1.Introduction

The rapid progress of nanotechnology now enables us to fabricate ultra-fast electronic devices with increasingly smaller size and higher performance. When the devices go down to nanoscale, quantum effects become obvious, which open new challenges not only in device engineering but also in fundamental science. Thus theoretical investigations of the electrical transport behaviors of nanomaterials, under quantum mechanics framework, is of great significance for giving guidance to experimental measurements of nanodevice characteristics and for realizing novel nanodevices that exploit quantum effects.

Recently, transient transport, especially the current dynamics immediately after applying the bias voltage, has attracted much attention in connection with investigations of high frequency applications of prototype nanoscopic ultrafast electrical devices. An experimental breakthrough is the fabrication of the quantum capacitor, a benchmark system for understanding ultrafast quantum dynamics of nanoscale devices [1]. Theoretical simulation based on a simple model [2] succeeded in reproducing the observed transient current dynamics. However, theoretical study, as well as our understanding, on the nanoscopic transient transport is yet insufficient, because only very simple models have been examined well so far and important effects on it, such as the carrier scattering by phonons and the resultant heating, have hardly been studied. In order to deepen our understanding on the relation between materials properties and their transient transport behaviors, in this dissertation, we aim at clarifying the following aspects closely related to device engineering in transient transport. Firstly, we examine the behaviors of elastic transient electrical currents of multi-level molecular systems. Next, the elastic transient energy currents as well as the energy balance within the nanoscale junctions is examined, which is an important step before calculating the heating effect in the nanojunction. Finally, the electron-phonon interaction is added to the scattering region and the inelastic phonon scattering effect on transient current as well as the amount of heat generated due to such an effect is examined.

2.Model and method

In our study, the transient transport behaviors of nanostructures have been examined, taking single-level quantum dot and small molecules as examples. These nanostructures are described by the tight-binding Hamiltonians. As shown in Fig. 1, at time t<0, the nanostructure is symmetrically coupled to two electrodes, i.e. Г(L)=Г(R), and the whole system is in the equilibrium state with the Fermi level εF set to zero. The bias voltage is applied oppositely on two electrodes at t=0 so that eV(L)=-eV(R). The transient electrical currents as well as the energy currents are calculated using the Nonequilibrium Green's function method within the wide-band-limit approximation.

3.Transient current dynamics of multi-level molecular systems

The elastic transient current dynamics of multi-level molecular systems weakly coupled to two electrodes have been investigated. Noticing that the concept of molecular orbital is still valid for small coupling Г, we show that the total current can be decomposed into nearly-independent current components, each corresponding to an eigen level, in case that no degenerate eigen levels exist in the molecule and the bias voltage is not large enough to excite transitions between different levels. By using such decomposition technique, we have clarified the transient current behavior of a hydrogen molecule (two-level system) and an octatetraene molecule (eight-level system) connected to two electrodes [3]. The transient current of the hydrogen molecular system is characterized by two current components with the same relaxation times and different oscillation periods. On the other hand, the current of octatetraene molecular system is decomposed into eight components. Five single levels, either sharing large coupling strengths or with larger initial electron densities, dominate the total current behavior at the initial stage (Fig. 2).

4.Transient energy transport

The elastic transient energy currents in a single-level quantum dot system have been examined for four different coupling strengths. The results show that the transient energy currents have the same relaxation times and oscillation behaviors as the corresponding electrical currents, and their physical meaning is explained as the power carried by electrons in the deep insides of electrodes. In contrast to the steady state, the transient energy currents in the two electrodes do not balance each other, and their sum reflects the real-time electron redistribution in energy domain in the region between deep insides of two electrodes (Fig. 3). In addition, the amplitude of the energy change, which is the time integral of the sum of energy currents, does not vary monotonically with the coupling strength between the dot and the electrode, in contrast to the relaxation time, which is inversely proportional to the coupling strength. The transfer Hamiltonian, which is the interaction between the quantum dot and the electrode, plays an important role in understanding the transient energy transport. Its time derivative, which is equal to zero in the steady state, is understood as the change of the energies of the electrons in time domain.

5.Inelastic transient transport and phonon heating

We have obtained the time-dependent power transfer between electrons and a single phonon mode in a single-level quantum dot system as the sum of inelastic part of the energy currents in two electrodes, subtracting the change of electron energy due to the time-dependent bias voltage. The subtraction is done using two methods, that is, the extraction of the energy conserved part of electron-phonon scattering term corresponding to the equilibrium state (t<0) and steady state (t→∞), respectively. The time-dependent phonon number, which determines the system temperature and the heating effect on inelastic current, is calculated using a phenomenological method employing the time-dependent power transfer. We have shown that the two methods provide qualitatively similar dynamical behavior of system temperature, which can be classified into two classes: when the energy corresponding to the applied bias voltage is smaller than or equal to the phonon energy, temperature first increases due to the phonon emission, and then decreases due to the phonon absorption process (Fig. 4(A)); On the other hand in the case in which the energy corresponding to the bias voltage is larger than the phonon energy, temperature increases continuously until reaching its highest value in the steady state (Fig. 4(B)). The total electrical current is suppressed by the phonon heating, while the heat transfer between the dot and environment helps decrease the system temperature and relieve the current suppression.

6.Concluding remark

In the present dissertation, the followings have been achieved on nanoscopic transient transport: (1) the current decomposition method have been proposed, which is useful in understanding the complex behaviors of the transient electrical currents in multi-level molecular systems; (2) the elastic transient energy currents in a single-level quantum dot system have been examined, with keeping the energy conservation of the whole nanocircuit; (3) the real-time phonon heating and its effect on the inelastic electrical current in a single-level quantum dot system has been clarified. We believe that the results obtained in this dissertation provide important guidance to the experimental studies, and expect that they will be confirmed in the near future.

Fig. 1. Model systems of our calculation.

Fig. 2. Total transient current (solid line) and five dominate components (dotted, dash-dot-dotted, dash-dotted, short-dashed, and short-dotted lines) of the octatetraene molecular system. Dashed line denotes the current summation of the five dominant levels. The vertical axis is current scaled by I(0)=0.387´10(-4)A. The horizontal axis is time scaled by t(0)=0.658x10(-15)s, which is also applied to Figs. 3 and 4.

Fig. 3: Time-dependent dynamics of the sum of energy currents in two electrodes for four different coupling strengths: 2.5 eV (solid), 0.5 eV (dash-dot-dotted), 0.1 eV (short-dotted), and 0.05 eV (dashed). The unit of energy current is J(0)=0.242x10(15) eV/s.

Fig. 4: Real-time temperature of the single-level quantum dot system in case of (A) voltage not larger than the phonon energy and (B) voltage larger than the phonon energy. The solid and dotted line denote the temperature estimated by two methods.

ナノスケール構造体の電子輸送特性の研究は、電子デバイスの微細化とともにその重要性が一段と増していることから、盛んに研究されている。しかし、これまでの研究の多くは定常状態における電子輸送特性に関するものである。実用的なデバイスが既にGHzオーダーの周波数で動作していることを考えれば過渡的輸送特性の理解の重要性は明らかであるにもかかわらず、これまでナノスケール構造体の過渡的電子輸送特性の研究は少なく、あまり理解が進んでいなかった。本論文は、ナノスケール構造体の過渡的電子輸送特性のいくつかの重要な側面について理論計算による解析を行い、その振舞いの解明を目指したものである。本論文は6章からなる。

第1章は緒言であり、ナノスケール電子輸送特性に関するこれまでの実験的・理論的研究を概観している。定常状態の研究に比べて過渡応答特性の研究は少なく、理論計算についてはコヒーレント輸送特性についても複数準位を持つ構造については理解が十分とは言えないこと、電子輸送特性への電子‐フォノン散乱の影響については過渡応答特性に関しても先行理論研究はあるものの、エネルギー移動に対する考慮が不十分で非物理的な振る舞いが見られていること、電子‐フォノン散乱による温度変化まで含めた過渡応答特性の理論解析はまだなされていないこと等を指摘して、本研究の目的を明確にした。

第2章では、本研究の計算手法である非平衡グリーン関数法を述べている。まず本研究で対象とする2つの電極に接続したナノ構造体の計算モデルを説明し、またある時刻に両電極にバイアス電圧を印加し、この印加電圧の立ち上がり時間が無限小である場合を考えることを述べた後、非平衡グリーン関数法の概略を述べている。さらに、この方法での電流およびエネルギー流の表式を、定常状態と過渡状態、電子-フォノン散乱を考慮しない場合とした場合のそれぞれについて提示している。

第3章では、複数準位から成るナノ構造体における過渡応答特性を計算した結果とそれに対する考察を述べている。また、この場合の過渡電流をほとんど独立な電流成分に分解する手法を提案している。この方法は電極とナノ構造体との結合が弱い場合に特に有効であり、複雑な挙動を示す過渡電流を、より単純な振る舞いを示す電流成分の和として理解することを可能にする。この方法をナノ構造体が水素分子およびオクタテトラエン分子の場合に適用し、過渡電流が単純な減衰振動挙動を示す成分に分解できることを示した。さらに、オクタテトラエンの場合には、8個の成分全部でなく、その中の数個が全電流の挙動を決めていることを明らかにした。

第4章では、弾性的な(すなわち電子‐フォノン非弾性散乱のない場合の)エネルギー流の過渡特性を、ナノ構造が単一準位の場合に対して考察している。エネルギー流が電流と同じ緩和時間と振動周期を持つ一方、その位相は電流と逆になる場合があることを示した。また、定常状態においては2つの電極におけるエネルギー流は必ず釣り合っているのに対し、電圧印加直後の過渡応答領域では、電圧印加に伴う電子エネルギー分布の変化によりこのバランスが崩れ得ることを明確化した。

第5章では、電子‐フォノン散乱を考慮した過渡応答特性を、電子‐フォノン散乱による温度変化も含めて計算し、考察している。温度変化を考慮するためには電子系とフォノン系との間のエネルギー移動を正しく考慮する必要があるが、バイアス電圧印加による電子エネルギーの変化があるため、この評価は容易ではない。ここでは、第4章の結果をもとに、このエネルギー移動について、平衡状態(電圧印加前の状態)で正しい表式と定常状態(電圧印加から十分時間が経過した後の状態)で正しい表式とを導いた。そして、これらの表式を用いることにより、先行研究でみられた、フォノン数がゼロであってもフォノン系からのエネルギー放出が起こってしまうという非物理的な振る舞いを除去することができることを示した。さらに、2つの表式で得られる結果の差が数%にとどまることから、これらの表式によって少なくとも定性的には過渡応答特性や局所温度上昇を評価できることを示した。また、定常状態ではフォノンエネルギーより低いエネルギーに対応するバイアス電圧では電子‐フォノン散乱が起こりえないのに対し、過渡領域においてはこのようなバイアス電圧に対しても電子‐フォノン散乱とそれによる温度上昇が起こること、そしてこの温度上昇により過渡電流が抑制されることを示した。

第6章は総括である。

以上のように、本論文は、ナノ構造体における過渡的電子輸送特性を理論計算により解析した。複雑な挙動を示す過渡電流をより単純な挙動の成分に分解する手法を提案し、またエネルギー流および電子系-フォノン系間のエネルギー移動を正しく考慮した上で電子‐フォノン散乱とそれによる温度変化が過渡的電子輸送特性に及ぼす影響を明らかにし、ナノスケール電気特性を理解する上で有用な知見を得た。よって本論文のナノスケール電子物性学、計算マテリアル工学への寄与は大きい。

よって本論文は博士(工学)の学位請求論文として合格と認められる。