# Phonon-assisted tunneling in asymmetric resonant tunneling structures

**ABSTRACT** Based on the dielectric continuum model, we calculated the phonon assisted

tunneling (PAT) current of general double barrier resonant tunneling structures

(DBRTSs) including both symmetric and antisymmetric ones. The results indicate

that the four higher frequency interface phonon modes (especially the one which

peaks at either interface of the emitter barrier) dominate the PAT processes,

which increase the valley current and decrease the PVR of the DBRTSs. We show

that an asymmetric structure can lead to improved performance.

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**ABSTRACT:**We present a theory of phonon-assisted tunneling in GaAs/AlAs double-barrier structures, which treats the electrons in the emitter as a two-dimensional electron gas. A complete set of confined and interface optical-phonon modes is calculated using a dielectric-continuum model, and we derive a general expression for the electron-phonon Hamiltonian valid for all optical-phonon modes in an arbitrary heterostructure. The electronic wave functions relevant to phonon-assisted tunneling are found by self-consistently solving the Schrödinger and Poisson equations in both the well and the emitter. Five different phonon modes are predicted to dominate the phonon-assisted tunneling current: three LO-like interface modes, the half-space modes in the emitter, and the confined modes in the well.Physical review. B, Condensed matter 10/1994; 50(12):8423-8432. · 3.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We propose an approach to study one-dimensional electron tunneling in an arbitrarily shaped barrier in the presence of electron-optical phonon scattering. An independent-boson model is used for the electron-phonon interaction. Our result for a double-barrier structure shows the occurrence of phonon-assisted resonant tunneling.Physical Review Letters 08/1989; 63(4):418-421. · 7.73 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The exact resonant-tunneling transmission probability for an electron interacting with phonons is presented in the limit that the elastic coupling to the leads is independent of energy. The phonons produce transmission sidebands but do not affect the integrated transmission probability or the escape rate of the electron from the resonant site. In the Appendixes, we evaluate the Green function that appears in the expression for the transmission probability.Physical review. B, Condensed matter 01/1990; 40(17):11834-11850. · 3.77 Impact Factor

Page 1

arXiv:cond-mat/9810376v1 [cond-mat.mes-hall] 28 Oct 1998

Phonon-assisted tunneling in asymmetric resonant tunneling structures

Jun-jie Shi∗,1,3,a, Barry C. Sanders1,band Shao-hua Pan1,2,3,c

1Department of Physics, Macquarie University, Sydney, New South Wales 2109, Australia

2Institute of Physics, Chinese Academy of Sciences, P. O. Box 603, Beijing 100080, P. R. China

3China Center of Advanced Science and Technology (World Laboratory), P. O. Box 8730, Beijing 100080, P. R. China

∗On leave from Department of Physics, Henan Normal University, Xinxiang 453002, Henan, P. R. China

(February 1, 2008)

Based on the dielectric continuum model, we calculated the phonon assisted tunneling (PAT) cur-

rent of general double barrier resoant tunneling structures (DBRTSs) including both symmetric and

asymmetric ones. The results indicate that the four higher frequency interface phonon modes (es-

pecially the one which peaks at either interface of the emitter barrier) dominate the PAT processes,

which increase the valley current and decrease the PVR of DBRTSs. We show that an asymmetric

structure can lead to improved performance.

73.40.Gk, 73.50.Bk

I. INTRODUCTION

The double barrier resonant tunneling structure

(DBRTS) [1] continues to attract attention both for its

potential applications to electronic devices and for its

value in exploring fundamental phenomena [2], including

tests of electron-phonon coupling theories [3–13]. Despite

extensive study of the DBRTS, challenging problems con-

tinue to exist with respect to practical applications and

also for modeling the characteristics of the DBRTS [2].

Our objective here is to fully analyze the phonon-assisted

tunneling (PAT) current, especially in the valley region,

and to demonstrate that there can be important advan-

tages in designing a DBRTS with an asymmetric struc-

ture. We include all of the phonon modes and provide

a complete analysis of the relative importance of these

modes, particularly for the asymmetric DBRTSs where a

limited investigation has been performed to date [12,13].

It is important to consider all of the phonon modes when

the asymmetric DBRTS is considered [14–16].

The motivations for considering the asymmetric

DBRTS are the possibility for improving performance,

tunability of the negative differential resistance (NDR)

[17], and tuning of charge accumulation in the quantum

well (QW) [18,19]. This asymmetry can be introduced

by varying the material compositions in the two barriers

or by creating two barriers of unequal width, and we will

consider both possibilities here. Our treatment includes

a detailed study of all of the phonon modes, assuming the

dielectric continuum theory and incorporating the effects

of subband nonparabolicity.

Our main concern is with DBRTS performance.

large NDR is particularly important for high-frequency

resonant tunneling and rapid switching devices and is

usually quantified by the peak-to-valley ratio (PVR) of

the current-to-voltage characteristic curve. A large PVR

and low valley current are desirable for most resonant

tunneling diodes (RTD) device applications. Generally,

PAT, Γ − X intervalley tunneling, impurity scattering,

the interface roughness scattering and the tunneling of

A

quasi-two-dimensional subband electron in the pseudo-

triangular well in the emitter can cause the valley cur-

rent [10]. For a polar semiconductor DBRTS, the effects

of phonon scattering on resonant tunneling are very im-

portant and inevitable especially at room temperature.

The electrons in the DBRTS may emit phonons during

the resonant tunneling process. In general, the contribu-

tion by the PAT current is small compared with the co-

herent resonant tunneling process but can be large com-

pared to the other contributions to current. Hence the

total tunneling current density can be approximated by

J = Jc+ Jp, for Jcthe coherent tunneling current den-

sity and Jpthe PAT current density. Although the PAT

current has been investigated theoretically [3–11,13], we

provide a more rigorous investigation of electron-phonon

scattering and the PAT current in general DBRTS in-

cluding both symmetric and asymmetric ones, with the

contribution of all of the phonon modes accounted for

quantitatively. Hence the PAT physical picture is fur-

ther clarified.

II. THEORY

Working within the framework of the dielectric con-

tinuum model and treating the electron-phonon inter-

action via the Fr¨ ohlich-like Hamiltonian, we can calcu-

late the electron-phonon scattering rate W according to

the Fermi golden rule for two cases [15]: scattering by

interface phonons W(i→f)

int

(?ki,Ez) and by confined LO

phonons W(i→f)

LO

(?ki,Ez).The scattering rate depends

explicitly on the phonon occupation number Nphwhich

is temperature-dependent according to the Planck dis-

tribution [12]. As the width of the final resonant state

is very narrow, the final state is treated as a com-

pletely localized state in the well [3–5,8,13,20]. We have

recently presented the expressions of these scattering

rates W(i→f)

DBRTSs [12].

int,LO(?ki,Ez) for both symmetric and asymmetric

1

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Electron tunneling in a DBRTS depends sensitively on

the bias voltage V . The coherent tunneling current den-

sity Jcand PAT current density Jpare given by

Jc= en?T(Ez)vz(Ez)? (2.1)

and

Jp= eN?W?/A. (2.2)

Here e is the absolute value of the electron charge, N

the total electron number in the electron reservoir with

volume Ω, n = N/Ω the electron density (assumed to

be constant), T(Ez) the electron transmission coefficient,

and vz(Ez) the longitudinal electron velocity, A the cross-

sectional area of the structure, and W the electron-

phonon scattering rate [12]. ?? represents averaging on

quantum states.

From Eq. (2.1) we obtain [1]

Jc= Jc→− Jc←, (2.3)

with

Jc→=em?kBT

2π2¯ h3

ln{1 + exp[(EF− Ez)/kBT]}dEz,

?∞

0

T(Ez)

(2.4)

and

Jc←=em?kBT

2π2¯ h3

ln{1 + exp[(EF− eV − Ez)/kBT]}dEz.

?∞

0

T(Ez)

(2.5)

Here Jc→ (Jc←) denotes the tunneling current density

from the emitter (collector) to the collector (emitter),

and Jc→ is critical for the NDR of the DBRTS, which

controls the coherent tunneling current density. EF and

EF−eV are, respectively, the local Fermi energy levels in

the emitter and collector, and m?is the electron effective

mass in the x − y plane parallel to the interfaces of the

DBRTS.

From Eq. (2.2) we derive

Jp=em?kBTLe

2π2¯ h3

?mz

2

?1/2 ?

p=ν,m

?∞

Ew+¯ hωp

E−1/2

z

×ln

?

1 + e(EF−Ez)/kBT?

Wp(Ez)dEz,(2.6)

with Lethe emitter length, mzthe electron effective mass

for longitudinal motion in the emitter, ν = 2,3,4 the in-

dex for confined bulk-like LO phonon modes in the left-

barrier, the well and the right-barrier layers, respectively,

and m (1 to 8) the index for the eight interface phonon

modes. The electron-phonon scattering rate in the νth

layer is Wν(Ez), and Wm(Ez) is the scattering rate be-

tween electron and the mthinterface phonon mode. Ex-

plicit expresssions for Wν(Ez) and Wm(Ez) have been

presented in our recent paper [12].

In obtaining Eq. (2.6) we have assumed W(?ki,Ez).=

W(0,Ez) ≡ W(Ez) and have ignored the contribution

from collector-to-emitter tunneling via a phonon absorp-

tion [3–5].

Despite theapparent

Eqs. (2.4) and (2.6), simulating Jpis more costly in com-

puter time compared to simulating Jc because Wp(Ez)

is more complicated than T(Ez) [12]. Our calculations

show that Jpmainly comes from the electron scattering

by the higher frequency interface phonons (especially the

interface phonons localized at either interface of the left

barrier).

minordifference between

III. NUMERICAL RESULTS AND DISCUSSION

Numerical calculations have been performed for asym-

metric and symmetric DBRTSs A(x,d) defined as

A(x,d) ≡ n+GaAs(1000˚ A)/AlxGa1−xAs(30˚ A)/GaAs(60˚ A)

/Al0.3Ga0.7As(d)/n+GaAs(1000˚ A), (3.1)

with m?= mzand equal doping concentration 1018cm−3

in emitter and collector. The physical parameters used

are the same as in [16].

Figure 1 presents the dispersion calculated from

Eq. (4) of [15] for the eight interface modes of structure

A(0.25,20˚ A), which reveals that the four lower-frequency

modes occupy a much narrower frequency band than the

four higher-frequency modes. Moreover, the dispersion of

the interface modes is significant for the case k ≤ 0.1˚ A−1

and negligible for k > 0.1˚ A−1. The electron–interface-

phonon coupling function Γ(k,z) [15] is a complicated

function of both z and k in a DBRTS. The Γ(k,z) − z

relation in Fig. 2 reveals the strength of the electron in-

teraction with different interface modes peaks at differ-

ent interfaces. For example, the electron interaction with

mode 7 (denoted e-p(7)) peaks at the z = 0 and 30˚ A in-

terfaces (i.e., either interface of the left barrier). We also

find that |Γ(k,z)| decreases rapidly as a function of k for

0 < k ≤ 0.01˚ A−1, slowly for 0.01˚ A−1< k < 0.08˚ A−1,

and then rapidly for k > 0.08˚ A−1for the e-p(7) inter-

action. The 7thinterface mode is much more important

than the other modes in the asymmetric DBRTS, and

the four higher-frequency modes produce intensive po-

larisation in the DBRTS, resulting in a significant inter-

action with electrons. On the contrary, the four lower-

frequency modes produce a weak interaction with elec-

trons compared with the higher-frequency modes, which

can be ignored [12,14–16]. In the following calculations,

we will thus only consider the contribution of the four

higher-frequency modes to the scattering rate and the

PAT current for simplicity.

Figure 3 shows the scattering rate W vs incident elec-

tron energy Ezfor A(0.25,20˚ A). For n = 1018cm−3, the

Fermi energy level EF = 42.5 meV at T = 300 K. We

observe that the contribution due to interface phonons

is larger than that due to LO bulk-like phonons for the

2

Page 3

electron with lower incident energies. We know from the

Fermi distribution function that the emitter states are

appreciably populated only for Ez ≤ EF+ kBT = 68.3

meV (as T = 300 K). Hence, we can infer that the in-

terface phonons contribute much more than the confined

bulk-like LO phonons to the PAT current.

We present the electron-phonon scattering rate, in-

clouding the subband nonparabolicity, as the dash-dot-

dot line in Fig. 3, which shows that the subband non-

parabolicity has a large influence on the electron-phonon

scattering. The peak position shifts to a lower energy,

and its value decreases under the influence of the sub-

band nonparabolicity.

In the case of including subband nonparabolicity, PAT

current-to-voltage curves are shown in Fig. 4 at T = 300

K for the structure A(0.25,20˚ A). Figure 4(a) shows the

tunneling current assisted by the four higher-frequency

interface phonon modes and their sum. We can see from

Fig. 4(a) that the 7thinterface mode, which is localized

at either interface of the left barrier, is the most im-

portant of all of the interface modes, and this result is

consistent with the results shown in Fig. 2. The total

interface PAT current is a complicated function of the

applied voltage and has two peaks for increasing voltage.

Figure 4(b) gives the confined bulk-like LO PAT current

density in structure A(0.25,20˚ A). This figure indicates

that the PAT current from the LO phonons in the well is

much larger than those from the LO phonons in the two

barrier layers with a complicated behavior for increasing

bias voltage. Figure 4(c) presents the total PAT current

density including the interface and the confined bulk-like

LO phonons. We can see from Fig. 4(c) that the inter-

face PAT current is larger by one order of magnitude than

the confined LO PAT current, confirming that interface-

phonon scattering dominates over confined LO phonon

scattering (c.f. Fig. 3). Moreover, Fig. 4(c) also shows

that the total PAT current is a very complicated function

of the applied voltage and has two peaks, similar to the

results based on the Green function method [6]. This is

completely due to the complexity of the contribution of

the phonon modes to the PAT current in our DBRTS, as

shown in Figure 4(a). Figure 4 also indicates that the

PAT current is mainly determined by scattering between

electrons and higher frequency interface phonons (espe-

cially the interface phonons localized at either interface

of the left barrier), showing a clear physical picture for

the PAT process in general DBRTS.

Figure 5 shows the total current-to-voltage curve at

room temperature for A(0.25,20˚ A), including coherent

and PAT currents. This figure shows that PAT increases

the valley current and decreases the PVR. The result

shown in Fig. 5 is similar to those obtained by the Wigner

function method [7] and the Wannier function envelope

equation method [10].

We have also studied the current-voltage characteris-

tics for DBRTSs A(0.3,d) with d = 20, 30 and 40˚ A. The

calculated results show that when the right-barrier thick-

ness d increases, the peak current decreases, the PVR

increases, and the peak position shifts towards higher

bias voltage. These three characteristics are in agree-

ment with recent experimental results [18]. Moreover,

we have also studied the current-voltage characteristics

for A(x,30˚ A) with x = 0.2, 0.3 and 0.4. The calculated

results show that when the Al composition x, and thus

the height of the left barrier, increases, the peak current

decreases, the PVR increases, and the peak position re-

mains at the same bias voltage. These theoretical results,

which show that an asymmetric DBRTS can lead to im-

proved performance, await experimental confirmation.

IV. SUMMARY

Employing the dielectric continuum model, including

all the phonon modes and treating conduction band non-

parabolicity, the PAT process is investigated in detail.

We show that for a DBRTS, no matter it is symmet-

ric or asymmetric one, the four higher frequency modes,

which reduce to two symmetric and two antisymmetric

higher frequency modes for a symmetric structure, dom-

inate the interface PAT process. In particular, for the

interface PAT process in a symmetric DBRTS, the two

symmetric higher frequency modes are most important,

the two antisymmetric higher frequency modes are less

important, and the two symmetric and antisymmetric

lower frequency modes are negligible. The above opinion

is different from that of Refs. [3–5].

In general, the confined LO phonons in the well layer

are more important than those in the two barrier layers,

and the four higher frequency interface modes (i.e., mode

5 to 8, especially the 7thone which peaks at either inter-

face of the left barrier) dominates over the four lower

frequency interface modes and all of the confined LO

phonon modes to electron-phonon scattering and PAT

current. The PAT increases the valley current and de-

creases the PVR of DBRTS. It is worth mentioning that

the mode 7 becomes the highest frequency symmetric in-

terface mode for a symmetric DBRTS. The PAT physical

picture stated in the above is useful and important for

further understanding PAT process in DBRTSs and for

designing better RTD devices.

Subband nonparabolicity has a significant influence

on electron-phonon scattering, PAT, and the current-to-

voltage characteristic of a DBRTS.

We find that the peak current is reduced, the position

of peak current is shifted to a higher voltage, and the

PVR is enlarged if the right-barrier width is increased

when the two barriers have the same height. The peak

current is reduced and the PVR is increased by suitably

increasing the left-barrier height when the two barriers

have the same width. An asymmetric DBRTS with a

suitably designed structure has a larger PVR than its

commonly-used symmetric counterpart. The results ob-

tained in this paper are useful for analysing coherence-

breaking phonon scattering and for potentially important

3

Page 4

resonant tunneling device applications.

ACKNOWLEDGMENTS

Jun-jie Shi has been supported by an Overseas Post-

graduate Research Scholarship and a Macquarie Univer-

sity International Postgraduate Research Award. This

work has been supported by an Australian Research

Council Large Grant and by a Macquarie University Re-

search Grant. We have benefitted from many useful dis-

cussions with L. Tribe, E. M. Goldys, and D. J. Skellern.

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FIG. 1.

structure A(0.25,20˚ A).

The dispersion curves of the interface modes for

FIG. 2.

functions (¯ he2/Aε0)−1/2Γ(k,z) for the structure A(0.25,20˚ A)

(k = 0.01˚ A−1). Here the numbers by the curves represent the

interface-phonon frequency in order of increasing magnitude:

(a) for the four lower-frequency modes and (b) for the four

higher-frequency modes with the interfaces localized at z =0,

30, 90 and 110˚ A, respectively.

Spatial dependence of the normalized coupling

FIG. 3.

rate W/(Nph + 1) vs incident electron energy Ez for

A(0.25,20˚ A) at the bias 150 mV. The dashed line and

dash-dot line represent, respectively, the contribution of the

interface phonons and confined bulk-like LO phonons, and the

solid line is their sum in the absence of subband nonparabolic-

ity. The dash-dot-dot line is the total scattering rate including

subband nonparabolicity.

The normalized electron-phonon scattering

FIG. 4.

ture for A(0.25,20˚ A), including the subband nonparabolicity:

(a) Interface PAT: the dashed line represents the 5thinterface

mode contribution for the tunneling current, the dash-dot-dot

line for the 6thmode, the thin solid line for the 7thmode,

the dash-dot line for the 8thmode and the heavy solid line

is their sum. (b) LO PAT: the dashed line represents the

confined LO phonon in the left barrier (Al0.25Ga0.75As) con-

tribution for the tunneling current, the dash-dot line for the

LO phonon in the right barrier (Al0.3Ga0.7As), the thin solid

line for the confined bulk-like LO phonon in the well (GaAs)

and the heavy solid line is their sum. (c) Interface PAT cur-

rent (thin solid line), confined LO PAT current (dashed line)

and their sum (heavy solid line).

PAT current-to-voltage curves at room tempera-

FIG. 5.

at room temperature considering the subband nonparabolic-

ity for A(0.25,20˚ A). The dashed line stands for the coherent

tunneling current density. The solid line is the total tunnel-

ing current density combining coherent tunneling and PAT

currents.

The total current-to-voltage characteristic curve

4

Page 5

Shi et al.

EPJ B

32.9

33.2

33.3

00.05 0.10.15

34

35

36

37

33

33.1

(meV)

_hω

k(A )

-1

Fig. 1

Page 6

Shi et al.

EPJ B

-0.1

0

0.1

1

2

3

4

(a)

-4

-200 -1000

z(A)

100 200 300

-3

-2

-1

0

1

2

3

(b)

5

6

7

8

Γ

Fig. 2

(A meV) 1/2

Page 7

E (meV)

z

0

100

200

300

0.05

0.1

0

W(ps )

-1

Shi et al.

EPJ B

Fig. 3

Page 8

4

Current Density (10 A/cm )

2

Shi et al.

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100

150200

0.15

0.3

0

0.45

0.02

0.04

0.06

0

0.15

0.3

0.45

0

(a)

(c)

(b)

Applied Voltage (mV)

Fig. 4

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Current Density (10 A/cm )

2

0 50100 150 200

6

12

18

0

Applied Voltage (mV)

Shi et al.

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Fig. 5

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