stimulatedelectron distribution is changed for h̄� = 8.2 eV,since the mechanism enabling these electrons tocross the potential barrier is changed from tunneling

FIGURE 5. Total energy distribution (TED) of the electrons. The energy of the photons is 2.05 eV (left), 4.1 eV (center),and 8.2 eV (right). The power flux density is 1012 W/m2. The extraction bias and the distance between the tip holder andthe conducting grid are, respectively, 8 V and 3 nm.

822 VOL. 80, NO. 4 / 5


to ballistic motion. The bottom of this distribution isat an energy value equal to the height of the poten-tial barrier encountered by the electrons at the Fermilevel in the supporting metal.

Conclusion

A stable and efficient technique, enabling thesimulation of photon-stimulated field emission withconsideration of three-dimensional aspects, was de-veloped. This technique was illustrated by a situa-tion characterized with an axial symmetry, i.e., theelectronic emission from a 1-nm-long elliptical tipby an extraction bias of 8 V. The curves illustrat-ing the current relative increase due to photonicstimulation point out a resonance that happens ata radiation wavelength of 0.3 µm. This resonance isexplained by the photon-stimulated electrons hav-ing the closed line integral of momentum betweenthe classical turning points equal to an integralmultiple of Planck’s constant. It is confirmed byone-dimensional simulations and preliminary ex-periments.

ACKNOWLEDGMENTS

A.M. was supported by the Belgian NationalFund for Scientific Research (FNRS). A.M. and J.-P.V.acknowledge the national program on the Interuni-versity Research Project (PAI) and the use of theNamur Scientific Computing Facility, a commonproject between the FNRS, IBM-Belgium, and theFUNDP.

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