## 1 Introduction  
Since the 19 th century, it has been found that a Bordeaux mixture (CuSO4) can prevent plants from bacterial infection. A tremendous number of metals or metal oxides were applied to agriculture production. Nowadays, nanotechnology, especially metal or metal oxide nanoparticles (NPs), provides an efficient way to promote growth and yield [1; 2] and resist pests and diseases [3] in crops, as determined by their particle size effects [4]. Thus, in recent years, metal or metal oxide NPs have been commonly used in agriculture as novel growth regulators, nanofertilizers, and nanopesticides [5; 6]. Copper (Cu) is one of the essential nutrients for plant growth and development. It is involved in redox reactions of enzymes [7] and participates in hormone signaling pathways, metabolism, and biochemical processes such as photosynthetic electron transfer, mitochondrial respiration, and cell wall metabolism in plants [8]. CuO NPs are among the most common nanoparticles used in industry and agriculture because of their antibacterial activity and bio-availability. The application doses of CuO NPs have positive or negative impacts on plants. For example, CuO NPs at 0.1 mg/L significantly promoted the germination of wheat seeds and subsequent growth of roots [9]; when the use dose was beyond 200 mg/L, CuO NPs inhibited the growth of mung beans and wheat [10]. Therefore, prior to the large-scale application of CuO NPs in agriculture, it is important to find an idealistic use dose to avoid potential damage to plants caused by high doses of CuO NPs [11]. Additionally, plant species is another factor that can determine the doses of CuO NPs. For example, exposure to 150 ug/L CuO NPs inhibited the growth of duckweed [12], but CuO NPs at 125 mg/kg exhibited no impact on the yield of bell pepper [13]. The application routes can also determine the impact of CuO NPs on crop growth. Foliar application of 40 ug/mL CuO NPs inhibited the germination rate of lettuce and subsequent plant growth [14], while no adverse effect was found in the root application route [15]. Few studies have explored the differential regulation of plant growth by CuO NPs as a function of particle size. For example, as compared to granular CuO, bulk CuO caused cell membrane damage to corn seedlings [16]. Soybean is an important crop and widely cultivated [17], and it also plays a crucial role in the human food supply by providing rich proteins [18]. Additionally, it contributes approximately 40% of the annual oilseed crop yield worldwide [19]. In terms of economics, soybeans' contribution to a variety of industries is worth USD 30 billion worldwide [20]. Hence, enhancing soybean growth and nitrogen fixation is considered one of the most effective ways to increase yield and global nitrogen cycling. Nitrogen (N) is an essential nutrient element for plant growth and development [21], and N deficiency can limit agriculture production on a global scale [22]. N metabolism is one of the basic physiological processes determining crop growth, quality, and yield [23], and N deficiency is a limiting factor that can inhibit plant growth [24]. Although N\({}_{2}\) in the atmosphere is sufficient, it must be converted into ammonia (NH\({}_{4}\)+) or nitrate (NO\({}_{3}\)\({}^{-}\)) for plant utilization [25]. NH\({}_{4}\)+ and NO\({}_{3}\)\({}^{-}\), as signaling molecules, participate in a series of physiological metabolism processes and play an essential role in plant growth and development [26]. Most chemical forms of nitrogen can be converted to NO\({}_{3}\)\({}^{-}\) rapidly in aerobic soil, and NO\({}_{3}\)\({}^{-}\) is also the main form that plants can take up from soil [27]. Once accumulating in root systems, NO\({}_{3}\)\({}^{-}\) can be assimilated into NH\({}_{4}\)+ and organic matters with the participation of nitrate reductase (NR) and nitrite reductase (NiR), and then the NH\({}_{4}\)\({}^{+}\) is converted into nitrogenous organics by the GS/GOGAT cycle [28].