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J. Phys. A: Math. Gen. 17 (1984) 1829- 1842. Printed in the UK

Tunnel transitions and vacuum polarisation in the potential well under the influence of an electric field
V P Oleinik and Ju D Arepjev Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, prospect Nauki 115, Kiev, 252028, USSR Received 30 March 1983 Abstract The exact solution of the problem of the vacuum polarisation and tunnel transitions of an electron out of the potential well under the influence of an electric field is obtained. The conclusion drawn earlier concerning the existence of the intense vacuum polarisation mechanism by the external field, for which the number of electron- positron pairs created in the field is proportional to the energy levels width of elementary electron- positron excitations, is confirmed. The estimates of the pair creation probability allow us to think that the atom ionisation in a steady electric field is accompanied by creation of electron- positron pairs which can be quite well registered experimentally at the relatively weak fields. It is shown that at the sudden switching on of an electric field, the value of the tunnel current of electron emission out of the well is affected appreciably by allowance for the positron band.

1. Introductio n
In this paper the non- stationary problem of the vacuum polarisation and tunnel transitions out of the potential well under the influence of an electric field E switched on instantaneously at some moment of time is considered. The tunnel current occurring both in the one- electron state which corresponds to the bound electron state in the potential well before switching on the field E and in the vacuum state is calculated. The electron tunnelling out of the potential well and the creation of electron- positron pairs are due to the decay of quasistationary states being formed in an electric field. When considering these phenomena we use the Drukarjev's method ( Drukarjev 1951 ), which offers the most rigorous and consistent way of describing the decay processes. As is seen from a comparison of the results obtained and the non- relativistic tunneling theory results ( Oleinik and Arepjev 1983 ), when considering the electron tunnelling out of the well in the case of sudden switching on of an electric field it is important to take into consideration the positron band which markedly



The main ideas of the present paper are briefly outlined in Oleinik and Belousov ( 1983 )


2

affects the value of tunnel current. For the one- level well the maximum value of tunnel current is considerably larger than that in the non- relativistic theory. The exact solution of the vacuum polarisation problem given in the paper confirms the conclusion drawn earlier ( Oleinik 1981 ) concerning the existence of the pair creation mechanism for which the number of pairs created in an externa l field is proportional to the energy level width of elementary excitations of the electron- positron field. According to numerical estimates presented, the vacuum polarisation due to the appearance and decay of quasi- stationary states is considerably stronger than that caused by the steady and uniform electric field in the absence of the potential well ( Schwinger 1951 ). In § 2 the general formulae for the electron current density in one- electron and vacuum states are given. In § 3 the correction to the energy and the width of the quasi- stationary electron level in the well are calculated. The formula for the electric current of electron- positron pairs created in an electric field is derived. The calculation of the electron tunnel current out of the well is given in § 4. The formulae for the wavefunctions of a relativistic electron in the well and in an electric field are presented in the appendix.

2. General formulae for the electric current density
Let us consider the electron- positron in the potential well V0 (z ) = -V0 ( z + L) (- z ), V0 > 0 (1)

where V0 and L are the depth and the width of the well. The electron- positron field operator of the system in the SchrÆdinger picture may be presented in the form ( for simplicity, we confine ourselves to investigating the one- dimensional model ): 0 ( z, t ) =




dp 0 m

p 0



(+ )

p 0

( z)e

- ip0 t

+

-m

-



dp0

+ p0



(- )

p0

( z)e

-ip0 t

+ (2)

+ In this expression
(± )
p 0


n

n

(+ )
n

( z)e

- ip0 nt

(z )
(+ )
n

and

(+ )
n

(z)

are the solution of the Dirac equation in the field (1) ( see

appendices ); the signs `+' and `- ` correspond to the electron and positron states, respectively; the functions
(± )
p 0

and p
0

describe the continuous spectra states and the bound states of electron in the

well, respectively;

and are the energy and the spin quantum number, ( = ±1) is the quantum

number characterising the doubly degenerate states with the fixed values of p0 and ; n is the quantum number pertaining to the discrete electron levels with the energy p
0n

in the well;

p 0

,

+ p0

and

n

are


3

the second quantisation operators for the fermi- particles obeying the ordinary anticommutation relations and equalities
p 0

0 =

p 0

0 =

n

0 = 0 ( 0 is the vacuum ket- vector ).

Let us suppose that at the moment of time t = 0 in the rang z f 0 the electric field with intensity E is switched on. The total potential energy of the system is given by V( z, t ) = V0 ( z ) - e0 E z( z)(t ),

(e0
(- )

E>0

)

(3)

The electron- positron field operator in the electric field is now written as follows ( z, t ) =




dp0 m

p0

+

(+ )

p0

( z, t ) +
n
(+ )
n

-m

dp0

+ p 0



p 0

( z, t )




+ (4)

-


n

(z, t )

Here

(± )
p 0

( z, t )

and

(+ )
n

(z, t )
=

are the Dirac equation solutions in the field with the potential energy (3)

satisfying the initial conditions:
(± )
p 0

( z,0)

(± )
p 0

( z ),



(+ )
n

( z,0 )

=

(+ )
n

( z)
+ n

(5) 0 , corresponding to the

The electric current density in the vacuum state 0 and in the state bound electron state in the well is defined by the formulae j v ( z, t ) = e 0 + ( z, t ) z ( z, t ) 0 j
n

( z, t )

= e 0 n + ( z, t ) z ( z, t )

+ n

0,

z 0

z

(6)

Substituting the field operator expression (4) into (6) we arrive at the following relationships: j v ( z, t ) = e dp0
- -m

[

(- )

p 0

( z, t )

]

+

z z

(-)

p 0

( z, t )
(7)

j

n

( z, t )

= j v ( z, t ) + e

[ ( ) ( z, t )]
+ n

+

(+ )
n

( z, t )
)

With the aid of the formulae in the appendices one can easily prove the equalities

[
It is seen from (7) and (8), that

(- )

p 0

( z )]+
+ n

z
+

(- )

p 0

(z )
(+ )
n

= - 2

(

(1 1

)

-1

(8)

[ ( ) ( z)]
j v ( z, t ) = j
n

z

( z)

=0

( z, t )

= 0 at

t 0

(9)


4

The quantities

j

v

and

j

n

have the following physical meaning:

j v is the electric current density j
n

produced by electron- positron pairs created under the influence of the electric field, current density (the emission current) out of the potential well.

is the tunnel

According to (7) the calculation of the electron current density reduces to evaluating the wavefunctions
(- )
p 0

( z, t )

and

(+ )
n

(z, t )

. We expand these functions in terms of the exact solutions of

the Dirac equation in the field V( z) = V0 ( z) - e0 E z( z

) ( z),

(10)

(see appendices). The expansions mentioned above are of the form ( at t 0 )
(- )
p 0

( z, t ) =

(+)
n

+m

dp e 0
+m

- ip 0 t

-m




a

p 0

(

p0 )

p 0

(11)

(z, t ) =

dp e 0

-i p t 0

-m




a

p 0

(n) p ( z),
0

(12)

where a

p 0

(

p 0

)

and a

p 0

(n)

are the constant coefficients defined by the initial conditions (5), the

functions

p 0

(z )

being determined by the formulae (A2.2). In the expansions (11)- (12) we have retained
p 0

only the eigenfunctions

(z )

with the energy p0 ( - m, m) .Allowance for the eigenfunctions with

energies lying outside the above mentioned interval would allow one can to calculate the wavefunction part which describes the spreading out of the wavepacket in time (Drukarjev 1951, Oleinik and Arepjev 1983); in this paper, however, we shall not be interested in the spreading out of the wavepacket in time. In this case the main contribution to the electric current density comes from the poles of integrands in (11)- (12) which occur only at Re p0 ( - m, m) .

3. Electron-positron pair creation
Let us consider the wavefunction formulae a
p 0

(- )

p 0

( z, t )

. The expansion coefficients (11) are expressed by the

(

p0 ) =

+

-



dz

+ p 0

(z )

(- )

p 0

(z)
int

=

1 p0 - p



00



dz

+ p 0

(z)

H

int

( z )

(- )
p0

( z);

(13)

H

-e0 E z
p0

Making use of the formulae in the appendices for the wavefunctions may be transformed into the form

and

(- )

p

the coefficients (13)


5

a

p 0

(

p0 ) = - +r




2m 2 * d p0 - p0 *
(+ )
1



{r ( )[c
- 1

*

(1- ) ( p , p 0 ) + c * 0 2
0

(- )
2

(

p0 , p

0

)]

+ (14)

(+ )

[c

1

(

p0 , p0 ) + c* (2+ ) ( p , p 2 0

)]}, (i )


where the following notation is used
(± )
1

( (

p , p 0

0

) )

= e 0E dzD 0

i

(i)

+e

i 4

p 0 ± k1 2e 0 E

D

i -1

ze

mik1 z

(15)



(± )
2

p0 , p

0

= e 0E dzD 0


-i -1

( )
;

+e

i 4

p0 ± k1 1 D 2e0 E

- i

()

ze

m ik1z

(16)

r1( m ) = (11) + * ± 1

( -1 )
m1

= e r
(m -1 )

- i 4

2 ( p0 + e0E z), 2e0 E

= 2 (11) + ±

( -1 )
m1

;
(-)
p 0

Taking into account the equalities (14) and (A2.2) the wavefunction, follows ( at z > 0 ): â {r(- ) c1 *
(- )
p 0

may be written as

( z, t )

= - (2

)

-1

d e

2

+m

-m



dp exp (- i p0 t )( p0 - p 0

0

)

-1

â
0

[

(- )
1

(

- p , p 0 ) + c * (2 ) ( p , p 0 2 0

0

)

]

+ + r( + ) c1 * (1+ ) ( p , p0 ) + c * (2 ) ( p , p 0 2 0

[

)]}â
(17)
0

1 d â * u i D dz c1

-i

()

+

d u i D * c dz 2

i -1

(i)

by p 0

where the prime ( ) means that in the corresponding quantity one ought to replace p ( for example c 1 = c
1p 0

=p 0

).

The integrand poles in (17) coincide with zeros of the functions c 1 * and c * . Taking into account 2 (A2.3), we represent the equalities c n* = 0 (n = 1, 2 ~* + ~* 1 -1 k2 ~ * - * = V - ia * ; ~ 1 -1 0 n where a 1 = -e
-i 4

)

in the form

(

n = 1, 2

) )

(18)

2e0 E 2e0 E

d ln D d0 d ln D d0

i -1

(

i

0

a 2 = -e

- i 4

- i

( 0 )

(19)

We confine ourselves to investigating the weak electric field, for which the condition


6

>> 1 is fulfilled. In addition to (20) we shall henceforth assume that >> - ~ 0 >> 1, p2
-
1 2

(20)

(

-~ p

2 0

)

3

2

>> 1,

~ p p0

0

2e0 E

(21)

Assuming the inequalities (20)- (21) to be satisfied, we turn to computing the asymptotic formulae for the functions D
-i

( 0 )

and D

i -1

(i0 )

. These functions satisfy the equation (22)

d 2 dy 2 where

y2 + 4 - b = 0

y = 2 ~0 , b = + i 2 . According to Abramowitz and Stegun (1964) the linearly independent p

solutions of (22) are of the form
(± )

( 0)

~ p

=

4

(b

1 -~ p
2 0

)

exp

± -

(b

~ p

-~ p

0

2 0

)

- b arcsin

~ p

1 + b8
0

d

(b

3

-~ p

23 0

)



, (23)
-i

1 ~3 p d 3 - 0 - b~0 . p 6 b

The formulae (23) may be derived by the JWKB method. We represent the functions D and D
i -1

( 0 )

(i0 )

in the form of linear combinations of the functions (± ) ; in particular D
-i

( 0 )

= 1

(- )

( ~0 ) p

+ 2

(+ )

( ~0 ) p

(24)

where the constants n (n = 1,2

)

may be defined by using the known expression for the parabolic

cylinder functions at ~0 0 . A simple calculation leads to the following formulae p 1 = e 2
-

2

- i 2

+ i 2
4

i2 1 + 2



;

2 = 2 1 (e



- 1 ). 2

(25)

Making use of the presented relationships and retaining throughout only the largest terms, we obtain a1 = -
2

e 0E d ln 2 d~0 p

(+ )

me

-

(- ) 2 (+ )

Then with the aid of the equalities (23) we find: a 1 = 1 + a
2 ±


7

4 eE a ± = - 0 2 ( p 0 - i 1 ) ± i 1 exp - 2 1 3 The substitution of (26) into (18) gives G( p
0

(

-~ p

23 0

)



(26)

)(
±

2 2 k2 - 1 - V02 )sin k2 L - 2 1 k2 cos k2 L =

= ( a

)* [k

2

cos k2 L + (1 - iV0 ) sin k2 L]
0

(27)

If an electric field is absent the last equality goes over into the dispersion equation G ( p

)

= 0 (A1.8)

defining the energy levels of the bound electron states in the well. Denoting the roots of the equation (27) by p
0n

=p

(0 )
0n

+ p

0n

- i

(± )
n

, where p

(0 )
0n

are the roots of the dis persion equation (A1.8), we obtain the
0n

following equation for defining the quantities p dG( p dp0
0

and

(± )
n

:
p0= p(0) 0n

)
p0 = p ( 0 ) 0n

(

p0 n - in(± ) = (a

)

±

)* [k

2

cos k2 L + (1 - iV0 ) sin k2 L]

(28)

From (28) we get p
0n

1 1eE k22 p + V0 1 + 1 L 0 =- 0 2 8 1 V0 ( p0 + V0 2) p0 + V0

2

-1

(29)
p0 = p ( 0 ) 0n -1

n

(± )

2 1 k22 =± exp - 4 mV0 ( p0 + V0 2)

[

4 3

(

- ~ p

2 0

)

3

2



-

1

2

]

1 p + V0 1 + 1 L 0 2 p0 + V0

2

p0= p(0) 0n

The quantities p

0n

and



(+ )
n

represent the shift and the width of energy level of a quasi-

stationary state appeared in the electric field. In order to compare these quantities and the analogous ones ( E
n

and n ) of the non- relativistic theory (Oleinik and Arepjev 1983), we pass to the non- relativistic

energy reading by putting,
p 0 = m - V0 + E

(30)
2 mE

Then

1 = 2m(V0 - E ),

k2 =

(

-~ p

23 0

)

=

2m e 0E

(V0

-E

)

3

~ 3 2

(31)

Using the last formulae and retaining only the largest terms, we obtain (in the ordinary system of units) p
0n

=-

he0 E 2E 2 (1 + 1 1L 2 m(V0 - E ) 8V0

)-

1 E =E (0)
n

(32)


8

n(± ) = ±(V0 - E The quantities p
0n

)

E ~ exp - 2 (1 + 1 1 L 2 V0

(

)

)-1

( E =E n 0 )

and

(+ )
n

coincide exactly with the non- relativistic ones ( E

n

and n ).

At t - z > 0 the integration path in (17) may be closed in the lower half- plane of the complex variable p 0 . Taking into account that it is only the function

(c )
* 1

-

1

that has singularities in the lower half-

plane of p 0 and retaining in (17) only the singular terms, we obtain âr
(- )
p 0

( z, t )
2

= id e

2



Re s e p = p0n 0
(+ )
2

- ip t 0

(

p - p 0

0

)

-1

c 2* â c1 *

(33)

[



( - ) (- )



(

p , p 0

0

)

+ r(+ )

(

p , p 0

0

)

]u



d i D dz

- i

()



where the symbol Re s means the residue in the pole defined by the formula
p0

=

p0 n

p

0n

=p

(0 )
0n

+p

0n

- i

(+ )
n

For convenience, we represent the expression (33) in the form g
(- )

p 0

( z, t ) =
n

g

n

( p 0 )e
)

- ip0 n t

d u n i D dz
0

-i

( n )

(34) â ~ 2

n

( p0 )

= -id e â

2

e

i ( ln - - 4

(

p0 n - p

)-1 (1

+ 1 1 n L 2

)-

1

2 2 1 n k2 n (- ) ( - ) r 2 ( p 0 n , p0 ) + r(+ ) (2+ ) ( p0 n , p 2 2m V0

[

0

)]exp

(-

n

)
)
3

(35)



1n

=

1 p =p 0

0n

,

k

2n

=k

2 p =p 0

0n

,

~ 2 n = 3 ; n =

(

- ~02 p

p0 = p

0n

d u n i D dz

-i

( n )

d = u i D dz

- i

( )
=
n

p0 = p

0n

In deriving the relationships (34)- (35) we have used the equality c* Re s 2 = - e p0 = p 0 n c * 1
i ( ln - - 4

)

~ 2 2 1n k2 n exp - 2 n 2m 2V0 1 + 1 1 n L 2

(

)

Let us calculate the electric current density in the state (34) keeping only one term in the sum over
n:

[ ( ) ]

+ - p 0

z

(-)
p0

= 2m 2 e

- 2

g

n

( p0 )

2

e

( - 2 n + ) (t - z

)

(36)


9

Here the equality D is taken into account. Further calculation of the vacuum current density reduces to the calculation of the coefficients (2± ) ( p , p 0
0

-i

( n )

2

-D

-i -1

( n )

2

=e

- 2

e

( 2 n +) z

e0 E z m >> 1

)

, defined by (15). With the aid of the asymptotic formulae (23) and (24) we obtain the
14 ~ ~ ( + 1 i ) 2 1 + p 0 ± k1 I ( 1 (i + 1)) 2

following representation: where z I 1 = dz - ~0 p 0


(± )
2

(

p , p 0

0

)

= e 0E e

i 4

2

- i 2

1

(37)

(

2 14 z

)

exp - 2

~ p0

dx(
z

- x2 + 1 i 2

)

12

0

m ik1 z

(38)

~ = ( p + e E z)(2e E p0 z 0 0 0

)

-1 2

;

~ k1 = k1 (2e 0E

)

-1 2

The integration in (38) by parts using the equality
e
G (z

)

dG ( dz = dz

z)

-

1 G (z

de

)

where G( z) = -2 yields
~ p0

dx(
z

- x2 + 1 i 2

)

12

m ik1 z

0

(
0

p 0 1 2 exp - 2 dx - x 2 + 1 i 2 z exp G( z ) 1 0 dz = 14 12 ~2 ~ 2 1 4 2e 0E - p0 z - ~0 2 p - ~ 2 p0 ± ik1

~

(

)

)

(

) [(

)

]

(39)

Taking into consideration the equalities (37)- (39) we arrive at the following representation for the coefficients (2± ) ( p , p 0
2
0

)

:
0

(± )

(

p0 , p

)

1 = e 22

i 4

2

-i 2

e

- 2

~ ~0 ± k1 1 + p â (1 + i )
-1 1 2

32 1 4 exp 2 -1 2 - ~0 2 p 3 â 14 12 ~ - ~ 2 p0 - ~ 2 ± i k1 p0

(

) [(

[

(

)

)

] ]

2

(40)


10

With the aid of relationships (35), (360 and (40) we find j
p0 n



[
, 2

(- )

p 0

]

+

z
4

(-)

p 0

=

1 mk 4 = 2 2 2n 8 V0 1n â

k1 p +p 0 0n
2

1n 2 p -p 0
+

2 0n

e

~ -2

n

(1
+

+ 1 1n L 2 t - z)

)-

2

â

= ±1



d

r(- )Q
(±)
n

(- n

)

+ r(+ )Qn(
(± )

)

2

exp - 2n(

[

)

(

]

(41)

Here the following notation is used: Q
(± )

Q

(

p0 n , p

0

)
2

Q

(

p , p 0

0

)

k = ±1 m i 1 1

1 m k1 ; p0 + m
En = p
(0 )
0n

~ 2 2m n = (V0 - E 3 e0 E

n

)

32

,

- m + V0

Making use of the equalities (16), (A1.12) and (A1.13), one can readily derive the formula

= ±1



d

2

r (- )Q

(- )
n

+ r( + )Q 1
(1 )
1 2

(+ n

)

2

= + c.c.

p 0 + k1 ( = Q 2 8m k1 n The quantity j
p 0n

-

)

2

p -k + 0 21 m
-
0 6

(

)

2

Q

(+ )
n

2

+

[Q

(+ )
n

Q

* (- n

) (1 )

-1

* (1 1

)

]

(42)

decreases as p

2 at p 0 >> m 2 .

Let us confine ourselves to the non- relativistic values of the quantity p 0 :
p
0

- m V0 - E

n

In this case the expression in curly brackets in (42) slightly depends on p 0 and equals unity in the order of magnitude. To obtain simplifications we obtain: j 1 = En 8V 0
2

the numerical estimate, we replace this expression by unity. After some

p0 n

p0 - m V - E n 0



1 2

V0 - E p -p 0

n 0n

e

4

~ -2

n

(1

+ 1 1 n L 2

)-

2

( exp - 2n + ) (t - z

[

)]

(43)

According to (43) the dependence of the quantity j

p 0n

on the field is of the form:


11

j

p0 n

(e0 E ) n(
2

+

)

The appearance of the factor exp - 2
j
p 0n

[

(+)
n

( t - z ) ] in (43) allows one to interpret the quantity

dp

0

in the following way. This quantity is a flux of electrons created in pairs together with positrons in

the well under the influence of an electric field E and moving away from the barrier in the direction
z + at the velocity of light, the energy of positrons formed in the field lying in the range

(

- p 0 , - p 0 - dp

0

)

-

. It is obvious that the quantity - e

0

-



m

j

p 0n

dp 0 Q at t - z 0 , z + is the total

electric charge of electrons created per unit time in the field under consideration. The peculiar feature of the given model consists in the fact that in the range z - the positron flux does not occur; the total electric charge of positrons equal to - Q is concentrated near the boundary z = - L of the potential well. Let us estimate the quantity j
-m - (m +

Vn



dp0 j
)

p 0n

,

assuming that the condition << V0 - E n is satisfied. We may approximately put:
-m -(m +



)

p -m dp 0 0 V - E n 0



1 2

V0 - En 3 2 2 p - p 3 (V0 - E 0n 0

4

n

)

-1 2

Estimate the vacuum current for the following values of parameters of the problem E = 5 10 6 v cm , Write out some auxiliary quantities b V0 e0 E = 0.2 10 -6 cm; - ~ p
2 0n

V0 = 1ev,

E

n

V0 =

1 2

,

V0 - E n =

1 5

(44)

= m 2 c 3 2he0 E = 1.3 10 9 ; ~ 2 n = 4.8

= 0.3 10 4 ;

Putting additionally 1 L 1 , we obtain jVn 10
-10

sec

-1

(45)

It should be emphasised that the pair creation probability in electric field in the presence of the potential well is by no means exponentially small. Remember that according to Schwinger (Schwinger 1951) the pair


12
-2

creation probability in an electric field in the absence of the well is proportional to the exponent e which equals exp ( - 8 10 ~ exp - 2
9

)

for the chosen parameters (44). Compare this quantity with the exponent

(

n

)

involved in (43): for the same parameters the latter exponent is equal to exp ( - 4 .8 ) !

4. Electron emission
Now we turn to calculating the tunnel current of electron emission defined by the wavefunction (12). The coefficient a
p 0

(+ )
n

(z, t )

(n)
p 0

in (12) is expressed by = 1 p - p 0


a

(n )

0n

(0 )


0

dz

+ p 0

( z)

H

int

(z )(n+) (z )

Using the formulae in the appendix we obtain the following representation a
p 0

(n)

~~ ~ = -2m 2 * 0 n ( 1n + p â c1* ( p ) 1 ( p , p 0 0

-1n

)(

p - p 0

( 0 ) -1 0n

)

â
(0 ) 0n

[

(0 0n

)

)

* + c2 ( p ) 2 ( p , p 0 0

)]



(46)

1 p , p 0

(

(0)
0n

) )

= e0 E dz D
0



[

i

(i)

+~ p

(

(0 ) 0n

- i~

1n

)e
1n

i 4

D

i -1

(i)

]ze

-1 n z

2 p , p 0

(

(0 )
0n

= e 0E dz D
0

[

- i -1

()

+~ p

(

(0 ) 0n

- i~

)

1

e

i 4

D

- i

( )]ze

- 1n z

(47)

where

~ p

0n

(0 )

=
<