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Tunnel Current in MIM System

1.2 Tunnel Current in MIM System

John G. Simmons Formula

Now, using the Sommerfeld model (see chapter "Metal Energy-Band Structure") and WKB approximation (see chapter "Tunneling Effect in Quasiclassical Approximation") and assuming that T = 0, potential barrier is of arbitrary shape and the mass of electrons is isotropic in space, we can derive an expression for the tunneling current flowing in a metal-insulator-metal (MIM) system.

Fig. 1. Diagram of MIM system in equilibrium.

j1 and j2 – work function of the left and right metals, respectively

Fig. 2. Model of MIM system with an arbitrary shape potential

barrier. Positive potential is applied to the right metal

Consider two metal electrodes with an insulator of thickness L between them. If electrodes are under the same potential, the system is in thermodynamic equilibrium (see chapter "Metal Energy-Band Structure") and Fermi levels of electrodes coincide (Fig. 1). However, if electrodes are under different potentials, current flow between them is available. Fig. 2 shows the energy diagram of electrodes with applied bias energy eV. Potential barrier width for electrons occupying the Fermi level is denoted as δz = z2 – z1. Consider that all the current flowing in the system is due to the tunneling effect. Probability D(Ez) of the electron transmission through the po-tential barrier of height U(z) is given by expression (4) in chapter "Tunneling Effect in Quasiclassical Approximation". For the number of electrons N1 tunneling through the barrier from electrode 1 into electrode 2, we can write [1], [2]

∫

∫ ∫ ∫∞

=

=+−=∞

∞−

∞

∞−

∞

∞−

E

zzz

zyxzz

dEpnED

dpdpdpEDeVEfEfm

pN

0

21331

)()(

)())(1)((4 hπ

(1)

where

∫ ∫∞

∞−

∞

∞−

+−= yxz dpdpeVEfEfpn ))(1)((4

1)( 2133hπ (2)

and Em – maximum energy of tunneling electrons. Integration of expression (2) can be performed in polar coordinates. Because in the model under consideration , and total energy is

222yxr ppp += mpE rr 2/2=

rz EEE += , changing variables θcosrx pp = , θsinry pp = , we get

1.2 Tunnel Current in MIM System

∫

∫ ∫∞

∞

+−=

=+−=

02132

2

0 02133

))(1)((2

))(1)((4

1)(

r

rrz

dEeVEfEfm

ddppeVEfEfpn

h

h

π

θπ

π

(3)

Substituting (3) in (1), we obtain

∫∫∞

++−+=∞

021

0321 ))(1)(()(

2 rrzrz

E

zz dEeVEEfEEfdEEDmNhπ

(4)

The number of electrons N2 tunneling back from electrode 2 into electrode 1 is calculated in the same way. According to (4) from chapter "Tunneling Effect in Quasiclassical Approximation", the potential barrier transparency in the given case will be such as if positive voltage V is applied to electrode 1 relative to electrode 2. In this case

∫∫∞

+−++=∞

012

0322 ))(1)(()(

2 rrzrz

E

zz dEEEfeVEEfdEEDmNhπ

(5)

Net electrons flow N through the barrier is obviously N = N1 – N2. Let us denote

∫∞

+−=0

21321 ))(1)((2

)( rz dEeVEfEfmeEhπ

ξ

∫ −+=0

12322 ))(1)((2

)( rz dEEfeVEfmeEhπ

ξ∞

[∫ +−=−=0

213221 )()(2

),( rz dEeVEfEfmeeVEhπ

ξξξ ]∞

(6)

Then, the tunneling current density J is

∫∞

=E

zzz dEeVEEDJ0

),()( ξ (7)

According Fig. 2, U(z) can be written in the form )()( zzU ϕμ += . Then, integrating (4) from chapter "Tunneling Effect in Quasiclassical Approximation" and using expression (A5) from Appendix, we get

{ }zzz EzAED −+−∞ )(exp)( ϕμδ (8)

where ϕ – average barrier height relative to Fermi

level of the negative electrode; ∫=2

1

)(1 z

zz

dzzφδ

φ ;

2

22h

mA β= , β – dimensionless factor defined in

the Appendix (A6). At T = 0 K

[ ][ ]

⎪⎪⎩

⎪⎪⎨

⎧

>

−∈−

−∈

=

μ

μμμ

μ

πξ

z

zz

z

z

Eпри

eVEприE

eVEприeVmeE

,0

;,

;;0,

2)( 32h

(9)

Introducing (8) and (9) into (7), we obtain

[ ]⎪⎩

⎪⎨⎧

+−+−= ∫−eV

zzz dEEAeVmeJμ

φμδπ 0

32 exp2 h

[ ]⎪⎭

⎪⎬⎫

−+−−+ ∫−

μ

μ

φμδμeV

zzzz dEEAE exp)( (10)

Integrating (10), we get

( ) ( ) [ ]{ }eVAeVAJ zzz

+−+−−= φδφφδφδα expexp2

(11)

where . h224/ βπα e= Thus, expression (11) approximates the tunneling current in the MIM system for arbitrary barrier shape.

Summary

− The general expression (7) to calculate the tunneling current in the MIM system was derived in this chapter.

− The analytic approximate solution (11) of tunneling current in the MIM system was calculated.

1.2 Tunnel Current in MIM System

References

1. Burshtein E., Lundquist S. Tunneling phenomena in solid bodies. Mir, 1973 (in Russian) 2. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 1793. 3. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 238.

• Appendix

Let us integrate an arbitrary function )(zf from z1 to z2.

∫2

1

)(z

z

dzzf (A1)

Defining f as

∫=2

1

)(1 z

zz

dzzffδ

(A2)

where f – average value of a function f on the interval from z1 to z2, 21 zzz −=δ . Then equation (A1) can be rewritten as

[ ]∫∫

−+=

2

1

2

1

)(1)(z

z

z

z

dzffzffdzzf (A3)

Considering a Taylor series expansion of the integrand (A3) in and neglecting [ ]3/))(( ffzf − and higher order members, we get

[ ] [ ]∫∫ ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

−−

+=2

1

2

1

2

2

8)(

2)(1)(

z

z

z

z

dzffzf

ffzffdzzf

(A4)

The second term in (A4) vanishes upon integration, therefore (A4) can be expressed as

z

z

z

fdzzf δβ=∫2

1

)( (A5)

where the correction factor is

[ ]∫ −−=2

1

2

2 )(8

11z

zz

dzfzff δ

β (A6)

John G. Simmons Formula in a Case of Small, Intermediate and High Voltage (Field Emission Mode)

According chapter John G. Simmons Formula, the approximate expression for the tunneling current in the MIM system can be written as [1]:

[ ]{ }eVAeVAJ zzz

+−+−−= φδφφδφδα exp)()exp(2

(1)

where , h224/ βπα e= 2

22h

mA β= , φ – average

barrier height, zδ – barrier width, V– voltage between electrodes.

Small voltage

At low voltages eV>>φ , expression (1) can be simplified [1]

)exp( φδδφγ

zz

AV

J −= (2)

where 2242hβπ

γ me= . Since eV>>φ , we can

consider that φ doesn't depend on V. Thus, in the case of small applied voltage, the tunneling current proportionate to V. Energy diagram of the MIM system then eV>>φ is shown on Fig. 1.

1.2 Tunnel Current in MIM System

Fig. 1. Potential barrier in the MIM system then V ~ 0.

1φ and 2φ – work function of the left and right metals, respectively

In this case, as shown on Fig. 1, Lz =δ and

2/)( 21 φφφ +=

Intermediate voltage

If 2φ<eV , then Lz =δ and 2/)( 21 eV−+= φφφ (Fig. 2).

Fig. 2. Potential barrier in the MIM system then 2φ<eV .

1φ and 2φ – work function of the left and right metals, respectively

In [2] it is shown, that for this case the tunneling current-voltage relation is given by

))(exp( 3VVAJ zz

σφδδφγ

+−= (3)

where 23

2

2

2

3296)(

φδδφσ

zz

AeAe−=

High voltage – Field emission mode

The case when 2φ>eV corresponds to energy diagram shown in Fig. 3 and to the following

)/( 211 eVLz +−= φφφδ , 2/1φφ = .

Fig. 3. Potential barrier in the MIM system then 2φ>eV .

1φ and 2φ – work function of the left and right metals, respectively.

Substituting zδ and φ into equation (1), we obtain

⎪⎭

⎪⎬⎫⎥⎦

⎤⎢⎣

⎡+−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

⎪⎩

⎪⎨⎧

−⎥⎦

⎤⎢⎣

⎡−=

1

231

1

2312

12

23

2122exp21

22exp8

φφβ

φ

φββφπ

eVmeF

eV

meF

FeJ

h

hh (4)

where LVF /= – electric field strength.

1.2 Tunnel Current in MIM System

At high applied voltage )( 1 μφ +>eV the Fermi level of electrode 2 is lower than the conduction band bottom of electrode 1. Under such conditions, electrons can not tunnel from electrode 2 into electrode 1 because of lack of empty states. An inverse situation is for electrons tunneling from electrode 1 into empty states of electrode 2. This process is similar to autoelectronic emission from a metal into vacuum. Thus, since μφ +> 1eV , the second summand in (4) can be neglected and for the current we get

⎥⎦

⎤⎢⎣

⎡−=

Fm

eFeJ

231

1

2

22

3 22exp8

φβφβπ hh

(5)

where coefficient β = 23/24. This result agrees qualitatively with an analytical expression for the field emission current density [3]. Thus, using formulas (2)–(5), we can compute the tunnel current at given system parameters and plot current-voltage characteristics. Fig. 4 shows theoretical tunneling current-applied voltage plot in case of carbon electrode 1 ( 1φ = 4,7 эВ) and platinum electrode 2 ( 2φ = 5,3 эВ) at zδ = 5 Å and contact area S = 10–17 m2.

Fig. 4. Current-voltage characteristic for carbon electrode 1

and platinum electrode 2 at zδ = 5 Å and contact area 10–17 m2. Parts of J(V) curve correspond to the following expressions:

AB – (22), BC – (23), CD – (24), DE – (25).

Summary

− Depend upon magnitude of applied voltage, formula (1) can be simplified (2)–(5).

− It is possible to describe the experimental tunneling current dependences by approximated expressions (2)–(5) in accordance with magnitude of applied voltage.

References

1. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 1793. 2. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 238. 3. Dobretzov L.N., Gomounova M.V. Emission electronics. Nauka, 1966 (in Russian)