Phonon-assisted resonant tunneling

Solid State Communications, Vol. 66, No. 1, pp. 65-69, 1988. Printed in Great Britain.

0038-1098/88 $3.00 + .00 Pergamon Press plc

PHONON-ASSISTED RESONANT TUNNELING L.I. Glazman Institute of Problems of Mlcroelectronics Technology and Superpure Materials, USSR Academy of Sciences, 142432 Chernogolovka, Moscow District, USSR and R.I. Shekhter Physico-Technical Institute for Low Temperatures, Kharkov, USSR

(Received 6 November 1987 by F.M. Agranovich) The influence of phonons on the effect of resonant electron tunneling through a dielectric layer involving local states is considered. An expression for the tunneling transmission coefficient has been obtained which is valid at arbitrary values of temperature and the strength of the electron-phonon interaction (EPI). The ratio of polaron shift on the centre to Debye frequency is shown to be the only relevant dimensionless parameter determining the contribution of EPI to the tunneling effect. The results presented enable interpretation of the experiments [3, 4] on the conductivity of tunneling contacts with amorphous silicon layers.

CHARGE TRANSFER IN disordered media is determined by phonon-assisted electron hops between localized states. The processes involving emission (or absorption) of a single phonon dominate upon weak electron-phonon interaction (EPI) [1], the temperature dependence of conductivity being insensitive to the EPI value. When EPI is strong and the polaron shift on the local centre exceeds a characteristic phonon (Debye) frequency, electron transition is accompanied by multiphonon processes which affects the temperature dependence of conductivity [2]. Strong EPI can be realized for deep centres in amorphous materials. Tunnel junctions involving thin layers of disordered material [3, 4] are convenient for the mvestigation of the electron-phonon interaction on centres. In this case EPI processes considerably influence electron resonant tunneling through individual local centres in the barrier region. Studies of the I-V characteristics of this interaction may serve as a method of EPI spectroscopy as is done in nonelastic tunneling spectroscopy of organic molecules [5]. The paper reports on the results of the phononassisted resonant tunneling theory which holds for any intensity of EPI. The temeperature dependence of conductivity and nonlinearities of I-V characteristics of amorphous layer contacts has been obtained. The results of weak EPI enable interpretation of the experiments performed by Beasley and co-workers [3, 4].

The investigation of EPI as part of the resonant tunneling problem calls for consideration of the effect of hybridization of localized electron states with phonon degrees of freedom. The discrete energy level of the electron state is split into the polaron state band [6]. The occurring modification of electron resonant transmission can be analyzed within a simple model in which the resonant state is formed by a rectangular potential U(x) (Fig. 1). With the EPI on the centre being VF/S (VF is the Fermi velocity, s is the sound velocity) times more effective than the EPI for band electrons, the Hamiltonian of the system of electron and phonon degrees of freedom can be written as

:2 -

2m + U(x) + ~p~(P, ~) + A fi-- Otx(ao a

x)],

(1)

where P, x are the momentum and coordinate of the electron;/5, ~ are corresponding phonon operators, 0 is the Heaviside step function. In Hamiltonian (1) we consider the interaction of electrons with a single phonon mode defined by the deformation potential constant A; a is the lattice constant. The tunneling problem can be analyzed by means of the solutions of the Schrrdinger equation for each of the three barrier-separated regions and by further 65

Vol. 66, No. 1

PHONON-ASSISTED RESONANT T U N N E L I N G

66

T(e, --+ < )

=

2 FiF2 ~-~

; dte.(~ ~2),~ oc

× ; dll e-qW+~(q-e'°)]ltt

D2

D~

o

l I

)

X

; d/2e-t2[r-t(~2 e0)]~ 0

x ao

x Sp 00exp tva ~,~ ph ,2]

Fig. 1. Model potential forming the resonant state for a tunneling electron. D~, D2 are transparencies of barriers 1, 2. The electron energy may be affected by the interaction with phonons in the interval 0 < x <

a0• matching using transfer matrices for the barriers. If reflection (rL2) and transmission (tl,2) amplitudes are introduced for the barriers (see Fig. 1), the probability of electron transition from the initial state i in the left half-space into its final state f in the right half-space can be expressed as (D,. 2 = It, 212"): o0

W,.f

Ol D2

x exp

i ~(A)t, + t - t2) - ~-~phU)

hv r

=

r I --t- r 2 ,

El2 -

~aa0

D12.(4)

e,£,~o

The trace in (4) is readily calculated, since ~,,~(A)ph,~p~ are the quadratic forms of phonon operators (similar calculations were made in the Raman light scattering x (~o;/~,)(~,/~,) (2) theory [7]). Formula (4) is easily generalized to the case of many phonon modes defined by wave vectors q. The Here ~o,, q~I are the eigenfunctlons of the phonon EPI value therewith is determined by the dimensionoscillator described by the Hamlltonian .,p(0). ~'g ph ' ~'t are .a~(A) ~z~(0) _+_ less ratio of the polaron shift at level %h and Debye the elgenfunctlons corresponding to ~ph = ~ph A~/a. At a2o]k2 - ~ ] <~ I the momentum of the mcom- frequency. ing electron is related to that of the interbarrier elec). : %.~coD, tron (k = 2 x ~ and/~t, respectively) by Ahq I (5) ~, = k + ~vv (E~d~ - E;2)' v = hk/m, (3) q l~=o 1 - rl r2 exp (2i~ao)

2.

where ~0°, E~A ° are eigenvalues corresponding to ¢p,, qJ~. In formula (2) allowance is made for the fact that the initial and final states are the products of pure electron and phonon states, the total energy of the electron and phonon being conserved. The value w,~ should be averaged by the initial state of the phonon system which is taken to be equilibrium defined by the density matrix 00 "~ exp { - f l ~(o) -~ ph J ' Provided I(e - eo)/eol ~ 1 (~o = n2ta2/2ma~ is the level of spatial quantization in the interbarrier region) expression (2) can be simplified by representing the ratio In (2) as a sum (by the index n) of a geometric series. If DL2 ~ 1, it is possible to change from summation over n to integration over "delay time" t = (2ao/v)n. These transformations yield the transmission coefficient T(g~ --+ e2) as

(M is the mass of lattice atoms, coq is the phonon frequency spectrum)• Relationship (4) determines probabilities of both elastic (e, = e2) and inelastic (e~ ¢: e2) tunneling processes at an arbitrary value 2. It follows from (4) that the integral probability of tunnehng is independent of the EPI value' f dg, f de2T(e, ~ e2) - f delT,(e,)

-

4nFl F2 F, + F2" (6)

However, when EPI is strong (2 >> 1), tunneling proceeds mainly in inelastic channels. The maximum probability of elastic tunneling Te~ is estimated by F~F2 Tc,(e , ~ el) ~" e2A(T), hO2q ezA(T) = Z l % [ 2 c ° t h ~ - , q

Vol. 66, No. 1

PHONON-ASSISTED RESONANT T U N N E L I N G I~, -

e0[ ~ e^(Z),

(7)

and constitutes a small fraction ,~ F/eA(T ) < 1 of the total probability T,(e). In the case of weak EPI (2 < 1) tunneling is largely elastic. The EPI reduces the probability of elastic tunneling and gives rise to channels of nonelastic tunneling accompamed by emission or absorption of a single phonon

67

,

/

70

Averaged

-

~te:°:uCgP'oCns

/resono.nt

tunneLing

2~D=r~ct__

4 F l 1"2

T(el -~ e2) = (el -- eo)2 + F 2 6(el

e2)

--

r

I

tunnetmg

4F11"2 "1-

(e I __ F.~0)2 Aft F 2

4r, r2

+ ( ~ _ ~o)2 + r~ x

[A~-6(~,

+ Ag6(el -

(A~

-

x

e2

e~)l;

=

1 .

(8)

In the following we use the above results for interpretation of the experiments [3, 4]. In those works resonant tunnehng occurred through amorphous dielectric layers of thickness d comprising a large amount of localized states with energies distributed over a wide range. In the actual energy region near the Fermi level the local state density may be assumed as constant: g(e) = g(ee). Estimations show [6, 8] that if

d < x/~o,

So = mao/g(eF)fl 2,

(9)

then the processes of resonant tunneling through isolated impurity centres play a leading role in resonance transmission. Under the conditions of the experiment [4] (g ,-~ 10+19eV-lcm -3, a0 "~ 10A) we obtain ~ 0 - 300/k, and inequality (9) is fulfilled for the used thicknesses of the dielectric layer d < 100 A. The conductance G of the contact is determined by the total contribution of resonance transmission through different impurities in the contact plane. In contacts with a sufficiently large area S satisfying the relation

In(S/So)> m a n { d ,

(S/So)

Fig. 2. Different cases of contact conductivity depending on the parameters d and S.

~q 2

coth-~_+

kn

q

la~q)

-

I

!,n (m---"ff~o T) x

+ hO)q)

+ ~)6(el 1

×

e,, -

1

I

Ln (SOKF2 )

ln[h2/maETl},

(10)

there is an impurity realizing resonant tunneling for each electron in the actual energy regmn. Hence, under these conditions the conductance of the contact

does not suffer mesoscop~c fluctuations and is defined by the value (T(el --* e2)) averaged by position and energy of local centres. Once inequality (10) is violated, the characteristics of the contact depend on specific realization of impurity distribution, G fluctuations being significant. The change of the sign in mequality (10) corresponds to the solid line in Fig. 2 illustrating different cases of contact conductivity. In the region under the line G(d) is a self-averaging value. The tunneling process (either direct or resonant) that makes a major contribution to G determines the G(d) relationship. In the absence of EPI resonant tunneling dominates if ffnke is the Fermi electron momentum within the contact): d > do,

do = a01n(S0k2F).

(11)

The region in which inequality (11) is satisfied is above the dotted line shown in Fig. 2. Behind this hne crossover occurs, i.e. the dependence G(d) ~ exp ( - 2 d / a0) specific for direct tunneling, is substituted for G(d) ,,~ exp ( - d / a o ) realized in the resonant case. The crossover in the G(d) relationship holds for the case of weak EPI. But at strong EPI the resonance elastic transmission coefficient is proportional to the product F 1F2 ~ exp (-2d/ao) according to formula (7). Hence, at 2 >> 1 there is no crossover in the In G(d) relationship. A strong nonlinearity of the I-V characteristic of the contact should be simultaneously observed at eV ,-~ eA(T). This is associated with a relatively small value of the probability of elastic tunneling (7) which governs conductivity at a low bias (eV < eA(T)) and a large value of conductivity determined by the integral transmission intensity (6) at eV >~ e h ( T ) (see Fig. 3).

PHONON-ASSISTED RESONANT TUNNELING

68 G (V) G (0)

Weak EPI opens nonelastic tunneling channels which increases the differential conductivity dI/dV on the area following the current step [4]. Formula (8) allows obtaining the relation between the gain 6(dI/dV) and the constant 2'

~^(T) F I

l(d,)

I

t I

v

~^(T) Fig. 3. Schematic representation of differential conductivlty G(V) = dI/dV as a function of V for "large" contacts (see (10)) in the case of strong EPI (2 ~> 1). The temperature dependence of conductance is also governed by EPI intensity on the centre. Analysis of (4) indicates [6] that at 2 ~> 1 the temperature correction 6G(T) to the conductance G(T = 0) is 6G(T) G(0)

T fie)o - -

fi~Oo T

In - -

(12)

In the case of a weak interaction equations (5), (8) yield a T 2 law for &G(T):

G(0)

2 ~

.

(13)

The experiments [4] provided results for "large" (S~ --~ 10-*cm 2) and small ($2 ~ 10-9cm 2) contacts. For the conditions in [4] ln(S0k2F)- 9, ln(Si/ So) -~ 16 and ln(S2/So) ~- 4.6 (lines 1 and 2 reveal the position of the investigated contacts with the areas S~ and $2 on the diagram of Fig. 2). In the G(d) relationship for "large" contacts the crossover was observed at thickness d ~- 55A which is in good agreement with the estimation made by formula (11): do ~- 70 A. The T 2 dependence of corrections fiG(T) to conductivity G(0) was reported [3] for large area contacts. In accordance with the theory these two facts point to a weak EPI on the local centres involved m resonant tunneling. Estimations of the parameter 2 are made below. Relationships (4)-(8) can be used to analyze the I - V characteristics in mesoscopic contacts of a small area. In the following we shall find how EPI affects the structure of the contribution of a resonance centre to the I - V characteristic. In the absence of EPI the resonant tunneling effect manifests Itself on the I(V) plot as a step at V = Vc, its height given by alo -

2er, r2

Vol. 66, No. 1

(14)

~coo) 2

Expression (15) is valid for eV, < he) D. In the opposite case V, in (15) should be substituted for V - V, which results in a step on the d2I/dV 2 plot. Note that if the contribution of a single phonon frequency ~o = ~o0 is most prominent in the EPI spectrum, the phonon replicas are associated with step (14) at voltages V+ = 2h~oo/e + V~. The value of additional steps is 6Ivh ~ 2610. The positions of the three steps are interrelated by V+ - V

-- 2V,.

(16)

If eV, > he) 0, there is no recurrence at V = V . The position of the first three peaks on the upper curve of Fig. 2(c) [4] obeys equation (16). I f two of them are assumed to be due to phonon recurrence, their ratio can be used to estimate the EPI constant 2 -~ 0.4 and obtain a reasonable energy value fi~o0 -~ 6.9 meV. On the other hand, comparison of formula (15) with Fig. 2(b) [4] enables an independent estimation of the constant 2 - 1.8. The spread of 2 does not seem to be significant because the numerical constants in the foregoing relationships were omitted. In conclusion it should be noted that the theory suggested is basically different from the Lee-Stone approach [9]. The latter was used to interpret inelastic processes as additional channels for electron escape from the resonant state. The analysis we have done showed that EPI occurring on a single centre does not lead to an increase of the resonance width in the Breit-Wignet formula, but forms satellite lines to the nonwidened resonance peak. This is easily seen in the case of weak EPI (see (8)). The difference between the two approaches is likely to be more pronounced in description of mesoscoplc contacts (see formulae (15), (16)). Our approach adequately fits the case of rather thin tunnel barriers. The parameter 2 characterizing EPI intensity does not involve exponential sensitivity to barrier thickness d being defined solely by the material constants. This fact seems to be important for tunneling spectroscopy of EPI on local centres in the experiments on resonant tunneling.

Acknowledgements - - Authors are grateful to Y.B. Levlnson for pointing out the analogy between electron tunneling and R a m a n scattering.

Vol. 66, No. 1

PHONON-ASSISTED RESONANT TUNNELING REFERENCES

1. 2. 3. 4. 5.

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L.I. Glazman & R.I. Shekhter, Zh. Eksp. Teor. Fiz. 94, No. 1, (1988). Translation to appear m Soy. Phys. JETP. V. Hizhnyakov, Light Scatterlng in Solids. p. 269, (Edited by J.L. Birman, H.Z. Cummin and K.K. Rebane), Plenum Press, N.Y.-London, (1979). A.I. Larkm & K.A. Matveev, Zh. Eksp. Teor. Phys. 93, 1030, (1987). A.D. Stone & P.A. Lee, Phys. Rev. Lett. 54, 1196 (1985).